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# Finite State Machines, Automata, Transducers

This module adds support for finite state machines, automata and transducers. See classes `Automaton` and `Transducer` (or the more general class `FiniteStateMachine`) and the examples below for details creating one.

## Contents

### `FiniteStateMachine` and derived classes `Transducer` and `Automaton`

#### Accessing parts of a finite state machine

`state()` | Get a state by its label
`states()` | List of states
`iter_states()` | Iterator over the states
`initial_states()` | List of initial states
`iter_initial_states()` | Iterator over initial states
`final_states()` | List of final states
`iter_final_states()` | Iterator over final states
`transition()` | Get a transition by its states and labels
`transitions()` | List of transitions
`iter_transitions()` | Iterator over the transitions
`predecessors()` | List of predecessors of a state
`induced_sub_finite_state_machine()` | Induced sub-machine
`accessible_components()` | Accessible components
`final_components()` | Final components (connected components which cannot be left again)

#### (Modified) Copies

`empty_copy()` | Returns an empty deep copy
`deepcopy()` | Returns a deep copy
`relabeled()` | Returns a relabeled deep copy

#### Manipulation

`add_state()` | Add a state
`add_states()` | Add states
`delete_state()` | Delete a state
`add_transition()` | Add a transition
`add_transitions_from_function()` | Add transitions
`on_duplicate_transition` | Hook for handling duplicate transitions
`add_from_transition_function()` | Add transitions by a transition function
`delete_transition()` | Delete a transition
`remove_epsilon_transitions()` | Remove epsilon transitions (not implemented)
`split_transitions()` | Split transitions with input words of length `> 1`
`determine_alphabets()` | Determines input and output alphabets
`construct_final_word_out()` | Construct final output by implicitly reading trailing letters; cf. `with_final_word_out()`

#### Properties

`has_state()` | Checks for a state
`has_initial_state()` | Checks for an initial state
`has_initial_states()` | Checks for initial states
`has_final_state()` | Checks for an final state
`has_final_states()` | Checks for final states
`has_transition()` | Checks for a transition
`is_deterministic()` | Checks for a deterministic machine
`is_complete()` | Checks for a complete machine
`is_connected()` | Checks for a connected machine
`is_Markov_chain()` | Checks for a Markov chain
`is_monochromatic()` | Checks whether the colors of all states are equal
`asymptotic_moments()` | Main terms of expectation and variance of sums of labels
`epsilon_successors()` | Epsilon successors of a state

#### Operations

`disjoint_union()` | Disjoint union (not implemented)
`concatenation()` | Concatenation (not implemented)
`Kleene_closure()` | Kleene closure (not implemented)
`Automaton.intersection()` | Intersection of automata
`Transducer.intersection()` | Intersection of transducers
`Transducer.cartesian_product()` | Cartesian product of a transducer with another finite state machine
`product_FiniteStateMachine()` | Product of finite state machines
`composition()` | Composition (output of other is input of self)
`__call__()` | Composition with other finite state machine
`input_projection()` | Input projection (output is deleted)
`output_projection()` | Output projection (old output is new input)
`projection()` | Input or output projection
`transposition()` | Transposition (all transitions are reversed)
`with_final_word_out()` | Machine with final output constructed by implicitly reading trailing letters, cf. `construct_final_word_out()` for inplace version
`Automaton.determinisation()` | Determinisation of an automaton
`process()` | Process input
`__call__()` | Process input with shortened output
`Automaton.process()` | Process input of an automaton (output differs from general case)
`Transducer.process()` | Process input of a transducer (output differs from general case)
`iter_process()` | Return process iterator

#### Simplification

`prepone_output()` | Prepone output where possible
`equivalence_classes()` | List of equivalent states
`quotient()` | Quotient with respect to equivalence classes
`merged_transitions()` | Merge transitions while adding input
`markov_chain_simplification()` | Simplification of a Markov chain
`Automaton.minimization()` | Minimization of an automaton
`Transducer.simplification()` | Simplification of a transducer

#### Conversion

`adjacency_matrix()` | (Weighted) adjacency `matrix`
`graph()` | Underlying `DiGraph`
`plot()` | Plot

##### LaTeX output

`latex_options()` | Set options
`set_coordinates()` | Set coordinates of the states
`default_format_transition_label()` | Default formatting of words in transition labels
`format_letter_negative()` | Format negative numbers as overlined number
`format_transition_label_reversed()` | Format words in transition labels in reversed order

### `FSMState`

`final_word_out` | Final output of a state
`is_final` | Describes whether a state is final or not
`is_initial` | Describes whether a state is initial or not
`label()` | Label of a state
`relabeled()` | Returns a relabeled deep copy of a state
`fully_equal()` | Checks whether two states are fully equal (including all attributes)

### `FSMTransition`

`from_state` | State in which transition starts
`to_state` | State in which transition ends
`word_in` | Input word of the transition
`word_out` | Output word of the transition
`deepcopy()` | Returns a deep copy of the transition

### `FSMProcessIterator`

`next()` | Makes one step in processing the input tape
`preview_word()` | Reads a word from the input tape
`result()` | Returns the finished branches during process

### Helper Functions

`equal()` | Checks whether all elements of `iterator` are equal
`full_group_by()` | Group iterable by values of some key
`startswith()` | Determine whether list starts with the given prefix
`FSMLetterSymbol()` | Returns a string associated to the input letter
`FSMWordSymbol()` | Returns a string associated to a word
`is_FSMState()` | Tests whether an object inherits from `FSMState`
`is_FSMTransition()` | Tests whether an object inherits from `FSMTransition`
`is_FiniteStateMachine()` | Tests whether an object inherits from `FiniteStateMachine`
`duplicate_transition_ignore()` | Default function for handling duplicate transitions
`duplicate_transition_raise_error()` | Raise error when inserting a duplicate transition
`duplicate_transition_add_input()` | Add input when inserting a duplicate transition

## Examples

We start with a general `FiniteStateMachine`. Later there will be also an `Automaton` and a `Transducer`.

### A simple finite state machine

We can easily create a finite state machine by

    sage: fsm = FiniteStateMachine()
    sage: fsm
    Finite state machine with 0 states

By default this is the empty finite state machine, so not very interesting. Let’s create and add some states and transitions:

    sage: day = fsm.add_state('day')
    sage: night = fsm.add_state('night')
    sage: sunrise = fsm.add_transition(night, day)
    sage: sunset = fsm.add_transition(day, night)

Let us look at `sunset` more closely:

    sage: sunset
    Transition from 'day' to 'night': -|-

Note that could also have created and added the transitions directly by:

    sage: fsm.add_transition('day', 'night')
    Transition from 'day' to 'night': -|-

This would have had added the states automatically, since they are present in the transitions.

Anyhow, we got the following finite state machine:

    sage: fsm
    Finite state machine with 2 states

We can also obtain the underlying directed graph by

    sage: fsm.graph()
    Digraph on 2 vertices

To visualize a finite state machine, we can use `latex()` and run the result through LaTeX, see the section on LaTeX output below.

Alternatively, we could have created the finite state machine above simply by

    sage: FiniteStateMachine([('night', 'day'), ('day', 'night')])
    Finite state machine with 2 states

See `FiniteStateMachine` for a lot of possibilities to create finite state machines.

### A simple Automaton (recognizing NAFs)

We want to build an automaton which recognizes non-adjacent forms (NAFs), i.e., sequences which have no adjacent non-zeros. We use \(0\), \(1\), and \(-1\) as digits:

    sage: NAF = Automaton(
    ....:     {'A': [('A', 0), ('B', 1), ('B', -1)], 'B': [('A', 0)]})
    sage: NAF.state('A').is_initial = True
    sage: NAF.state('A').is_final = True
    sage: NAF.state('B').is_final = True
    sage: NAF
    Automaton with 2 states

Of course, we could have specified the initial and final states directly in the definition of `NAF` by `initial_states=['A']` and `final_states=['A', 'B']`.

So let’s test the automaton with some input:

    sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False  # activate new output behavior
    sage: NAF([0])
    True
    sage: NAF([0, 1])
    True
    sage: NAF([1, -1])
    False
    sage: NAF([0, -1, 0, 1])
    True
    sage: NAF([0, -1, -1, -1, 0])
    False
    sage: NAF([-1, 0, 0, 1, 1])
    False

Alternatively, we could call that by

    sage: NAF.process([0, -1, 0, 1])
    (True, 'B')

which gives additionally the state in which we arrived.

### LaTeX output

We can visualize a finite state machine by converting it to LaTeX by using the usual function `latex()`. Within LaTeX, TikZ is used for typesetting the graphics, see the Wikipedia article PGF/TikZ.

    sage: print latex(NAF)
    \begin{tikzpicture}[auto, initial text=, >=latex]
    \node[state, accepting, initial] (v0) at (3.000000, 0.000000) {$\text{\texttt{A}}$};
    \node[state, accepting] (v1) at (-3.000000, 0.000000) {$\text{\texttt{B}}$};
    \path[->] (v0) edge[loop above] node {$0$} ();
    \path[->] (v0.185.00) edge node[rotate=360.00, anchor=north] {$1, -1$} (v1.355.00);
    \path[->] (v1.5.00) edge node[rotate=0.00, anchor=south] {$0$} (v0.175.00);
    \end{tikzpicture}

We can turn this into a graphical representation.

    sage: view(NAF) # not tested

To actually see this, use the live documentation in the Sage notebook and execute the cells in this and the previous section.

Several options can be set to customize the output, see `latex_options()` for details. In particular, we use `format_letter_negative()` to format \(-1\) as \(\overline{1}\).

    sage: NAF.latex_options(
    ....:     coordinates={'A': (0, 0),
    ....:                  'B': (6, 0)},
    ....:     initial_where={'A': 'below'},
    ....:     format_letter=NAF.format_letter_negative,
    ....:     format_state_label=lambda x:
    ....:         r'\mathcal{%s}' % x.label()
    ....: )
    sage: print latex(NAF)
    \begin{tikzpicture}[auto, initial text=, >=latex]
    \node[state, accepting, initial, initial where=below] (v0) at (0.000000, 0.000000) {$\mathcal{A}$};
    \node[state, accepting] (v1) at (6.000000, 0.000000) {$\mathcal{B}$};
    \path[->] (v0) edge[loop above] node {$0$} ();
    \path[->] (v0.5.00) edge node[rotate=0.00, anchor=south] {$1, \overline{1}$} (v1.175.00);
    \path[->] (v1.185.00) edge node[rotate=360.00, anchor=north] {$0$} (v0.355.00);
    \end{tikzpicture}
    sage: view(NAF) # not tested

### A simple transducer (binary inverter)

Let’s build a simple transducer, which rewrites a binary word by iverting each bit:

    sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
    ....:     initial_states=['A'], final_states=['A'])

We can look at the states and transitions:

    sage: inverter.states()
    ['A']
    sage: for t in inverter.transitions():
    ....:     print t
    Transition from 'A' to 'A': 0|1
    Transition from 'A' to 'A': 1|0

Now we apply a word to it and see what the transducer does:

    sage: inverter([0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1])
    [1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0]

`True` means, that we landed in a final state, that state is labeled `'A'`, and we also got an output.

### A transducer which performs division by \(3\) in binary

Now we build a transducer, which divides a binary number by \(3\). The labels of the states are the remainder of the division. The transition function is

    sage: def f(state_from, read):
    ....:     if state_from + read <= 1:
    ....:         state_to = 2*state_from + read
    ....:         write = 0
    ....:     else:
    ....:         state_to = 2*state_from + read - 3
    ....:         write = 1
    ....:     return (state_to, write)

which assumes reading a binary number from left to right. We get the transducer with

    sage: D = Transducer(f, initial_states=[0], final_states=[0],
    ....:                input_alphabet=[0, 1])

Let us try to divide \(12\) by \(3\):

    sage: D([1, 1, 0, 0])
    [0, 1, 0, 0]

Now we want to divide \(13\) by \(3\):

    sage: D([1, 1, 0, 1])
    Traceback (most recent call last):
    ...
    ValueError: Invalid input sequence.

The raised `ValueError` means \(13\) is not divisible by \(3\).

### Gray Code

The Gray code is a binary numeral system where two successive values differ in only one bit, cf. the Wikipedia article Gray_code. The Gray code of an integer \(n\) is obtained by a bitwise xor between the binary expansion of \(n\) and the binary expansion of \(\lfloor n/2\rfloor\); the latter corresponds to a shift by one position in binary.

The purpose of this example is to construct a transducer converting the standard binary expansion to the Gray code by translating this construction into operations with transducers.

For this construction, the least significant digit is at the left-most position. Note that it is easier to shift everything to the right first, i.e., multiply by \(2\) instead of building \(\lfloor n/2\rfloor\). Then, we take the input xor with the right shift of the input and forget the first letter.

We first construct a transducer shifting the binary expansion to the right. This requires storing the previously read digit in a state.

    sage: def shift_right_transition(state, digit):
    ....:     if state == 'I':
    ....:         return (digit, None)
    ....:     else:
    ....:         return (digit, state)
    sage: shift_right_transducer = Transducer(
    ....:     shift_right_transition,
    ....:     initial_states=['I'],
    ....:     input_alphabet=[0, 1],
    ....:     final_states=[0])
    sage: shift_right_transducer.transitions()
    [Transition from 'I' to 0: 0|-,
     Transition from 'I' to 1: 1|-,
     Transition from 0 to 0: 0|0,
     Transition from 0 to 1: 1|0,
     Transition from 1 to 0: 0|1,
     Transition from 1 to 1: 1|1]
    sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
    sage: shift_right_transducer([0, 1, 1, 0])
    [0, 1, 1]
    sage: shift_right_transducer([1, 0, 0])
    [1, 0]

The output of the shifts above look a bit weird (from a right-shift transducer, we would expect, for example, that `[1, 0, 0]` was mapped to `[0, 1, 0]`), since we write `None` instead of the zero at the left. Further, note that only \(0\) is listed as a final state as we have to enforce that a most significant zero is read as the last input letter in order to flush the last digit:

    sage: shift_right_transducer([0, 1, 0, 1])
    Traceback (most recent call last):
    ...
    ValueError: Invalid input sequence.

Next, we construct the transducer performing the xor operation. We also have to take `None` into account as our `shift_right_transducer` waits one iteration until it starts writing output. This corresponds with our intention to forget the first letter.

    sage: def xor_transition(state, digits):
    ....:    if digits[0] is None or digits[1] is None:
    ....:        return (0, None)
    ....:    else:
    ....:        return (0, digits[0].__xor__(digits[1]))
    sage: from itertools import product
    sage: xor_transducer = Transducer(
    ....:    xor_transition,
    ....:    initial_states=[0],
    ....:    final_states=[0],
    ....:    input_alphabet=list(product([None, 0, 1], [0, 1])))
    sage: xor_transducer.transitions()
    [Transition from 0 to 0: (None, 0)|-,
     Transition from 0 to 0: (None, 1)|-,
     Transition from 0 to 0: (0, 0)|0,
     Transition from 0 to 0: (0, 1)|1,
     Transition from 0 to 0: (1, 0)|1,
     Transition from 0 to 0: (1, 1)|0]
    sage: xor_transducer([(None, 0), (None, 1), (0, 0), (0, 1), (1, 0), (1, 1)])
    [0, 1, 1, 0]
    sage: xor_transducer([(0, None)])
    Traceback (most recent call last):
    ...
    ValueError: Invalid input sequence.

The transducer computing the Gray code is then constructed as a `cartesian product` between the shifted version and the original input (represented here by the `shift_right_transducer` and the `identity transducer`, respectively). This cartesian product is then fed into the `xor_transducer` as a `composition` of transducers.

As described in `Transducer.cartesian_product()`, we have to temporarily set `finite_state_machine.FSMOldCodeTransducerCartesianProduct` to `False` in order to disable backwards compatible code.

    sage: sage.combinat.finite_state_machine.FSMOldCodeTransducerCartesianProduct = False
    sage: product_transducer = shift_right_transducer.cartesian_product(transducers.Identity([0, 1]))
    sage: sage.combinat.finite_state_machine.FSMOldCodeTransducerCartesianProduct = True
    sage: Gray_transducer = xor_transducer(product_transducer)

We use `construct_final_word_out()` to make sure that all output is written; otherwise, we would have to make sure that a sufficient number of trailing zeros is read.

    sage: Gray_transducer.construct_final_word_out([0])
    sage: Gray_transducer.transitions()
    [Transition from (('I', 0), 0) to ((0, 0), 0): 0|-,
     Transition from (('I', 0), 0) to ((1, 0), 0): 1|-,
     Transition from ((0, 0), 0) to ((0, 0), 0): 0|0,
     Transition from ((0, 0), 0) to ((1, 0), 0): 1|1,
     Transition from ((1, 0), 0) to ((0, 0), 0): 0|1,
     Transition from ((1, 0), 0) to ((1, 0), 0): 1|0]

There is a `prepackaged transducer` for Gray code, let’s see whether they agree. We have to use `relabeled()` to relabel our states with integers.

    sage: constructed = Gray_transducer.relabeled()
    sage: packaged = transducers.GrayCode()
    sage: constructed == packaged
    True

Finally, we check that this indeed computes the Gray code of the first 10 non-negative integers.

    sage: for n in srange(10):
    ....:     Gray_transducer(n.bits())
    []
    [1]
    [1, 1]
    [0, 1]
    [0, 1, 1]
    [1, 1, 1]
    [1, 0, 1]
    [0, 0, 1]
    [0, 0, 1, 1]
    [1, 0, 1, 1]

### Using the hook-functions

Let’s use the previous example “divison by 3” to demonstrate the optional state and transition parameters `hook`.

First, we define what those functions should do. In our case, this is just saying in which state we are and which transition we take

    sage: def state_hook(process, state, output):
    ....:     print "We are now in State %s." % (state.label(),)
    sage: from sage.combinat.finite_state_machine import FSMWordSymbol
    sage: def transition_hook(transition, process):
    ....:     print ("Currently we go from %s to %s, "
    ....:            "reading %s and writing %s." % (
    ....:                transition.from_state, transition.to_state,
    ....:                FSMWordSymbol(transition.word_in),
    ....:                FSMWordSymbol(transition.word_out)))

Now, let’s add these hook-functions to the existing transducer:

    sage: for s in D.iter_states():
    ....:     s.hook = state_hook
    sage: for t in D.iter_transitions():
    ....:     t.hook = transition_hook

Rerunning the process again now gives the following output:

    sage: D.process([1, 1, 0, 1], check_epsilon_transitions=False)
    We are now in State 0.
    Currently we go from 0 to 1, reading 1 and writing 0.
    We are now in State 1.
    Currently we go from 1 to 0, reading 1 and writing 1.
    We are now in State 0.
    Currently we go from 0 to 0, reading 0 and writing 0.
    We are now in State 0.
    Currently we go from 0 to 1, reading 1 and writing 0.
    We are now in State 1.
    (False, 1, [0, 1, 0, 0])

The example above just explains the basic idea of using hook-functions. In the following, we will use those hooks more seriously.

Warning

The hooks of the states are also called while exploring the epsilon successors of a state (during processing). In the example above, we used `check_epsilon_transitions=False` to avoid this (and also therefore got a cleaner output).

Warning

The arguments used when calling a hook have changed in trac ticket #16538 from `hook(state, process)` to `hook(process, state, output)`. If you are using an old-style hook, a deprecation warning is displayed.

### Detecting sequences with same number of \(0\) and \(1\)

Suppose we have a binary input and want to accept all sequences with the same number of \(0\) and \(1\). This cannot be done with a finite automaton. Anyhow, we can make usage of the hook functions to extend our finite automaton by a counter:

    sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition
    sage: C = FiniteStateMachine()
    sage: def update_counter(process, state, output):
    ....:     l = process.preview_word()
    ....:     process.fsm.counter += 1 if l == 1 else -1
    ....:     if process.fsm.counter > 0:
    ....:         next_state = 'positive'
    ....:     elif process.fsm.counter < 0:
    ....:         next_state = 'negative'
    ....:     else:
    ....:         next_state = 'zero'
    ....:     return FSMTransition(state, process.fsm.state(next_state),
    ....:                          l, process.fsm.counter)
    sage: C.add_state(FSMState('zero', hook=update_counter,
    ....:             is_initial=True, is_final=True))
    'zero'
    sage: C.add_state(FSMState('positive', hook=update_counter))
    'positive'
    sage: C.add_state(FSMState('negative', hook=update_counter))
    'negative'

Now, let’s input some sequence:

    sage: C.counter = 0; C([1, 1, 1, 1, 0, 0])
    (False, 'positive', [1, 2, 3, 4, 3, 2])

The result is False, since there are four \(1\) but only two \(0\). We land in the state `positive` and we can also see the values of the counter in each step.

Let’s try some other examples:

    sage: C.counter = 0; C([1, 1, 0, 0])
    (True, 'zero', [1, 2, 1, 0])
    sage: C.counter = 0; C([0, 1, 0, 0])
    (False, 'negative', [-1, 0, -1, -2])

See also methods `Automaton.process()` and `Transducer.process()` (or even `FiniteStateMachine.process()`), the explanation of the parameter `hook` and the examples in `FSMState` and `FSMTransition`, and the description and examples in `FSMProcessIterator` for more information on processing and hooks.

AUTHORS:

- Daniel Krenn (2012-03-27): initial version

- Clemens Heuberger (2012-04-05): initial version

- Sara Kropf (2012-04-17): initial version

- Clemens Heuberger (2013-08-21): release candidate for Sage patch

- Daniel Krenn (2013-08-21): release candidate for Sage patch

- Sara Kropf (2013-08-21): release candidate for Sage patch

- Clemens Heuberger (2013-09-02): documentation improved

- Daniel Krenn (2013-09-13): comments from trac worked in

- Clemens Heuberger (2013-11-03): output (labels) of determinisation,

    product, composition, etc. changed (for consistency), representation of state changed, documentation improved

- Daniel Krenn (2013-11-04): whitespaces in documentation corrected

- Clemens Heuberger (2013-11-04): full_group_by added

- Daniel Krenn (2013-11-04): next release candidate for Sage patch

- Sara Kropf (2013-11-08): fix for adjacency matrix

- Clemens Heuberger (2013-11-11): fix for prepone_output

- Daniel Krenn (2013-11-11): comments from trac #15078 included:

    docstring of FiniteStateMachine rewritten, Automaton and Transducer inherited from FiniteStateMachine

- Daniel Krenn (2013-11-25): documentation improved according to

    comments from trac #15078

- Clemens Heuberger, Daniel Krenn, Sara Kropf (2014-02-21–2014-07-18): A huge bunch of improvements. Details see #15841, #15847, #15848, #15849, #15850, #15922, #15923, #15924, #15925, #15928, #15960, #15961, #15962, #15963, #15975, #16016, #16024, #16061, #16128, #16132, #16138, #16139, #16140, #16143, #16144, #16145, #16146, #16191, #16200, #16205, #16206, #16207, #16229, #16253, #16254, #16255, #16266, #16355, #16357, #16387, #16425, #16539, #16555, #16557, #16588, #16589, #16666, #16668, #16674, #16675, #16677.

ACKNOWLEDGEMENT:

- Clemens Heuberger, Daniel Krenn and Sara Kropf are supported by the Austrian Science Fund (FWF): P 24644-N26.

## Methods

class `sage.combinat.finite_state_machine.``Automaton`(*args, **kwargs)

  Bases: `sage.combinat.finite_state_machine.FiniteStateMachine`

  This creates an automaton, which is a finite state machine, whose transitions have input labels.

  An automaton has additional features like creating a deterministic and a minimized automaton.

  See class `FiniteStateMachine` for more information.

  EXAMPLES:

  We can create an automaton recognizing even numbers (given in binary and read from left to right) in the following way:

      sage: A = Automaton([('P', 'Q', 0), ('P', 'P', 1),
      ....:                ('Q', 'P', 1), ('Q', 'Q', 0)],
      ....:               initial_states=['P'], final_states=['Q'])
      sage: A
      Automaton with 2 states
      sage: A([0])
      True
      sage: A([1, 1, 0])
      True
      sage: A([1, 0, 1])
      False

  Note that the full output of the commands can be obtained by calling `process()` and looks like this:

      sage: A.process([1, 0, 1])
      (False, 'P')

  TESTS:

      sage: Automaton()
      Automaton with 0 states

  `cartesian_product`(other, only_accessible_components=True)

    Returns a new automaton which accepts an input if it is accepted by both given automata.

    INPUT:

    - `other` – an automaton
    - `only_accessible_components` – If `True` (default), then the result is piped through `accessible_components()`. If no `new_input_alphabet` is given, it is determined by `determine_alphabets()`.

    OUTPUT:

    A new automaton which computes the intersection (see below) of the languages of `self` and `other`.

    The set of states of the new automaton is the cartesian product of the set of states of both given automata. There is a transition \(((A, B), (C, D), a)\) in the new automaton if there are transitions \((A, C, a)\) and \((B, D, a)\) in the old automata.

    The methods `intersection()` and `cartesian_product()` are the same (for automata).

    EXAMPLES:

        sage: aut1 = Automaton([('1', '2', 1),
        ....:                   ('2', '2', 1),
        ....:                   ('2', '2', 0)],
        ....:                  initial_states=['1'],
        ....:                  final_states=['2'],
        ....:                  determine_alphabets=True)
        sage: aut2 = Automaton([('A', 'A', 1),
        ....:                   ('A', 'B', 0),
        ....:                   ('B', 'B', 0),
        ....:                   ('B', 'A', 1)],
        ....:                  initial_states=['A'],
        ....:                  final_states=['B'],
        ....:                  determine_alphabets=True)
        sage: res = aut1.intersection(aut2)
        sage: (aut1([1, 1]), aut2([1, 1]), res([1, 1]))
        (True, False, False)
        sage: (aut1([1, 0]), aut2([1, 0]), res([1, 0]))
        (True, True, True)
        sage: res.transitions()
        [Transition from ('1', 'A') to ('2', 'A'): 1|-,
         Transition from ('2', 'A') to ('2', 'B'): 0|-,
         Transition from ('2', 'A') to ('2', 'A'): 1|-,
         Transition from ('2', 'B') to ('2', 'B'): 0|-,
         Transition from ('2', 'B') to ('2', 'A'): 1|-]

    For automata with epsilon-transitions, intersection is not well defined. But for any finite state machine, epsilon-transitions can be removed by `remove_epsilon_transitions()`.

        sage: a1 = Automaton([(0, 0, 0),
        ....:                 (0, 1, None),
        ....:                 (1, 1, 1),
        ....:                 (1, 2, 1)],
        ....:                 initial_states=[0],
        ....:                 final_states=[1],
        ....:                 determine_alphabets=True)
        sage: a2 = Automaton([(0, 0, 0), (0, 1, 1), (1, 1, 1)],
        ....:                 initial_states=[0],
        ....:                 final_states=[1],
        ....:                 determine_alphabets=True)
        sage: a1.intersection(a2)
        Traceback (most recent call last):
        ...
        ValueError: An epsilon-transition (with empty input)
        was found.
        sage: a1.remove_epsilon_transitions()  # not tested (since not implemented yet)
        sage: a1.intersection(a2)  # not tested

  `determinisation`()

    Returns a deterministic automaton which accepts the same input words as the original one.

    INPUT:

    Nothing.

    OUTPUT:

    A new automaton, which is deterministic.

    The labels of the states of the new automaton are frozensets of states of `self`. The color of a new state is the frozenset of colors of the constituent states of `self`. Therefore, the colors of the constituent states have to be hashable.

    The input alphabet must be specified.

    EXAMPLES:

        sage: aut = Automaton([('A', 'A', 0), ('A', 'B', 1), ('B', 'B', 1)],
        ....:                 initial_states=['A'], final_states=['B'])
        sage: aut.determinisation().transitions()
        [Transition from frozenset(['A'])
                      to frozenset(['A']): 0|-,
         Transition from frozenset(['A'])
                      to frozenset(['B']): 1|-,
         Transition from frozenset(['B'])
                      to frozenset([]): 0|-,
         Transition from frozenset(['B'])
                      to frozenset(['B']): 1|-,
         Transition from frozenset([])
                      to frozenset([]): 0|-,
         Transition from frozenset([])
                      to frozenset([]): 1|-]

        sage: A = Automaton([('A', 'A', 1), ('A', 'A', 0), ('A', 'B', 1),
        ....:                ('B', 'C', 0), ('C', 'C', 1), ('C', 'C', 0)],
        ....:               initial_states=['A'], final_states=['C'])
        sage: A.determinisation().states()
        [frozenset(['A']), frozenset(['A', 'B']),
        frozenset(['A', 'C']), frozenset(['A', 'C', 'B'])]

        sage: A = Automaton([(0, 1, 1), (0, 2, [1, 1]), (0, 3, [1, 1, 1]),
        ....:                (1, 0, -1), (2, 0, -2), (3, 0, -3)],
        ....:               initial_states=[0], final_states=[0, 1, 2, 3])
        sage: B = A.determinisation().relabeled()
        sage: all(t.to_state.label() == 2 for t in
        ....:     B.state(2).transitions)
        True
        sage: B.state(2).is_final
        False
        sage: B.delete_state(2)  # this is a sink
        sage: sorted(B.transitions())
        [Transition from 0 to 1: 1|-,
         Transition from 1 to 0: -1|-,
         Transition from 1 to 3: 1|-,
         Transition from 3 to 0: -2|-,
         Transition from 3 to 4: 1|-,
         Transition from 4 to 0: -3|-]

    Note that colors of states have to be hashable:

        sage: A = Automaton([[0, 0, 0]], initial_states=[0])
        sage: A.state(0).color = []
        sage: A.determinisation()
        Traceback (most recent call last):
        ...
        TypeError: unhashable type: 'list'
        sage: A.state(0).color = ()
        sage: A.determinisation()
        Automaton with 1 states

    TESTS:

    This is from trac ticket #15078, comment 13.

        sage: D = {'A': [('A', 'a'), ('B', 'a'), ('A', 'b')],
        ....:      'C': [], 'B': [('C', 'b')]}
        sage: auto = Automaton(D, initial_states=['A'], final_states=['C'])
        sage: auto.is_deterministic()
        False
        sage: auto.process(list('aaab'))
        [(False, 'A'), (True, 'C')]
        sage: auto.states()
        ['A', 'C', 'B']
        sage: Ddet = auto.determinisation()
        sage: Ddet
        Automaton with 3 states
        sage: Ddet.is_deterministic()
        True
        sage: sorted(Ddet.transitions())
        [Transition from frozenset(['A']) to frozenset(['A', 'B']): 'a'|-,
         Transition from frozenset(['A']) to frozenset(['A']): 'b'|-,
         Transition from frozenset(['A', 'B']) to frozenset(['A', 'B']): 'a'|-,
         Transition from frozenset(['A', 'B']) to frozenset(['A', 'C']): 'b'|-,
         Transition from frozenset(['A', 'C']) to frozenset(['A', 'B']): 'a'|-,
         Transition from frozenset(['A', 'C']) to frozenset(['A']): 'b'|-]
        sage: Ddet.initial_states()
        [frozenset(['A'])]
        sage: Ddet.final_states()
        [frozenset(['A', 'C'])]
        sage: Ddet.process(list('aaab'))
        (True, frozenset(['A', 'C']))

  `intersection`(other, only_accessible_components=True)

    Returns a new automaton which accepts an input if it is accepted by both given automata.

    INPUT:

    - `other` – an automaton
    - `only_accessible_components` – If `True` (default), then the result is piped through `accessible_components()`. If no `new_input_alphabet` is given, it is determined by `determine_alphabets()`.

    OUTPUT:

    A new automaton which computes the intersection (see below) of the languages of `self` and `other`.

    The set of states of the new automaton is the cartesian product of the set of states of both given automata. There is a transition \(((A, B), (C, D), a)\) in the new automaton if there are transitions \((A, C, a)\) and \((B, D, a)\) in the old automata.

    The methods `intersection()` and `cartesian_product()` are the same (for automata).

    EXAMPLES:

        sage: aut1 = Automaton([('1', '2', 1),
        ....:                   ('2', '2', 1),
        ....:                   ('2', '2', 0)],
        ....:                  initial_states=['1'],
        ....:                  final_states=['2'],
        ....:                  determine_alphabets=True)
        sage: aut2 = Automaton([('A', 'A', 1),
        ....:                   ('A', 'B', 0),
        ....:                   ('B', 'B', 0),
        ....:                   ('B', 'A', 1)],
        ....:                  initial_states=['A'],
        ....:                  final_states=['B'],
        ....:                  determine_alphabets=True)
        sage: res = aut1.intersection(aut2)
        sage: (aut1([1, 1]), aut2([1, 1]), res([1, 1]))
        (True, False, False)
        sage: (aut1([1, 0]), aut2([1, 0]), res([1, 0]))
        (True, True, True)
        sage: res.transitions()
        [Transition from ('1', 'A') to ('2', 'A'): 1|-,
         Transition from ('2', 'A') to ('2', 'B'): 0|-,
         Transition from ('2', 'A') to ('2', 'A'): 1|-,
         Transition from ('2', 'B') to ('2', 'B'): 0|-,
         Transition from ('2', 'B') to ('2', 'A'): 1|-]

    For automata with epsilon-transitions, intersection is not well defined. But for any finite state machine, epsilon-transitions can be removed by `remove_epsilon_transitions()`.

        sage: a1 = Automaton([(0, 0, 0),
        ....:                 (0, 1, None),
        ....:                 (1, 1, 1),
        ....:                 (1, 2, 1)],
        ....:                 initial_states=[0],
        ....:                 final_states=[1],
        ....:                 determine_alphabets=True)
        sage: a2 = Automaton([(0, 0, 0), (0, 1, 1), (1, 1, 1)],
        ....:                 initial_states=[0],
        ....:                 final_states=[1],
        ....:                 determine_alphabets=True)
        sage: a1.intersection(a2)
        Traceback (most recent call last):
        ...
        ValueError: An epsilon-transition (with empty input)
        was found.
        sage: a1.remove_epsilon_transitions()  # not tested (since not implemented yet)
        sage: a1.intersection(a2)  # not tested

  `minimization`(algorithm=None)

    Returns the minimization of the input automaton as a new automaton.

    INPUT:

    - `algorithm` – Either Moore’s algorithm (by `algorithm='Moore'` or as default for deterministic automata) or Brzozowski’s algorithm (when `algorithm='Brzozowski'` or when the automaton is not deterministic) is used.

    OUTPUT:

    A new automaton.

    The resulting automaton is deterministic and has a minimal number of states.

    EXAMPLES:

        sage: A = Automaton([('A', 'A', 1), ('A', 'A', 0), ('A', 'B', 1),
        ....:                ('B', 'C', 0), ('C', 'C', 1), ('C', 'C', 0)],
        ....:               initial_states=['A'], final_states=['C'])
        sage: B = A.minimization(algorithm='Brzozowski')
        sage: B.transitions(B.states()[1])
        [Transition from frozenset([frozenset(['A', 'C', 'B']),
        frozenset(['C', 'B']), frozenset(['A', 'C'])]) to
        frozenset([frozenset(['A', 'C', 'B']), frozenset(['C', 'B']),
        frozenset(['A', 'C']), frozenset(['C'])]): 0|-,
        Transition from frozenset([frozenset(['A', 'C', 'B']),
        frozenset(['C', 'B']), frozenset(['A', 'C'])]) to
        frozenset([frozenset(['A', 'C', 'B']), frozenset(['C', 'B']),
        frozenset(['A', 'C'])]): 1|-]
        sage: len(B.states())
        3
        sage: C = A.minimization(algorithm='Brzozowski')
        sage: C.transitions(C.states()[1])
        [Transition from frozenset([frozenset(['A', 'C', 'B']),
        frozenset(['C', 'B']), frozenset(['A', 'C'])]) to
        frozenset([frozenset(['A', 'C', 'B']), frozenset(['C', 'B']),
        frozenset(['A', 'C']), frozenset(['C'])]): 0|-,
        Transition from frozenset([frozenset(['A', 'C', 'B']),
        frozenset(['C', 'B']), frozenset(['A', 'C'])]) to
        frozenset([frozenset(['A', 'C', 'B']), frozenset(['C', 'B']),
        frozenset(['A', 'C'])]): 1|-]
        sage: len(C.states())
        3

        sage: aut = Automaton([('1', '2', 'a'), ('2', '3', 'b'),
        ....:                  ('3', '2', 'a'), ('2', '1', 'b'),
        ....:                  ('3', '4', 'a'), ('4', '3', 'b')],
        ....:                  initial_states=['1'], final_states=['1'])
        sage: min = aut.minimization(algorithm='Brzozowski')
        sage: [len(min.states()), len(aut.states())]
        [3, 4]
        sage: min = aut.minimization(algorithm='Moore')
        Traceback (most recent call last):
        ...
        NotImplementedError: Minimization via Moore's Algorithm is only
        implemented for deterministic finite state machines

  `process`(*args, **kwargs)

    Warning

    The default output of this method is scheduled to change. This docstring describes the new default behaviour, which can already be achieved by setting `FSMOldProcessOutput` to `False`.

    Returns whether the automaton accepts the input and the state where the computation stops.

    INPUT:

    - `input_tape` – the input tape can be a list or an iterable with entries from the input alphabet. If we are working with a multi-tape machine (see parameter `use_multitape_input` and notes below), then the tape is a list or tuple of tracks, each of which can be a list or an iterable with entries from the input alphabet.
    - `initial_state` or `initial_states` – the initial state(s) in which the machine starts. Either specify a single one with `initial_state` or a list of them with `initial_states`. If both are given, `initial_state` will be appended to `initial_states`. If neither is specified, the initial states of the finite state machine are taken.
    - `list_of_outputs` – (default: `None`) a boolean or `None`. If `True`, then the outputs are given in list form (even if we have no or only one single output). If `False`, then the result is never a list (an exception is raised if the result cannot be returned). If `list_of_outputs=None` the method determines automatically what to do (e.g. if a non-deterministic machine returns more than one path, then the output is returned in list form).
    - `only_accepted` – (default: `False`) a boolean. If set, then the first argument in the output is guaranteed to be `True` (if the output is a list, then the first argument of each element will be `True`).
    - `full_output` – (default: `True`) a boolean. If set, then the full output is given, otherwise only whether the sequence is accepted or not (the first entry below only).
    - `always_include_output` – if set (not by default), always return a triple containing the (non-existing) output. This is in order to obtain output compatible with that of `FiniteStateMachine.process()`. If this parameter is set, `full_output` has no effect.
    - `format_output` – a function that translates the written output (which is in form of a list) to something more readable. By default (`None`) identity is used here.
    - `check_epsilon_transitions` – (default: `True`) a boolean. If `False`, then epsilon transitions are not taken into consideration during process.
    - `write_final_word_out` – (default: `True`) a boolean specifying whether the final output words should be written or not.
    - `use_multitape_input` – (default: `False`) a boolean. If `True`, then the multi-tape mode of the process iterator is activated. See also the notes below for multi-tape machines.

    OUTPUT:

    The full output is a pair (or a list of pairs, cf. parameter `list_of_outputs`), where

    - the first entry is `True` if the input string is accepted and
    - the second gives the state reached after processing the input tape (This is a state with label `None` if the input could not be processed, i.e., if at one point no transition to go on could be found.).

    If `full_output` is `False`, then only the first entry is returned.

    If `always_include_output` is set, an additional third entry `[]` is included.

    Note that in the case the automaton is not deterministic, all possible paths are taken into account. You can use `determinisation()` to get a deterministic automaton machine.

    By setting `FSMOldProcessOutput` to `False` the new desired output is produced.

    This function uses an iterator which, in its simplest form, goes from one state to another in each step. To decide which way to go, it uses the input words of the outgoing transitions and compares them to the input tape. More precisely, in each step, the iterator takes an outgoing transition of the current state, whose input label equals the input letter of the tape.

    If the choice of the outgoing transition is not unique (i.e., we have a non-deterministic finite state machine), all possibilites are followed. This is done by splitting the process into several branches, one for each of the possible outgoing transitions.

    The process (iteration) stops if all branches are finished, i.e., for no branch, there is any transition whose input word coincides with the processed input tape. This can simply happen when the entire tape was read.

    Also see `__call__()` for a version of `process()` with shortened output.

    Internally this function creates and works with an instance of `FSMProcessIterator`. This iterator can also be obtained with `iter_process()`.

    If working with multi-tape finite state machines, all input words of transitions are words of \(k\)-tuples of letters. Moreover, the input tape has to consist of \(k\) tracks, i.e., be a list or tuple of \(k\) iterators, one for each track.

    Warning

    Working with multi-tape finite state machines is still experimental and can lead to wrong outputs.

    EXAMPLES:

    In the following examples, we construct an automaton which accepts non-adjacent forms (see also the example on non-adjacent forms in the documentation of the module `finite_state_machine`) and then test it by feeding it with several binary digit expansions.

        sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False  # activate new output behavior
        sage: NAF = Automaton(
        ....:     {'_': [('_', 0), ('1', 1)], '1': [('_', 0)]},
        ....:     initial_states=['_'], final_states=['_', '1'])
        sage: [NAF.process(w) for w in [[0], [0, 1], [1, 1], [0, 1, 0, 1],
        ....:                           [0, 1, 1, 1, 0], [1, 0, 0, 1, 1]]]
        [(True, '_'), (True, '1'), (False, None),
         (True, '1'), (False, None), (False, None)]

    If we just want a condensed output, we use:

        sage: [NAF.process(w, full_output=False)
        ....:     for w in [[0], [0, 1], [1, 1], [0, 1, 0, 1],
        ....:               [0, 1, 1, 1, 0], [1, 0, 0, 1, 1]]]
        [True, True, False, True, False, False]

    It is equivalent to:

        sage: [NAF(w) for w in [[0], [0, 1], [1, 1], [0, 1, 0, 1],
        ....:                   [0, 1, 1, 1, 0], [1, 0, 0, 1, 1]]]
        [True, True, False, True, False, False]

    The following example illustrates the difference between non-existing paths and reaching a non-final state:

        sage: NAF.process([2])
        (False, None)
        sage: NAF.add_transition(('_', 's', 2))
        Transition from '_' to 's': 2|-
        sage: NAF.process([2])
        (False, 's')

    A simple example of a (non-deterministic) multi-tape automaton is the following: It checks whether the two input tapes have the same number of ones:

        sage: M = Automaton([('=', '=', (1, 1)),
        ....:                ('=', '=', (None, 0)),
        ....:                ('=', '=', (0, None)),
        ....:                ('=', '<', (None, 1)),
        ....:                ('<', '<', (None, 1)),
        ....:                ('<', '<', (None, 0)),
        ....:                ('=', '>', (1, None)),
        ....:                ('>', '>', (1, None)),
        ....:                ('>', '>', (0, None))],
        ....:               initial_states=['='],
        ....:               final_states=['='])
        sage: M.process(([1, 0, 1], [1, 0]), use_multitape_input=True)
        (False, '>')
        sage: M.process(([0, 1, 0], [0, 1, 1]), use_multitape_input=True)
        (False, '<')
        sage: M.process(([1, 1, 0, 1], [0, 0, 1, 0, 1, 1]),
        ....:           use_multitape_input=True)
        (True, '=')

    Alternatively, we can use the following (non-deterministic) multi-tape automaton for the same check:

        sage: N = Automaton([('=', '=', (0, 0)),
        ....:                ('=', '<', (None, 1)),
        ....:                ('<', '<', (0, None)),
        ....:                ('<', '=', (1, None)),
        ....:                ('=', '>', (1, None)),
        ....:                ('>', '>', (None, 0)),
        ....:                ('>', '=', (None, 1))],
        ....:               initial_states=['='],
        ....:               final_states=['='])
        sage: N.process(([1, 0, 1], [1, 0]), use_multitape_input=True)
        (False, '>')
        sage: N.process(([0, 1, 0], [0, 1, 1]), use_multitape_input=True)
        (False, '<')
        sage: N.process(([1, 1, 0, 1], [0, 0, 1, 0, 1, 1]),
        ....:           use_multitape_input=True)
        (True, '=')

    See also

    `FiniteStateMachine.process()`, `Transducer.process()`, `iter_process()`, `__call__()`, `FSMProcessIterator`.

`sage.combinat.finite_state_machine.``FSMLetterSymbol`(letter)

  Returns a string associated to the input letter.

  INPUT:

  - `letter` – the input letter or `None` (representing the empty word).

  OUTPUT:

  If `letter` is `None` the symbol for the empty word `FSMEmptyWordSymbol` is returned, otherwise the string associated to the letter.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import FSMLetterSymbol
      sage: FSMLetterSymbol(0)
      '0'
      sage: FSMLetterSymbol(None)
      '-'

class `sage.combinat.finite_state_machine.``FSMProcessIterator`(fsm, input_tape=None, initial_state=None, initial_states=[], use_multitape_input=False, check_epsilon_transitions=True, write_final_word_out=True, format_output=None, **kwargs)

  Bases: `sage.structure.sage_object.SageObject`, `_abcoll.Iterator`

  This class takes an input, feeds it into a finite state machine (automaton or transducer, in particular), tests whether this was successful and calculates the written output.

  INPUT:

  - `fsm` – the finite state machine on which the input should be processed.
  - `input_tape` – the input tape can be a list or an iterable with entries from the input alphabet. If we are working with a multi-tape machine (see parameter `use_multitape_input` and notes below), then the tape is a list or tuple of tracks, each of which can be a list or an iterable with entries from the input alphabet.
  - `initial_state` or `initial_states` – the initial state(s) in which the machine starts. Either specify a single one with `initial_state` or a list of them with `initial_states`. If both are given, `initial_state` will be appended to `initial_states`. If neither is specified, the initial states of the finite state machine are taken.
  - `format_output` – a function that translates the written output (which is in form of a list) to something more readable. By default (`None`) identity is used here.
  - `check_epsilon_transitions` – (default: `True`) a boolean. If `False`, then epsilon transitions are not taken into consideration during process.
  - `write_final_word_out` – (default: `True`) a boolean specifying whether the final output words should be written or not.
  - `use_multitape_input` – (default: `False`) a boolean. If `True`, then the multi-tape mode of the process iterator is activated. See also the notes below for multi-tape machines.

  OUTPUT:

  An iterator.

  In its simplest form, it behaves like an iterator which, in each step, goes from one state to another. To decide which way to go, it uses the input words of the outgoing transitions and compares them to the input tape. More precisely, in each step, the process iterator takes an outgoing transition of the current state, whose input label equals the input letter of the tape. The output label of the transition, if present, is written on the output tape.

  If the choice of the outgoing transition is not unique (i.e., we have a non-deterministic finite state machine), all possibilites are followed. This is done by splitting the process into several branches, one for each of the possible outgoing transitions.

  The process (iteration) stops if all branches are finished, i.e., for no branch, there is any transition whose input word coincides with the processed input tape. This can simply happen when the entire tape was read. When the process stops, a `StopIteration` exception is thrown.

  Warning

  Processing an input tape of length \(n\) usually takes at least \(n+1\) iterations, since there will be \(n+1\) states visited (in the case the taken transitions have input words consisting of single letters).

  An instance of this class is generated when `FiniteStateMachine.process()` or `FiniteStateMachine.iter_process()` of a finite state machine, an automaton, or a transducer is invoked.

  When working with multi-tape finite state machines, all input words of transitions are words of \(k\)-tuples of letters. Moreover, the input tape has to consist of \(k\) tracks, i.e., be a list or tuple of \(k\) iterators, one for each track.

  Warning

  Working with multi-tape finite state machines is still experimental and can lead to wrong outputs.

  EXAMPLES:

  The following transducer reads binary words and outputs a word, where blocks of ones are replaced by just a single one. Further only words that end with a zero are accepted.

      sage: T = Transducer({'A': [('A', 0, 0), ('B', 1, None)],
      ....:                 'B': [('B', 1, None), ('A', 0, [1, 0])]},
      ....:     initial_states=['A'], final_states=['A'])
      sage: input = [1, 1, 0, 0, 1, 0, 1, 1, 1, 0]
      sage: T.process(input)
      (True, 'A', [1, 0, 0, 1, 0, 1, 0])

  The function `FiniteStateMachine.process()` (internally) uses a `FSMProcessIterator`. We can do that manually, too, and get full access to the iteration process:

      sage: from sage.combinat.finite_state_machine import FSMProcessIterator
      sage: it = FSMProcessIterator(T, input_tape=input)
      sage: for current in it:
      ....:     print current
      process (1 branch)
      + at state 'B'
      +-- tape at 1, [[]]
      process (1 branch)
      + at state 'B'
      +-- tape at 2, [[]]
      process (1 branch)
      + at state 'A'
      +-- tape at 3, [[1, 0]]
      process (1 branch)
      + at state 'A'
      +-- tape at 4, [[1, 0, 0]]
      process (1 branch)
      + at state 'B'
      +-- tape at 5, [[1, 0, 0]]
      process (1 branch)
      + at state 'A'
      +-- tape at 6, [[1, 0, 0, 1, 0]]
      process (1 branch)
      + at state 'B'
      +-- tape at 7, [[1, 0, 0, 1, 0]]
      process (1 branch)
      + at state 'B'
      +-- tape at 8, [[1, 0, 0, 1, 0]]
      process (1 branch)
      + at state 'B'
      +-- tape at 9, [[1, 0, 0, 1, 0]]
      process (1 branch)
      + at state 'A'
      +-- tape at 10, [[1, 0, 0, 1, 0, 1, 0]]
      process (0 branches)
      sage: it.result()
      [(True, 'A', [1, 0, 0, 1, 0, 1, 0])]

      sage: T = Transducer([(0, 0, 0, 'a'), (0, 1, 0, 'b'),
      ....:                 (1, 2, 1, 'c'), (2, 0, 0, 'd'),
      ....:                 (2, 1, None, 'd')],
      ....:                initial_states=[0], final_states=[2])
      sage: T.process([0, 0, 1], format_output=lambda o: ''.join(o))
      [(False, 1, 'abcd'), (True, 2, 'abc')]
      sage: it = FSMProcessIterator(T, input_tape=[0, 0, 1],
      ....:                         format_output=lambda o: ''.join(o))
      sage: for current in it:
      ....:     print current
      process (2 branches)
      + at state 0
      +-- tape at 1, [['a']]
      + at state 1
      +-- tape at 1, [['b']]
      process (2 branches)
      + at state 0
      +-- tape at 2, [['a', 'a']]
      + at state 1
      +-- tape at 2, [['a', 'b']]
      process (2 branches)
      + at state 1
      +-- tape at 3, [['a', 'b', 'c', 'd']]
      + at state 2
      +-- tape at 3, [['a', 'b', 'c']]
      process (0 branches)
      sage: it.result()
      [(False, 1, 'abcd'), (True, 2, 'abc')]

  See also

  `FiniteStateMachine.process()`, `Automaton.process()`, `Transducer.process()`, `FiniteStateMachine.iter_process()`, `FiniteStateMachine.__call__()`, `next()`.

  TESTS:

      sage: T = Transducer([[0, 0, 0, 0]])
      sage: T.process([])
      Traceback (most recent call last):
      ...
      ValueError: No state is initial.

      sage: T = Transducer([[0, 1, 0, 0]], initial_states=[0, 1])
      sage: T.process([])
      [(False, 0, []), (False, 1, [])]

      sage: T = Transducer([[0, 0, 0, 0]],
      ....:                initial_states=[0], final_states=[0])
      sage: T.state(0).final_word_out = [42]
      sage: T.process([0])
      (True, 0, [0, 42])
      sage: T.process([0], write_final_word_out=False)
      (True, 0, [0])

  class `Current`

    Bases: `dict`

    This class stores the branches which have to be processed during iteration and provides a nicer formatting of them.

    This class is derived from `dict`. It is returned by the `next`-function during iteration.

    EXAMPLES:

    In the following example you can see the dict directly and then the nicer output provided by this class:

        sage: from sage.combinat.finite_state_machine import FSMProcessIterator
        sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:     initial_states=['A'], final_states=['A'])
        sage: it = FSMProcessIterator(inverter, input_tape=[0, 1])
        sage: for current in it:
        ....:     print dict(current)
        ....:     print current
        {((1, 0),): {'A': (tape at 1, [[1]])}}
        process (1 branch)
        + at state 'A'
        +-- tape at 1, [[1]]
        {((2, 0),): {'A': (tape at 2, [[1, 0]])}}
        process (1 branch)
        + at state 'A'
        +-- tape at 2, [[1, 0]]
        {}
        process (0 branches)

  `FSMProcessIterator.``accept_input`

    Is `True` if the reached state is accepted. This is only available at the end of the iteration process.

    Warning

    This attribute is deprecated and should not be used any longer (it may return a wrong result for non-deterministic finite state machines).

    TESTS:

        sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:     initial_states=['A'], final_states=['A'])
        sage: it = inverter.iter_process(input_tape=[0, 1, 1])
        sage: for _ in it:
        ....:     pass
        sage: it.result()
        [(True, 'A', [1, 0, 0])]
        sage: it.accept_input
        doctest:...: DeprecationWarning: This attribute will be removed
        in future releases. Use result() at the end of our iteration
        or the output of next().
        See http://trac.sagemath.org/16538 for details.
        True

  `FSMProcessIterator.``current_state`

    The current/reached state in the process.

    Warning

    This attribute is deprecated and should not be used any longer (it may return a wrong result for non-deterministic finite state machines).

    TESTS:

        sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:     initial_states=['A'], final_states=['A'])
        sage: it = inverter.iter_process(input_tape=[0, 1, 1])
        sage: for current in it:
        ....:     s = it.current_state
        ....:     print current
        ....:     print 'current state:', s
        doctest:...: DeprecationWarning: This attribute will be
        removed in future releases. Use result() at the end of our
        iteration or the output of next().
        See http://trac.sagemath.org/16538 for details.
        process (1 branch)
        + at state 'A'
        +-- tape at 1, [[1]]
        current state: 'A'
        process (1 branch)
        + at state 'A'
        +-- tape at 2, [[1, 0]]
        current state: 'A'
        process (1 branch)
        + at state 'A'
        +-- tape at 3, [[1, 0, 0]]
        current state: 'A'
        process (0 branches)
        current state: None

  `FSMProcessIterator.``next`()

    Makes one step in processing the input tape.

    INPUT:

    Nothing.

    OUTPUT:

    It returns the current status of the iterator (see below). A `StopIteration` exception is thrown when there is/was nothing to do (i.e. all branches ended with previous call of `__next__()`).

    The current status is a dictionary (encapsulated into an instance of `Current`). The keys are positions on the tape. The value corresponding to such a position is again a dictionary, where each entry represents a branch of the process. This dictionary maps the current state of a branch to a pair consisting of a tape cache and a list of output words, which were written during reaching this current state.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMProcessIterator
        sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:     initial_states=['A'], final_states=['A'])
        sage: it = FSMProcessIterator(inverter, input_tape=[0, 1])
        sage: next(it)
        process (1 branch)
        + at state 'A'
        +-- tape at 1, [[1]]
        sage: next(it)
        process (1 branch)
        + at state 'A'
        +-- tape at 2, [[1, 0]]
        sage: next(it)
        process (0 branches)
        sage: next(it)
        Traceback (most recent call last):
        ...
        StopIteration

    See also

    `FiniteStateMachine.process()`, `Automaton.process()`, `Transducer.process()`, `FiniteStateMachine.iter_process()`, `FiniteStateMachine.__call__()`, `FSMProcessIterator`.

    TESTS:

        sage: Z = Transducer()
        sage: s = Z.add_state(0)
        sage: s.is_initial = True
        sage: s.is_final = True
        sage: s.final_word_out = [1, 2]
        sage: Z.process([])
        (True, 0, [1, 2])
        sage: it = FSMProcessIterator(Z, input_tape=[])
        sage: next(it)
        process (0 branches)
        sage: next(it)
        Traceback (most recent call last):
        ...
        StopIteration

        sage: N = Transducer([(0, 0, 0, 1)], initial_states=[0])
        sage: def h_old(state, process):
        ....:     print state, process
        sage: N.state(0).hook = h_old
        sage: N.process([0, 0])
        doctest:...: DeprecationWarning: The hook of state 0 cannot
        be processed: It seems that you are using an old-style hook,
        which is deprecated.
        See http://trac.sagemath.org/16538 for details.
        (False, 0, [1, 1])
        sage: def h_new(process, state, outputs):
        ....:     print state, outputs
        sage: N.state(0).hook = h_new
        sage: N.process([0, 0], check_epsilon_transitions=False)
        0 [[]]
        0 [[1]]
        0 [[1, 1]]
        (False, 0, [1, 1])

  `FSMProcessIterator.``output_tape`

    The written output.

    Warning

    This attribute is deprecated and should not be used any longer (it may return a wrong result for non-deterministic finite state machines).

    TESTS:

        sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:     initial_states=['A'], final_states=['A'])
        sage: it = inverter.iter_process(input_tape=[0, 1, 1])
        sage: for current in it:
        ....:     t = it.output_tape
        ....:     print current
        ....:     print 'output:', t
        doctest:...: DeprecationWarning: This attribute will be removed
        in future releases. Use result() at the end of our iteration
        or the output of next().
        See http://trac.sagemath.org/16538 for details.
        process (1 branch)
        + at state 'A'
        +-- tape at 1, [[1]]
        output: [1]
        process (1 branch)
        + at state 'A'
        +-- tape at 2, [[1, 0]]
        output: [1, 0]
        process (1 branch)
        + at state 'A'
        +-- tape at 3, [[1, 0, 0]]
        output: [1, 0, 0]
        process (0 branches)
        output: None

  `FSMProcessIterator.``preview_word`(track_number=None, length=1, return_word=False)

    Reads a word from the input tape.

    INPUT:

    - `track_number` – an integer or `None`. If `None`, then a tuple of words (one from each track) is returned.
    - `length` – (default: `1`) the length of the word(s).
    - `return_word` – (default: `False`) a boolean. If set, then a word is returned, otherwise a single letter (in which case `length` has to be `1`).

    OUTPUT:

    A single letter or a word.

    An exception `StopIteration` is thrown if the tape (at least one track) has reached its end.

    Typically, this method is called from a hook-function of a state.

    EXAMPLES:

        sage: inverter = Transducer({'A': [('A', 0, 'one'),
        ....:                              ('A', 1, 'zero')]},
        ....:     initial_states=['A'], final_states=['A'])
        sage: def state_hook(process, state, output):
        ....:     print "We are now in state %s." % (state.label(),)
        ....:     print "Next on the tape is a %s." % (
        ....:         process.preview_word(),)
        sage: inverter.state('A').hook = state_hook
        sage: it = inverter.iter_process(
        ....:     input_tape=[0, 1, 1],
        ....:     check_epsilon_transitions=False)
        sage: for _ in it:
        ....:     pass
        We are now in state A.
        Next on the tape is a 0.
        We are now in state A.
        Next on the tape is a 1.
        We are now in state A.
        Next on the tape is a 1.
        We are now in state A.
        sage: it.result()
        [(True, 'A', ['one', 'zero', 'zero'])]

  `FSMProcessIterator.``result`(format_output=None)

    Returns the already finished branches during process.

    INPUT:

    - `format_output` – a function converting the output from list form to something more readable (default: output the list directly).

    OUTPUT:

    A list of triples `(accepted, state, output)`.

    See also the parameter `format_output` of `FSMProcessIterator`.

    EXAMPLES:

        sage: inverter = Transducer({'A': [('A', 0, 'one'), ('A', 1, 'zero')]},
        ....:     initial_states=['A'], final_states=['A'])
        sage: it = inverter.iter_process(input_tape=[0, 1, 1])
        sage: for _ in it:
        ....:     pass
        sage: it.result()
        [(True, 'A', ['one', 'zero', 'zero'])]
        sage: it.result(lambda L: ', '.join(L))
        [(True, 'A', 'one, zero, zero')]

    Using both the parameter `format_output` of `FSMProcessIterator` and the parameter `format_output` of `result()` leads to concatenation of the two functions:

        sage: it = inverter.iter_process(input_tape=[0, 1, 1],
        ....:                            format_output=lambda L: ', '.join(L))
        sage: for _ in it:
        ....:     pass
        sage: it.result()
        [(True, 'A', 'one, zero, zero')]
        sage: it.result(lambda L: ', '.join(L))
        [(True, 'A', 'o, n, e, ,,  , z, e, r, o, ,,  , z, e, r, o')]

class `sage.combinat.finite_state_machine.``FSMState`(label, word_out=None, is_initial=False, is_final=False, final_word_out=None, hook=None, color=None, allow_label_None=False)

  Bases: `sage.structure.sage_object.SageObject`

  Class for a state of a finite state machine.

  INPUT:

  - `label` – the label of the state.
  - `word_out` – (default: `None`) a word that is written when the state is reached.
  - `is_initial` – (default: `False`)
  - `is_final` – (default: `False`)
  - `final_word_out` – (default: `None`) a word that is written when the state is reached as the last state of some input; only for final states.
  - `hook` – (default: `None`) A function which is called when the state is reached during processing input. It takes two input parameters: the first is the current state (to allow using the same hook for several states), the second is the current process iterator object (to have full access to everything; e.g. the next letter from the input tape can be read in). It can output the next transition, i.e. the transition to take next. If it returns `None` the process iterator chooses. Moreover, this function can raise a `StopIteration` exception to stop processing of a finite state machine the input immediately. See also the example below.
  - `color` – (default: `None`) In order to distinguish states, they can be given an arbitrary “color” (an arbitrary object). This is used in `FiniteStateMachine.equivalence_classes()`: states of different colors are never considered to be equivalent. Note that `Automaton.determinisation()` requires that `color` is hashable.
  - `allow_label_None` – (default: `False`) If `True` allows also `None` as label. Note that a state with label `None` is used in `FSMProcessIterator`.

  OUTPUT:

  Returns a state of a finite state machine.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import FSMState
      sage: A = FSMState('state 1', word_out=0, is_initial=True)
      sage: A
      'state 1'
      sage: A.label()
      'state 1'
      sage: B = FSMState('state 2')
      sage: A == B
      False

  We can also define a final output word of a final state which is used if the input of a transducer leads to this state. Such final output words are used in subsequential transducers.

      sage: C = FSMState('state 3', is_final=True, final_word_out='end')
      sage: C.final_word_out
      ['end']

  The final output word can be a single letter, `None` or a list of letters:

      sage: A = FSMState('A')
      sage: A.is_final = True
      sage: A.final_word_out = 2
      sage: A.final_word_out
      [2]
      sage: A.final_word_out = [2, 3]
      sage: A.final_word_out
      [2, 3]

  Only final states can have a final output word which is not `None`:

      sage: B = FSMState('B')
      sage: B.final_word_out is None
      True
      sage: B.final_word_out = 2
      Traceback (most recent call last):
      ...
      ValueError: Only final states can have a final output word,
      but state B is not final.

  Setting the `final_word_out` of a final state to `None` is the same as setting it to `[]` and is also the default for a final state:

      sage: C = FSMState('C', is_final=True)
      sage: C.final_word_out
      []
      sage: C.final_word_out = None
      sage: C.final_word_out
      []
      sage: C.final_word_out = []
      sage: C.final_word_out
      []

  It is not allowed to use `None` as a label:

      sage: from sage.combinat.finite_state_machine import FSMState
      sage: FSMState(None)
      Traceback (most recent call last):
      ...
      ValueError: Label None reserved for a special state,
      choose another label.

  This can be overridden by:

      sage: FSMState(None, allow_label_None=True)
      None

  Note that `Automaton.determinisation()` requires that `color` is hashable:

      sage: A = Automaton([[0, 0, 0]], initial_states=[0])
      sage: A.state(0).color = []
      sage: A.determinisation()
      Traceback (most recent call last):
      ...
      TypeError: unhashable type: 'list'
      sage: A.state(0).color = ()
      sage: A.determinisation()
      Automaton with 1 states

  We can use a hook function of a state to stop processing. This is done by raising a `StopIteration` exception. The following code demonstrates this:

      sage: T = Transducer([(0, 1, 9, 'a'), (1, 2, 9, 'b'),
      ....:                 (2, 3, 9, 'c'), (3, 4, 9, 'd')],
      ....:                initial_states=[0],
      ....:                final_states=[4],
      ....:                input_alphabet=[9])
      sage: def stop(process, state, output):
      ....:     raise StopIteration()
      sage: T.state(3).hook = stop
      sage: T.process([9, 9, 9, 9])
      (False, 3, ['a', 'b', 'c'])

  `copy`()

    Returns a (shallow) copy of the state.

    INPUT:

    Nothing.

    OUTPUT:

    A new state.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A')
        sage: copy(A)
        'A'

  `deepcopy`(memo=None)

    Returns a deep copy of the state.

    INPUT:

    - `memo` – (default: `None`) a dictionary storing already processed elements.

    OUTPUT:

    A new state.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState((1, 3), color=[1, 2],
        ....:              is_final=True, final_word_out=3)
        sage: B = deepcopy(A)
        sage: B
        (1, 3)
        sage: B.label == A.label
        True
        sage: B.label is A.label
        False
        sage: B.color == A.color
        True
        sage: B.color is A.color
        False
        sage: B.is_final == A.is_final
        True
        sage: B.is_final is A.is_final
        True
        sage: B.final_word_out == A.final_word_out
        True
        sage: B.final_word_out is A.final_word_out
        False

  `final_word_out`

    The final output word of a final state which is written if the state is reached as the last state of the input of the finite state machine. For a non-final state, the value is `None`.

    `final_word_out` can be a single letter, a list or `None`, but for a final-state, it is always saved as a list.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A', is_final=True, final_word_out=2)
        sage: A.final_word_out
        [2]
        sage: A.final_word_out = 3
        sage: A.final_word_out
        [3]
        sage: A.final_word_out = [3, 4]
        sage: A.final_word_out
        [3, 4]
        sage: A.final_word_out = None
        sage: A.final_word_out
        []
        sage: B = FSMState('B')
        sage: B.final_word_out is None
        True

    A non-final state cannot have a final output word:

        sage: B.final_word_out = [3, 4]
        Traceback (most recent call last):
        ...
        ValueError: Only final states can have a final
        output word, but state B is not final.

  `fully_equal`(left, right, compare_color=True)

    Checks whether two states are fully equal, i.e., including all attributes except `hook`.

    INPUT:

    - `left` – a state.
    - `right` – a state.
    - `compare_color` – If `True` (default) colors are compared as well, otherwise not.

    OUTPUT:

    `True` or `False`.

    Note that usual comparison by `==` does only compare the labels.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A')
        sage: B = FSMState('A', is_initial=True)
        sage: A.fully_equal(B)
        False
        sage: A == B
        True
        sage: A.is_initial = True; A.color = 'green'
        sage: A.fully_equal(B)
        False
        sage: A.fully_equal(B, compare_color=False)
        True

  `is_final`

    Describes whether the state is final or not.

    `True` if the state is final and `False` otherwise.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A', is_final=True, final_word_out=3)
        sage: A.is_final
        True
        sage: A.is_final = False
        Traceback (most recent call last):
        ...
        ValueError: State A cannot be non-final, because it has a
        final output word. Only final states can have a final output
        word.
        sage: A.final_word_out = None
        sage: A.is_final = False
        sage: A.is_final
        False

  `is_initial`

    Describes whether the state is initial.

    EXAMPLES:

        sage: T = Automaton([(0,0,0)])
        sage: T.initial_states()
        []
        sage: T.state(0).is_initial = True
        sage: T.initial_states()
        [0]

  `label`()

    Returns the label of the state.

    INPUT:

    Nothing.

    OUTPUT:

    The label of the state.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('state')
        sage: A.label()
        'state'

  `relabeled`(label, memo=None)

    Returns a deep copy of the state with a new label.

    INPUT:

    - `label` – the label of new state.
    - `memo` – (default: `None`) a dictionary storing already processed elements.

    OUTPUT:

    A new state.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A')
        sage: A.relabeled('B')
        'B'

class `sage.combinat.finite_state_machine.``FSMTransition`(from_state, to_state, word_in=None, word_out=None, hook=None)

  Bases: `sage.structure.sage_object.SageObject`

  Class for a transition of a finite state machine.

  INPUT:

  - `from_state` – state from which transition starts.
  - `to_state` – state in which transition ends.
  - `word_in` – the input word of the transitions (when the finite state machine is used as automaton)
  - `word_out` – the output word of the transitions (when the finite state machine is used as transducer)

  OUTPUT:

  A transition of a finite state machine.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition
      sage: A = FSMState('A')
      sage: B = FSMState('B')
      sage: S = FSMTransition(A, B, 0, 1)
      sage: T = FSMTransition('A', 'B', 0, 1)
      sage: T == S
      True
      sage: U = FSMTransition('A', 'B', 0)
      sage: U == T
      False

  `copy`()

    Returns a (shallow) copy of the transition.

    INPUT:

    Nothing.

    OUTPUT:

    A new transition.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMTransition
        sage: t = FSMTransition('A', 'B', 0)
        sage: copy(t)
        Transition from 'A' to 'B': 0|-

  `deepcopy`(memo=None)

    Returns a deep copy of the transition.

    INPUT:

    - `memo` – (default: `None`) a dictionary storing already processed elements.

    OUTPUT:

    A new transition.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMTransition
        sage: t = FSMTransition('A', 'B', 0)
        sage: deepcopy(t)
        Transition from 'A' to 'B': 0|-

  `from_state`

    State from which the transition starts. Read-only.

  `to_state`

    State in which the transition ends. Read-only.

  `word_in`

    Input word of the transition. Read-only.

  `word_out`

    Output word of the transition. Read-only.

`sage.combinat.finite_state_machine.``FSMWordSymbol`(word)

  Returns a string of `word`. It may returns the symbol of the empty word `FSMEmptyWordSymbol`.

  INPUT:

  - `word` – the input word.

  OUTPUT:

  A string of `word`.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import FSMWordSymbol
      sage: FSMWordSymbol([0, 1, 1])
      '0,1,1'

class `sage.combinat.finite_state_machine.``FiniteStateMachine`(data=None, initial_states=None, final_states=None, input_alphabet=None, output_alphabet=None, determine_alphabets=None, with_final_word_out=None, store_states_dict=True, on_duplicate_transition=None)

  Bases: `sage.structure.sage_object.SageObject`

  Class for a finite state machine.

  A finite state machine is a finite set of states connected by transitions.

  INPUT:

  - `data` – can be any of the following:

    1. a dictionary of dictionaries (of transitions),
    2. a dictionary of lists (of states or transitions),
    3. a list (of transitions),
    4. a function (transition function),
    5. an other instance of a finite state machine.

  - `initial_states` and `final_states` – the initial and final states of this machine

  - `input_alphabet` and `output_alphabet` – the input and output alphabets of this machine

  - `determine_alphabets` – If `True`, then the function `determine_alphabets()` is called after `data` was read and processed, if `False`, then not. If it is `None`, then it is decided during the construction of the finite state machine whether `determine_alphabets()` should be called.

  - `with_final_word_out` – If given (not `None`), then the function `with_final_word_out()` (more precisely, its inplace pendant `construct_final_word_out()`) is called with input `letters=with_final_word_out` at the end of the creation process.

  - `store_states_dict` – If `True`, then additionally the states are stored in an interal dictionary for speed up.

  - `on_duplicate_transition` – A function which is called when a transition is inserted into `self` which already existed (same `from_state`, same `to_state`, same `word_in`, same `word_out`).

    This function is assumed to take two arguments, the first being the already existing transition, the second being the new transition (as an `FSMTransition`). The function must return the (possibly modified) original transition.

    By default, we have `on_duplicate_transition=None`, which is interpreted as `on_duplicate_transition=duplicate_transition_ignore`, where `duplicate_transition_ignore` is a predefined function ignoring the occurrence. Other such predefined functions are `duplicate_transition_raise_error` and `duplicate_transition_add_input`.

  OUTPUT:

  A finite state machine.

  The object creation of `Automaton` and `Transducer` is the same as the one described here (i.e. just replace the word `FiniteStateMachine` by `Automaton` or `Transducer`).

  Each transition of an automaton has an input label. Automata can, for example, be determinised (see `Automaton.determinisation()`) and minimized (see `Automaton.minimization()`). Each transition of a transducer has an input and an output label. Transducers can, for example, be simplified (see `Transducer.simplification()`).

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition

  See documentation for more examples.

  We illustrate the different input formats:

  1. The input-data can be a dictionary of dictionaries, where

     - the keys of the outer dictionary are state-labels (from-states of transitions),
     - the keys of the inner dictionaries are state-labels (to-states of transitions),
     - the values of the inner dictionaries specify the transition more precisely.

     The easiest is to use a tuple consisting of an input and an output word:

         sage: FiniteStateMachine({'a':{'b':(0, 1), 'c':(1, 1)}})
         Finite state machine with 3 states

     Instead of the tuple anything iterable (e.g. a list) can be used as well.

     If you want to use the arguments of `FSMTransition` directly, you can use a dictionary:

         sage: FiniteStateMachine({'a':{'b':{'word_in':0, 'word_out':1},
         ....:                          'c':{'word_in':1, 'word_out':1}}})
         Finite state machine with 3 states

     In the case you already have instances of `FSMTransition`, it is possible to use them directly:

         sage: FiniteStateMachine({'a':{'b':FSMTransition('a', 'b', 0, 1),
         ....:                          'c':FSMTransition('a', 'c', 1, 1)}})
         Finite state machine with 3 states

  2. The input-data can be a dictionary of lists, where the keys are states or label of states.

     The list-elements can be states:

         sage: a = FSMState('a')
         sage: b = FSMState('b')
         sage: c = FSMState('c')
         sage: FiniteStateMachine({a:[b, c]})
         Finite state machine with 3 states

     Or the list-elements can simply be labels of states:

         sage: FiniteStateMachine({'a':['b', 'c']})
         Finite state machine with 3 states

     The list-elements can also be transitions:

         sage: FiniteStateMachine({'a':[FSMTransition('a', 'b', 0, 1),
         ....:                          FSMTransition('a', 'c', 1, 1)]})
         Finite state machine with 3 states

     Or they can be tuples of a label, an input word and an output word specifying a transition:

         sage: FiniteStateMachine({'a':[('b', 0, 1), ('c', 1, 1)]})
         Finite state machine with 3 states

  3. The input-data can be a list, where its elements specify transitions:

         sage: FiniteStateMachine([FSMTransition('a', 'b', 0, 1),
         ....:                     FSMTransition('a', 'c', 1, 1)])
         Finite state machine with 3 states

     It is possible to skip `FSMTransition` in the example above:

         sage: FiniteStateMachine([('a', 'b', 0, 1), ('a', 'c', 1, 1)])
         Finite state machine with 3 states

     The parameters of the transition are given in tuples. Anyhow, anything iterable (e.g. a list) is possible.

     You can also name the parameters of the transition. For this purpose you take a dictionary:

         sage: FiniteStateMachine([{'from_state':'a', 'to_state':'b',
         ....:                      'word_in':0, 'word_out':1},
         ....:                     {'from_state':'a', 'to_state':'c',
         ....:                      'word_in':1, 'word_out':1}])
         Finite state machine with 3 states

     Other arguments, which `FSMTransition` accepts, can be added, too.

  4. The input-data can also be function acting as transition function:

     This function has two input arguments:

     1. a label of a state (from which the transition starts),
     2. a letter of the (input-)alphabet (as input-label of the transition).

     It returns a tuple with the following entries:

     1. a label of a state (to which state the transition goes),
     2. a letter of or a word over the (output-)alphabet (as output-label of the transition).

     It may also output a list of such tuples if several transitions from the from-state and the input letter exist (this means that the finite state machine is non-deterministic).

     If the transition does not exist, the function should raise a `LookupError` or return an empty list.

     When constructing a finite state machine in this way, some inital states and an input alphabet have to be specified.

         sage: def f(state_from, read):
         ....:     if int(state_from) + read <= 2:
         ....:         state_to = 2*int(state_from)+read
         ....:         write = 0
         ....:     else:
         ....:         state_to = 2*int(state_from) + read - 5
         ....:         write = 1
         ....:     return (str(state_to), write)
         sage: F = FiniteStateMachine(f, input_alphabet=[0, 1],
         ....:                        initial_states=['0'],
         ....:                        final_states=['0'])
         sage: F([1, 0, 1])
         (True, '0', [0, 0, 1])

  5. The input-data can be an other instance of a finite state machine:

         sage: FiniteStateMachine(FiniteStateMachine([]))
         Traceback (most recent call last):
         ...
         NotImplementedError

  The following examples demonstrate the use of `on_duplicate_transition`:

      sage: F = FiniteStateMachine([['a', 'a', 1/2], ['a', 'a', 1/2]])
      sage: F.transitions()
      [Transition from 'a' to 'a': 1/2|-]

      sage: from sage.combinat.finite_state_machine import duplicate_transition_raise_error
      sage: F1 = FiniteStateMachine([['a', 'a', 1/2], ['a', 'a', 1/2]],
      ....:                         on_duplicate_transition=duplicate_transition_raise_error)
      Traceback (most recent call last):
      ...
      ValueError: Attempting to re-insert transition Transition from 'a' to 'a': 1/2|-

  Use `duplicate_transition_add_input` to emulate a Markov chain, the input labels are considered as transition probabilities:

      sage: from sage.combinat.finite_state_machine import duplicate_transition_add_input
      sage: F = FiniteStateMachine([['a', 'a', 1/2], ['a', 'a', 1/2]],
      ....:                        on_duplicate_transition=duplicate_transition_add_input)
      sage: F.transitions()
      [Transition from 'a' to 'a': 1|-]

  Use `with_final_word_out` to construct final output:

      sage: T = Transducer([(0, 1, 0, 0), (1, 0, 0, 0)],
      ....:                initial_states=[0],
      ....:                final_states=[0],
      ....:                with_final_word_out=0)
      sage: for s in T.iter_final_states():
      ....:     print s, s.final_word_out
      0 []
      1 [0]

  TESTS:

      sage: a = FSMState('S_a', 'a')
      sage: b = FSMState('S_b', 'b')
      sage: c = FSMState('S_c', 'c')
      sage: d = FSMState('S_d', 'd')
      sage: FiniteStateMachine({a:[b, c], b:[b, c, d],
      ....:                     c:[a, b], d:[a, c]})
      Finite state machine with 4 states

  We have several constructions which lead to the same finite state machine:

      sage: A = FSMState('A')
      sage: B = FSMState('B')
      sage: C = FSMState('C')
      sage: FSM1 = FiniteStateMachine(
      ....:  {A:{B:{'word_in':0, 'word_out':1},
      ....:   C:{'word_in':1, 'word_out':1}}})
      sage: FSM2 = FiniteStateMachine({A:{B:(0, 1), C:(1, 1)}})
      sage: FSM3 = FiniteStateMachine(
      ....:  {A:{B:FSMTransition(A, B, 0, 1),
      ....:      C:FSMTransition(A, C, 1, 1)}})
      sage: FSM4 = FiniteStateMachine({A:[(B, 0, 1), (C, 1, 1)]})
      sage: FSM5 = FiniteStateMachine(
      ....:  {A:[FSMTransition(A, B, 0, 1), FSMTransition(A, C, 1, 1)]})
      sage: FSM6 = FiniteStateMachine(
      ....:  [{'from_state':A, 'to_state':B, 'word_in':0, 'word_out':1},
      ....:   {'from_state':A, 'to_state':C, 'word_in':1, 'word_out':1}])
      sage: FSM7 = FiniteStateMachine([(A, B, 0, 1), (A, C, 1, 1)])
      sage: FSM8 = FiniteStateMachine(
      ....:  [FSMTransition(A, B, 0, 1), FSMTransition(A, C, 1, 1)])

      sage: FSM1 == FSM2 == FSM3 == FSM4 == FSM5 == FSM6 == FSM7 == FSM8
      True

  It is possible to skip `FSMTransition` in the example above.

  Some more tests for different input-data:

      sage: FiniteStateMachine({'a':{'a':[0, 0], 'b':[1, 1]},
      ....:                     'b':{'b':[1, 0]}})
      Finite state machine with 2 states

      sage: a = FSMState('S_a', 'a')
      sage: b = FSMState('S_b', 'b')
      sage: c = FSMState('S_c', 'c')
      sage: d = FSMState('S_d', 'd')
      sage: t1 = FSMTransition(a, b)
      sage: t2 = FSMTransition(b, c)
      sage: t3 = FSMTransition(b, d)
      sage: t4 = FSMTransition(c, d)
      sage: FiniteStateMachine([t1, t2, t3, t4])
      Finite state machine with 4 states

  `__call__`(*args, **kwargs)

    Warning

    The default output of this method is scheduled to change. This docstring describes the new default behaviour, which can already be achieved by setting `FSMOldProcessOutput` to `False`.

    Calls either method `composition()` or `process()` (with `full_output=False`). See the documentation of these functions for possible parameters.

    By setting `FSMOldProcessOutput` to `False` the new desired output is produced.

    EXAMPLES:

        sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False  # activate new output behavior
        sage: binary_inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:                              initial_states=['A'], final_states=['A'])
        sage: binary_inverter([0, 1, 0, 0, 1, 1])
        [1, 0, 1, 1, 0, 0]

        sage: F = Transducer([('A', 'B', 1, 0), ('B', 'B', 1, 1),
        ....:                 ('B', 'B', 0, 0)],
        ....:                initial_states=['A'], final_states=['B'])
        sage: G = Transducer([(1, 1, 0, 0), (1, 2, 1, 0),
        ....:                 (2, 2, 0, 1), (2, 1, 1, 1)],
        ....:                initial_states=[1], final_states=[1])
        sage: H = G(F)
        sage: H.states()
        [('A', 1), ('B', 1), ('B', 2)]

    When only one successful path is found in a non-deterministic transducer, the result of that path is returned.

        sage: T = Transducer([(0, 1, 0, 1), (0, 2, 0, 2)],
        ....:                initial_states=[0], final_states=[1])
        sage: T.process([0])
        [(True, 1, [1]), (False, 2, [2])]
        sage: T([0])
        [1]

    See also

    `composition()`, `process()`, `Automaton.process()`, `Transducer.process()`.

    TESTS:

        sage: F = FiniteStateMachine([(0, 1, 1, 'a'), (0, 2, 2, 'b')],
        ....:                        initial_states=[0],
        ....:                        final_states=[1])
        sage: A = Automaton([(0, 1, 1), (0, 2, 2)],
        ....:               initial_states=[0],
        ....:               final_states=[1])
        sage: T = Transducer([(0, 1, 1, 'a'), (0, 2, 2, 'b')],
        ....:                initial_states=[0],
        ....:                final_states=[1])
        sage: F([1])
        (True, 1, ['a'])
        sage: A([1])
        True
        sage: T([1])
        ['a']
        sage: F([2])
        (False, 2, ['b'])
        sage: A([2])
        False
        sage: T([2])
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.
        sage: F([3])
        (False, None, None)
        sage: A([3])
        False
        sage: T([3])
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.

        sage: F = FiniteStateMachine([(11, 11, 1, 'a'), (11, 12, 2, 'b'),
        ....:                         (11, 13, 3, 'c'), (11, 14, 4, 'd'),
        ....:                         (12, 13, 3, 'e'), (12, 13, 3, 'f'),
        ....:                         (12, 14, 4, 'g'), (12, 14, 4, 'h'),
        ....:                         (12, 13, 2, 'i'), (12, 14, 2, 'j')],
        ....:                        initial_states=[11],
        ....:                        final_states=[13])
        sage: def f(o):
        ....:     return ''.join(o)
        sage: F([0], format_output=f)
        (False, None, None)
        sage: F([3], format_output=f)
        (True, 13, 'c')
        sage: F([4], format_output=f)
        (False, 14, 'd')
        sage: F([2, 2], format_output=f)
        Traceback (most recent call last):
        ...
        ValueError: Got more than one output, but only allowed to show
        one. Change list_of_outputs option.
        sage: F([2, 2], format_output=f, list_of_outputs=True)
        [(True, 13, 'bi'), (False, 14, 'bj')]
        sage: F([2, 3], format_output=f)
        Traceback (most recent call last):
        ...
        ValueError: Got more than one output, but only allowed to show
        one. Change list_of_outputs option.
        sage: F([2, 3], format_output=f, list_of_outputs=True)
        [(True, 13, 'be'), (True, 13, 'bf')]
        sage: F([2, 4], format_output=f)
        Traceback (most recent call last):
        ...
        ValueError: Got more than one output, but only allowed to show
        one. Change list_of_outputs option.
        sage: F([2, 4], format_output=f, list_of_outputs=True)
        [(False, 14, 'bg'), (False, 14, 'bh')]

        sage: A = Automaton([(11, 11, 1), (11, 12, 2),
        ....:                (11, 13, 3), (11, 14, 4),
        ....:                (12, 13, 3), (12, 14, 4),
        ....:                (12, 32, 3), (12, 42, 4),
        ....:                (12, 13, 2), (12, 14, 2)],
        ....:               initial_states=[11],
        ....:               final_states=[13, 32])
        sage: def f(o):
        ....:     return ''.join(o)
        sage: A([0], format_output=f)
        False
        sage: A([3], format_output=f)
        True
        sage: A([4], format_output=f)
        False
        sage: A([2, 2], format_output=f)
        True
        sage: A([2, 2], format_output=f, list_of_outputs=True)
        [True, False]
        sage: A([2, 3], format_output=f)
        True
        sage: A([2, 3], format_output=f, list_of_outputs=True)
        [True, True]
        sage: A([2, 4], format_output=f)
        False
        sage: A([2, 4], format_output=f, list_of_outputs=True)
        [False, False]

        sage: T = Transducer([(11, 11, 1, 'a'), (11, 12, 2, 'b'),
        ....:                 (11, 13, 3, 'c'), (11, 14, 4, 'd'),
        ....:                 (12, 13, 3, 'e'), (12, 13, 3, 'f'),
        ....:                 (12, 14, 4, 'g'), (12, 14, 4, 'h'),
        ....:                 (12, 13, 2, 'i'), (12, 14, 2, 'j')],
        ....:                initial_states=[11],
        ....:                final_states=[13])
        sage: def f(o):
        ....:     return ''.join(o)
        sage: T([0], format_output=f)
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.
        sage: T([3], format_output=f)
        'c'
        sage: T([4], format_output=f)
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.
        sage: T([2, 2], format_output=f)
        'bi'
        sage: T([2, 2], format_output=f, list_of_outputs=True)
        ['bi', None]
        sage: T([2, 2], format_output=f,
        ....:   list_of_outputs=True, only_accepted=True)
        ['bi']
        sage: T.process([2, 2], format_output=f, list_of_outputs=True)
        [(True, 13, 'bi'), (False, 14, 'bj')]
        sage: T([2, 3], format_output=f)
        Traceback (most recent call last):
        ...
        ValueError: Found more than one accepting path.
        sage: T([2, 3], format_output=f, list_of_outputs=True)
        ['be', 'bf']
        sage: T([2, 4], format_output=f)
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.
        sage: T([2, 4], format_output=f, list_of_outputs=True)
        [None, None]

  `Kleene_closure`()

    TESTS:

        sage: FiniteStateMachine().Kleene_closure()
        Traceback (most recent call last):
        ...
        NotImplementedError

  `accessible_components`()

    Returns a new finite state machine with the accessible states of self and all transitions between those states.

    INPUT:

    Nothing.

    OUTPUT:

    A finite state machine with the accessible states of self and all transitions between those states.

    A state is accessible if there is a directed path from an initial state to the state. If self has no initial states then a copy of the finite state machine self is returned.

    EXAMPLES:

        sage: F = Automaton([(0, 0, 0), (0, 1, 1), (1, 1, 0), (1, 0, 1)],
        ....:               initial_states=[0])
        sage: F.accessible_components()
        Automaton with 2 states

        sage: F = Automaton([(0, 0, 1), (0, 0, 1), (1, 1, 0), (1, 0, 1)],
        ....:               initial_states=[0])
        sage: F.accessible_components()
        Automaton with 1 states

    TESTS:

    Check whether input of length > 1 works:

        sage: F = Automaton([(0, 1, [0, 1]), (0, 2, 0)],
        ....:               initial_states=[0])
        sage: F.accessible_components()
        Automaton with 3 states

  `add_from_transition_function`(function, initial_states=None, explore_existing_states=True)

    Constructs a finite state machine from a transition function.

    INPUT:

    - `function` may return a tuple (new_state, output_word) or a list of such tuples.
    - `initial_states` – If no initial states are given, the already existing initial states of self are taken.
    - If `explore_existing_states` is True (default), then already existing states in self (e.g. already given final states) will also be processed if they are reachable from the initial states.

    OUTPUT:

    Nothing.

    EXAMPLES:

        sage: F = FiniteStateMachine(initial_states=['A'],
        ....:                        input_alphabet=[0, 1])
        sage: def f(state, input):
        ....:     return [('A', input), ('B', 1-input)]
        sage: F.add_from_transition_function(f)
        sage: F.transitions()
        [Transition from 'A' to 'A': 0|0,
        Transition from 'A' to 'B': 0|1,
        Transition from 'A' to 'A': 1|1,
        Transition from 'A' to 'B': 1|0,
        Transition from 'B' to 'A': 0|0,
        Transition from 'B' to 'B': 0|1,
        Transition from 'B' to 'A': 1|1,
        Transition from 'B' to 'B': 1|0]

    Initial states can also be given as a parameter:

        sage: F = FiniteStateMachine(input_alphabet=[0,1])
        sage: def f(state, input):
        ....:     return [('A', input), ('B', 1-input)]
        sage: F.add_from_transition_function(f,initial_states=['A'])
        sage: F.initial_states()
        ['A']

    Already existing states in the finite state machine (the final states in the example below) are also explored:

        sage: F = FiniteStateMachine(initial_states=[0],
        ....:                        final_states=[1],
        ....:                        input_alphabet=[0])
        sage: def transition_function(state, letter):
        ....:     return(1-state, [])
        sage: F.add_from_transition_function(transition_function)
        sage: F.transitions()
        [Transition from 0 to 1: 0|-,
         Transition from 1 to 0: 0|-]

    If `explore_existing_states=False`, however, this behavior is turned off, i.e., already existing states are not explored:

        sage: F = FiniteStateMachine(initial_states=[0],
        ....:                        final_states=[1],
        ....:                        input_alphabet=[0])
        sage: def transition_function(state, letter):
        ....:     return(1-state, [])
        sage: F.add_from_transition_function(transition_function,
        ....:                                explore_existing_states=False)
        sage: F.transitions()
        [Transition from 0 to 1: 0|-]

    TEST:

        sage: F = FiniteStateMachine(initial_states=['A'])
        sage: def f(state, input):
        ....:     return [('A', input), ('B', 1-input)]
        sage: F.add_from_transition_function(f)
        Traceback (most recent call last):
        ...
        ValueError: No input alphabet is given.
        Try calling determine_alphabets().

        sage: def transition(state, where):
        ....:     return (vector([0, 0]), 1)
        sage: Transducer(transition, input_alphabet=[0], initial_states=[0])
        Traceback (most recent call last):
        ...
        TypeError: mutable vectors are unhashable

  `add_state`(state)

    Adds a state to the finite state machine and returns the new state. If the state already exists, that existing state is returned.

    INPUT:

    - `state` is either an instance of `FSMState` or, otherwise, a label of a state.

    OUTPUT:

    The new or existing state.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: F = FiniteStateMachine()
        sage: A = FSMState('A', is_initial=True)
        sage: F.add_state(A)
        'A'

  `add_states`(states)

    Adds several states. See add_state for more information.

    INPUT:

    - `states` – a list of states or iterator over states.

    OUTPUT:

    Nothing.

    EXAMPLES:

        sage: F = FiniteStateMachine()
        sage: F.add_states(['A', 'B'])
        sage: F.states()
        ['A', 'B']

  `add_transition`(*args, **kwargs)

    Adds a transition to the finite state machine and returns the new transition.

    If the transition already exists, the return value of `self.on_duplicate_transition` is returned. See the documentation of `FiniteStateMachine`.

    INPUT:

    The following forms are all accepted:

        sage: from sage.combinat.finite_state_machine import FSMState, FSMTransition
        sage: A = FSMState('A')
        sage: B = FSMState('B')

        sage: FSM = FiniteStateMachine()
        sage: FSM.add_transition(FSMTransition(A, B, 0, 1))
        Transition from 'A' to 'B': 0|1

        sage: FSM = FiniteStateMachine()
        sage: FSM.add_transition(A, B, 0, 1)
        Transition from 'A' to 'B': 0|1

        sage: FSM = FiniteStateMachine()
        sage: FSM.add_transition(A, B, word_in=0, word_out=1)
        Transition from 'A' to 'B': 0|1

        sage: FSM = FiniteStateMachine()
        sage: FSM.add_transition('A', 'B', {'word_in': 0, 'word_out': 1})
        Transition from 'A' to 'B': {'word_in': 0, 'word_out': 1}|-

        sage: FSM = FiniteStateMachine()
        sage: FSM.add_transition(from_state=A, to_state=B,
        ....:                    word_in=0, word_out=1)
        Transition from 'A' to 'B': 0|1

        sage: FSM = FiniteStateMachine()
        sage: FSM.add_transition({'from_state': A, 'to_state': B,
        ....:                    'word_in': 0, 'word_out': 1})
        Transition from 'A' to 'B': 0|1

        sage: FSM = FiniteStateMachine()
        sage: FSM.add_transition((A, B, 0, 1))
        Transition from 'A' to 'B': 0|1

        sage: FSM = FiniteStateMachine()
        sage: FSM.add_transition([A, B, 0, 1])
        Transition from 'A' to 'B': 0|1

    If the states `A` and `B` are not instances of `FSMState`, then it is assumed that they are labels of states.

    OUTPUT:

    The new transition.

  `add_transitions_from_function`(function, labels_as_input=True)

    Adds one or more transitions if `function(state, state)` says that there are some.

    INPUT:

    - `function` – a transition function. Given two states `from_state` and `to_state` (or their labels if `label_as_input` is true), this function shall return a tuple `(word_in, word_out)` to add a transition from `from_state` to `to_state` with input and output labels `word_in` and `word_out`, respectively. If no such addition is to be added, the transition function shall return `None`. The transition function may also return a list of such tuples in order to add multiple transitions between the pair of states.
    - `label_as_input` – (default: `True`)

    OUTPUT:

    Nothing.

    EXAMPLES:

        sage: F = FiniteStateMachine()
        sage: F.add_states(['A', 'B', 'C'])
        sage: def f(state1, state2):
        ....:     if state1 == 'C':
        ....:         return None
        ....:     return (0, 1)
        sage: F.add_transitions_from_function(f)
        sage: len(F.transitions())
        6

    Multiple transitions are also possible:

        sage: F = FiniteStateMachine()
        sage: F.add_states([0, 1])
        sage: def f(state1, state2):
        ....:     if state1 != state2:
        ....:          return [(0, 1), (1, 0)]
        ....:     else:
        ....:          return None
        sage: F.add_transitions_from_function(f)
        sage: F.transitions()
        [Transition from 0 to 1: 0|1,
         Transition from 0 to 1: 1|0,
         Transition from 1 to 0: 0|1,
         Transition from 1 to 0: 1|0]

    TESTS:

        sage: F = FiniteStateMachine()
        sage: F.add_state(0)
        0
        sage: def f(state1, state2):
        ....:     return 1
        sage: F.add_transitions_from_function(f)
        Traceback (most recent call last):
        ...
        ValueError: The callback function for add_transitions_from_function
        is expected to return a pair (word_in, word_out) or a list of such
        pairs. For states 0 and 0 however, it returned 1,
        which is not acceptable.

  `adjacency_matrix`(input=None, entry=None)

    Returns the adjacency matrix of the underlying graph.

    INPUT:

    - `input` – Only transitions with input label `input` are respected.
    - `entry` – The function `entry` takes a transition and the return value is written in the matrix as the entry `(transition.from_state, transition.to_state)`. The default value (`None`) of entry takes the variable `x` to the power of the sum of the output word of the transition.

    OUTPUT:

    A matrix.

    If any label of a state is not an integer, the finite state machine is relabeled at the beginning. If there are more than one transitions between two states, then the different return values of `entry` are added up.

    EXAMPLES:

        sage: B = FiniteStateMachine({0:{0:(0, 0), 'a':(1, 0)},
        ....:                         'a':{2:(0, 0), 3:(1, 0)},
        ....:                         2:{0:(1, 1), 4:(0, 0)},
        ....:                         3:{'a':(0, 1), 2:(1, 1)},
        ....:                         4:{4:(1, 1), 3:(0, 1)}},
        ....:                        initial_states=[0])
        sage: B.adjacency_matrix()
        [1 1 0 0 0]
        [0 0 1 1 0]
        [x 0 0 0 1]
        [0 x x 0 0]
        [0 0 0 x x]

    This is equivalent to:

        sage: matrix(B)
        [1 1 0 0 0]
        [0 0 1 1 0]
        [x 0 0 0 1]
        [0 x x 0 0]
        [0 0 0 x x]

    It is also possible to use other entries in the adjacency matrix:

        sage: B.adjacency_matrix(entry=(lambda transition: 1))
        [1 1 0 0 0]
        [0 0 1 1 0]
        [1 0 0 0 1]
        [0 1 1 0 0]
        [0 0 0 1 1]
        sage: B.adjacency_matrix(1, entry=(lambda transition:
        ....:     exp(I*transition.word_out[0]*var('t'))))
        [      0       1       0       0       0]
        [      0       0       0       1       0]
        [e^(I*t)       0       0       0       0]
        [      0       0 e^(I*t)       0       0]
        [      0       0       0       0 e^(I*t)]
        sage: a = Automaton([(0, 1, 0),
        ....:                (1, 2, 0),
        ....:                (2, 0, 1),
        ....:                (2, 1, 0)],
        ....:               initial_states=[0],
        ....:               final_states=[0])
        sage: a.adjacency_matrix()
        [0 1 0]
        [0 0 1]
        [1 1 0]

  `asymptotic_moments`(variable=n)

    Returns the main terms of expectation and variance of the sum of output labels and its covariance with the sum of input labels.

    INPUT:

    - `variable` – a symbol denoting the length of the input, by default \(n\).

    OUTPUT:

    A dictionary consisting of

    - `expectation` – \(e n + \operatorname{Order}(1)\),
    - `variance` – \(v n + \operatorname{Order}(1)\),
    - `covariance` – \(c n + \operatorname{Order}(1)\)

    for suitable constants \(e\), \(v\) and \(c\).

    Assume that all input and output labels are numbers and that `self` is complete and has only one final component. Assume further that this final component is aperiodic. Furthermore, assume that there is exactly one initial state and that all states are final.

    Denote by \(X_n\) the sum of output labels written by the finite state machine when reading a random input word of length \(n\) over the input alphabet (assuming equidistribution).

    Then the expectation of \(X_n\) is \(en+O(1)\), the variance of \(X_n\) is \(vn+O(1)\) and the covariance of \(X_n\) and the sum of input labels is \(cn+O(1)\), cf. [HKW2014], Theorem 3.9.

    In the case of non-integer input or output labels, performance degrades significantly. For rational input and output labels, consider rescaling to integers. This limitation comes from the fact that determinants over polynomial rings can be computed much more efficiently than over the symbolic ring. In fact, we compute (parts) of a trivariate generating function where the input and output labels are exponents of some indeterminates, see [HKW2014], Theorem 3.9 for details. If those exponents are integers, we can use a polynomial ring.

    EXAMPLES:

    1. A trivial example: write the negative of the input:

           sage: T = Transducer([(0, 0, 0, 0), (0, 0, 1, -1)],
           ....:                initial_states=[0],
           ....:                final_states=[0])
           sage: T([0, 1, 1])
           [0, -1, -1]
           sage: moments = T.asymptotic_moments()
           sage: moments['expectation']
           -1/2*n + Order(1)
           sage: moments['variance']
           1/4*n + Order(1)
           sage: moments['covariance']
           -1/4*n + Order(1)

    2. For the case of the Hamming weight of the non-adjacent-form (NAF) of integers, cf. the Wikipedia article Non-adjacent_form and the example on recognizing NAFs, the following agrees with the results in [HP2007].

       We first use the transducer to convert the standard binary expansion to the NAF given in [HP2007]. We use the parameter `with_final_word_out` such that we do not have to add sufficiently many trailing zeros:

           sage: NAF = Transducer([(0, 0, 0, 0),
           ....:                   (0, '.1', 1, None),
           ....:                   ('.1', 0, 0, [1, 0]),
           ....:                   ('.1', 1, 1, [-1, 0]),
           ....:                   (1, 1, 1, 0),
           ....:                   (1, '.1', 0, None)],
           ....:                  initial_states=[0],
           ....:                  final_states=[0],
           ....:                  with_final_word_out=[0])

       As an example, we compute the NAF of \(27\) by this transducer.

           sage: binary_27 = 27.bits()
           sage: binary_27
           [1, 1, 0, 1, 1]
           sage: NAF_27 = NAF(binary_27)
           sage: NAF_27
           [-1, 0, -1, 0, 0, 1, 0]
           sage: ZZ(NAF_27, base=2)
           27

       Next, we are only interested in the Hamming weight:

           sage: def weight(state, input):
           ....:     if input is None:
           ....:         result = 0
           ....:     else:
           ....:         result = ZZ(input != 0)
           ....:     return (0, result)
           sage: weight_transducer = Transducer(weight,
           ....:                                input_alphabet=[-1, 0, 1],
           ....:                                initial_states=[0],
           ....:                                final_states=[0])

       At the moment, we can not use composition with `NAF`, because it has non-empty final output words:

           sage: NAFweight = weight_transducer.composition(
           ....:     NAF,
           ....:     algorithm='explorative')
           Traceback (most recent call last):
           ...
           NotImplementedError: Explorative composition is not
           implemented for transducers with non-empty final output
           words. Try the direct algorithm instead.

       Thus, we change `NAF`, then compose and again construct the final output words:

           sage: for s in NAF.final_states():
           ....:     s.final_word_out = []
           sage: NAFweight = weight_transducer.composition(
           ....:     NAF,
           ....:     algorithm='explorative').relabeled()
           sage: NAFweight.construct_final_word_out(0)
           sage: sorted(NAFweight.transitions())
           [Transition from 0 to 0: 0|0,
            Transition from 0 to 1: 1|-,
            Transition from 1 to 0: 0|1,0,
            Transition from 1 to 2: 1|1,0,
            Transition from 2 to 1: 0|-,
            Transition from 2 to 2: 1|0]
           sage: NAFweight(binary_27 + [0, 0])
           [1, 0, 1, 0, 0, 1, 0]

       Now, we actually compute the asymptotic moments:

           sage: moments = NAFweight.asymptotic_moments()
           sage: moments['expectation']
           1/3*n + Order(1)
           sage: moments['variance']
           2/27*n + Order(1)
           sage: moments['covariance']
           Order(1)

    3. This is Example 3.16 in [HKW2014], where a transducer with variable output labels is given. There, the aim was to choose the output labels of this very simple transducer such that the input and output sum are asymptotically independent, i.e., the constant \(c\) vanishes.

           sage: var('a_1, a_2, a_3, a_4')
           (a_1, a_2, a_3, a_4)
           sage: T = Transducer([[0, 0, 0, a_1], [0, 1, 1, a_3],
           ....:                 [1, 0, 0, a_4], [1, 1, 1, a_2]],
           ....:                initial_states=[0], final_states=[0, 1])
           sage: moments = T.asymptotic_moments()
           verbose 0 (...) Non-integer output weights lead to
           significant performance degradation.
           sage: moments['expectation']
           1/4*(a_1 + a_2 + a_3 + a_4)*n + Order(1)
           sage: moments['covariance']
           -1/4*(a_1 - a_2)*n + Order(1)

       Therefore, the asymptotic covariance vanishes if and only if \(a_2=a_1\).

    4. This is Example 4.3 in [HKW2014], dealing with the transducer converting the binary expansion of an integer into Gray code (cf. the Wikipedia article Gray_code and the example on Gray code):

           sage: moments = transducers.GrayCode().asymptotic_moments()
           sage: moments['expectation']
           1/2*n + Order(1)
           sage: moments['variance']
           1/4*n + Order(1)
           sage: moments['covariance']
           Order(1)

    5. This is the first part of Example 4.4 in [HKW2014], counting the number of 10 blocks in the standard binary expansion. The least significant digit is at the left-most position:

           sage: block10 = transducers.CountSubblockOccurrences(
           ....:     [1, 0],
           ....:     input_alphabet=[0, 1])
           sage: sorted(block10.transitions())
           [Transition from () to (): 0|0,
            Transition from () to (1,): 1|0,
            Transition from (1,) to (): 0|1,
            Transition from (1,) to (1,): 1|0]
           sage: moments = block10.asymptotic_moments()
           sage: moments['expectation']
           1/4*n + Order(1)
           sage: moments['variance']
           1/16*n + Order(1)
           sage: moments['covariance']
           Order(1)

    6. This is the second part of Example 4.4 in [HKW2014], counting the number of 11 blocks in the standard binary expansion. The least significant digit is at the left-most position:

           sage: block11 = transducers.CountSubblockOccurrences(
           ....:     [1, 1],
           ....:     input_alphabet=[0, 1])
           sage: sorted(block11.transitions())
           [Transition from () to (): 0|0,
            Transition from () to (1,): 1|0,
            Transition from (1,) to (): 0|0,
            Transition from (1,) to (1,): 1|1]
           sage: var('N')
           N
           sage: moments = block11.asymptotic_moments(N)
           sage: moments['expectation']
           1/4*N + Order(1)
           sage: moments['variance']
           5/16*N + Order(1)
           sage: correlation = (moments['covariance'].coefficient(N) /
           ....:                (1/2 * sqrt(moments['variance'].coefficient(N))))
           sage: correlation
           2/5*sqrt(5)

    7. This is Example 4.5 in [HKW2014], counting the number of 01 blocks minus the number of 10 blocks in the standard binary expansion. The least significant digit is at the left-most position:

           sage: block01 = transducers.CountSubblockOccurrences(
           ....:     [0, 1],
           ....:     input_alphabet=[0, 1])
           sage: sage.combinat.finite_state_machine.FSMOldCodeTransducerCartesianProduct = False
           sage: product_01x10 = block01.cartesian_product(block10)
           sage: block_difference = transducers.sub([0, 1])(product_01x10)
           sage: T = block_difference.simplification().relabeled()
           sage: sage.combinat.finite_state_machine.FSMOldCodeTransducerCartesianProduct = True
           sage: T.transitions()
           [Transition from 0 to 1: 0|-1,
            Transition from 0 to 0: 1|0,
            Transition from 1 to 1: 0|0,
            Transition from 1 to 0: 1|1,
            Transition from 2 to 1: 0|0,
            Transition from 2 to 0: 1|0]
           sage: moments = T.asymptotic_moments()
           sage: moments['expectation']
           Order(1)
           sage: moments['variance']
           Order(1)
           sage: moments['covariance']
           Order(1)

    8. The finite state machine must have a unique final component:

           sage: T = Transducer([(0, -1, -1, -1), (0, 1, 1, 1),
           ....:                 (-1, -1, -1, -1), (-1, -1, 1, -1),
           ....:                 (1, 1, -1, 1), (1, 1, 1, 1)],
           ....:                initial_states=[0],
           ....:                final_states=[0, 1, -1])
           sage: T.asymptotic_moments()
           Traceback (most recent call last):
           ...
           NotImplementedError: asymptotic_moments is only
           implemented for finite state machines with one final
           component.

       In this particular example, the first letter of the input decides whether we reach the loop at \(-1\) or the loop at \(1\). In the first case, we have \(X_n = -n\), while we have \(X_n = n\) in the second case. Therefore, the expectation \(E(X_n)\) of \(X_n\) is \(E(X_n) = 0\). We get \((X_n-E(X_n))^2 = n^2\) in all cases, which results in a variance of \(n^2\).

       So this example shows that the variance may be non-linear if there is more than one final component.

    TESTS:

    1. An input alphabet must be given:

           sage: T = Transducer([[0, 0, 0, 0]],
           ....:                initial_states=[0], final_states=[0],
           ....:                determine_alphabets=False)
           sage: T.asymptotic_moments()
           Traceback (most recent call last):
           ...
           ValueError: No input alphabet is given.
           Try calling determine_alphabets().

    2. The finite state machine must have a unique initial state:

           sage: T = Transducer([(0, 0, 0, 0)])
           sage: T.asymptotic_moments()
           Traceback (most recent call last):
           ...
           ValueError: A unique initial state is required.

    3. The finite state machine must be complete:

           sage: T = Transducer([[0, 0, 0, 0]],
           ....:                initial_states=[0], final_states=[0],
           ....:                input_alphabet=[0, 1])
           sage: T.asymptotic_moments()
           Traceback (most recent call last):
           ...
           NotImplementedError: This finite state machine is
           not complete.

    4. The final component of the finite state machine must be aperiodic:

           sage: T = Transducer([(0, 1, 0, 0), (1, 0, 0, 0)],
           ....:                initial_states=[0], final_states=[0, 1])
           sage: T.asymptotic_moments()
           Traceback (most recent call last):
           ...
           NotImplementedError: asymptotic_moments is only
           implemented for finite state machines whose unique final
           component is aperiodic.

    5. Non-integer input or output labels lead to a warning:

           sage: T = Transducer([[0, 0, 0, 0], [0, 0, 1, -1/2]],
           ....:                initial_states=[0], final_states=[0])
           sage: moments = T.asymptotic_moments()
           verbose 0 (...) Non-integer output weights lead to
           significant performance degradation.
           sage: moments['expectation']
           -1/4*n + Order(1)
           sage: moments['variance']
           1/16*n + Order(1)
           sage: moments['covariance']
           -1/8*n + Order(1)

       This warning can be silenced by `set_verbose()`:

           sage: set_verbose(-1, "finite_state_machine.py")
           sage: moments = T.asymptotic_moments()
           sage: moments['expectation']
           -1/4*n + Order(1)
           sage: moments['variance']
           1/16*n + Order(1)
           sage: moments['covariance']
           -1/8*n + Order(1)
           sage: set_verbose(0, "finite_state_machine.py")

    6. Check whether `word_out` of `FSMState` are correctly dealt with:

           sage: from sage.combinat.finite_state_machine import FSMState
           sage: s = FSMState(0, word_out=2,
           ....:              is_initial=True,
           ....:              is_final=True)
           sage: T = Transducer([(s, s, 0, 1)],
           ....:                initial_states=[s], final_states=[s])
           sage: T([0, 0])
           [2, 1, 2, 1, 2]
           sage: T.asymptotic_moments()['expectation']
           3*n + Order(1)

       The same test for non-integer output:

           sage: from sage.combinat.finite_state_machine import FSMState
           sage: s = FSMState(0, word_out=2/3)
           sage: T = Transducer([(s, s, 0, 1/2)],
           ....:                initial_states=[s], final_states=[s])
           sage: T.asymptotic_moments()['expectation']
           verbose 0 (...) Non-integer output weights lead to
           significant performance degradation.
           7/6*n + Order(1)

    7. All states of `self` have to be final:

           sage: T = Transducer([(0, 1, 1, 4)], initial_states=[0])
           sage: T.asymptotic_moments()
           Traceback (most recent call last):
           ...
           ValueError: Not all states are final.

    ALGORITHM:

    See [HKW2014], Theorem 3.9.

    REFERENCES:

    [HKW2014] | (1, 2, 3, 4, 5, 6, 7, 8, 9) Clemens Heuberger, Sara Kropf and Stephan Wagner, Variances and Covariances in the Central Limit Theorem for the Output of a Transducer, Arxiv 1404.3680v2.

    [HP2007] | (1, 2) Clemens Heuberger and Helmut Prodinger, The Hamming Weight of the Non-Adjacent-Form under Various Input Statistics, Periodica Mathematica Hungarica Vol. 55 (1), 2007, pp. 81–96, doi:10.1007/s10998-007-3081-z.

  `composition`(other, algorithm=None, only_accessible_components=True)

    Returns a new transducer which is the composition of `self` and `other`.

    INPUT:

    - `other` – a transducer

    - `algorithm` – can be one of the following

      - `direct` – The composition is calculated directly.

        There can be arbitrarily many initial and final states, but the input and output labels must have length 1.

        WARNING: The output of other is fed into self.

      - `explorative` – An explorative algorithm is used.

        At least the following restrictions apply, but are not checked: - both self and other have exactly one initial state - all input labels of transitions have length exactly 1

        The input alphabet of self has to be specified.

        This is a very limited implementation of composition.

        WARNING: The output of `other` is fed into `self`.

      If algorithm is `None`, then the algorithm is chosen automatically (at the moment always `direct`).

    OUTPUT:

    A new transducer.

    The labels of the new finite state machine are pairs of states of the original finite state machines. The color of a new state is the tuple of colors of the constituent states.

    EXAMPLES:

        sage: F = Transducer([('A', 'B', 1, 0), ('B', 'A', 0, 1)],
        ....:                initial_states=['A', 'B'], final_states=['B'],
        ....:                determine_alphabets=True)
        sage: G = Transducer([(1, 1, 1, 0), (1, 2, 0, 1),
        ....:                 (2, 2, 1, 1), (2, 2, 0, 0)],
        ....:                initial_states=[1], final_states=[2],
        ....:                determine_alphabets=True)
        sage: Hd = F.composition(G, algorithm='direct')
        sage: Hd.initial_states()
        [(1, 'B'), (1, 'A')]
        sage: Hd.transitions()
        [Transition from (1, 'B') to (1, 'A'): 1|1,
         Transition from (1, 'A') to (2, 'B'): 0|0,
         Transition from (2, 'B') to (2, 'A'): 0|1,
         Transition from (2, 'A') to (2, 'B'): 1|0]

        sage: F = Transducer([('A', 'B', 1, [1, 0]), ('B', 'B', 1, 1),
        ....:                 ('B', 'B', 0, 0)],
        ....:                initial_states=['A'], final_states=['B'])
        sage: G = Transducer([(1, 1, 0, 0), (1, 2, 1, 0),
        ....:                 (2, 2, 0, 1), (2, 1, 1, 1)],
        ....:                initial_states=[1], final_states=[1])
        sage: He = G.composition(F, algorithm='explorative')
        sage: He.transitions()
        [Transition from ('A', 1) to ('B', 2): 1|0,1,
         Transition from ('B', 2) to ('B', 2): 0|1,
         Transition from ('B', 2) to ('B', 1): 1|1,
         Transition from ('B', 1) to ('B', 1): 0|0,
         Transition from ('B', 1) to ('B', 2): 1|0]

    Also final output words are considered if `algorithm='direct'` or `None`:

        sage: F = Transducer([('A', 'B', 1, 0), ('B', 'A', 0, 1)],
        ....:                initial_states=['A', 'B'],
        ....:                final_states=['A', 'B'])
        sage: F.state('A').final_word_out = 0
        sage: F.state('B').final_word_out = 1
        sage: G = Transducer([(1, 1, 1, 0), (1, 2, 0, 1),
        ....:                 (2, 2, 1, 1), (2, 2, 0, 0)],
        ....:                initial_states=[1], final_states=[2])
        sage: G.state(2).final_word_out = 0
        sage: Hd = F.composition(G, algorithm='direct')
        sage: Hd.final_states()
        [(2, 'B')]

    Note that `(2, 'A')` is not final, as the final output \(0\) of state \(2\) of \(G\) cannot be processed in state `'A'` of \(F\).

        sage: [s.final_word_out for s in Hd.final_states()]
        [[1, 0]]

    Be aware that after composition, different transitions may share the same output label (same python object):

        sage: F = Transducer([ ('A','B',0,0), ('B','A',0,0)],
        ....:                initial_states=['A'],
        ....:                final_states=['A'])
        sage: F.transitions()[0].word_out is F.transitions()[1].word_out
        False
        sage: G = Transducer([('C','C',0,1)],)
        ....:                initial_states=['C'],
        ....:                final_states=['C'])
        sage: H = G.composition(F)
        sage: H.transitions()[0].word_out is H.transitions()[1].word_out
        True

    In the explorative algorithm, transducers with non-empty final output words are currently not implemented:

        sage: A = transducers.GrayCode()
        sage: B = transducers.abs([0, 1])
        sage: A.composition(B, algorithm='explorative')
        Traceback (most recent call last):
        ...
        NotImplementedError: Explorative composition is not
        implemented for transducers with non-empty final output
        words. Try the direct algorithm instead.

    Similarly, the explorative algorithm cannot handle non-deterministic finite state machines:

        sage: A = Transducer([(0, 0, 0, 0), (0, 1, 0, 0)])
        sage: B = transducers.Identity([0])
        sage: A.composition(B, algorithm='explorative')
        Traceback (most recent call last):
        ...
        NotImplementedError: Explorative composition is currently
        not implemented for non-deterministic transducers.
        sage: B.composition(A, algorithm='explorative')
        Traceback (most recent call last):
        ...
        NotImplementedError: Explorative composition is currently
        not implemented for non-deterministic transducers.

    TESTS:

    Due to the limitations of the two algorithms the following (examples from above, but different algorithm used) does not give a full answer or does not work.

    In the following, `algorithm='explorative'` is inadequate, as `F` has more than one initial state:

        sage: F = Transducer([('A', 'B', 1, 0), ('B', 'A', 0, 1)],
        ....:                initial_states=['A', 'B'], final_states=['B'],
        ....:                determine_alphabets=True)
        sage: G = Transducer([(1, 1, 1, 0), (1, 2, 0, 1),
        ....:                 (2, 2, 1, 1), (2, 2, 0, 0)],
        ....:                initial_states=[1], final_states=[2],
        ....:                determine_alphabets=True)
        sage: He = F.composition(G, algorithm='explorative')
        sage: He.initial_states()
        [(1, 'A')]
        sage: He.transitions()
        [Transition from (1, 'A') to (2, 'B'): 0|0,
         Transition from (2, 'B') to (2, 'A'): 0|1,
         Transition from (2, 'A') to (2, 'B'): 1|0]

    In the following example, `algorithm='direct'` is inappropriate as there are edges with output labels of length greater than 1:

        sage: F = Transducer([('A', 'B', 1, [1, 0]), ('B', 'B', 1, 1),
        ....:                 ('B', 'B', 0, 0)],
        ....:                initial_states=['A'], final_states=['B'])
        sage: G = Transducer([(1, 1, 0, 0), (1, 2, 1, 0),
        ....:                 (2, 2, 0, 1), (2, 1, 1, 1)],
        ....:                initial_states=[1], final_states=[1])
        sage: Hd = G.composition(F, algorithm='direct')

    In the following examples, we compose transducers and automata and check whether the types are correct.

        sage: from sage.combinat.finite_state_machine import (
        ....:     is_Automaton, is_Transducer)
        sage: T = Transducer([(0, 0, 0, 0)], initial_states=[0])
        sage: A = Automaton([(0, 0, 0)], initial_states=[0])
        sage: is_Transducer(T.composition(T, algorithm='direct'))
        True
        sage: is_Transducer(T.composition(T, algorithm='explorative'))
        True
        sage: T.composition(A, algorithm='direct')
        Traceback (most recent call last):
        ...
        TypeError: Composition with automaton is not possible.
        sage: T.composition(A, algorithm='explorative')
        Traceback (most recent call last):
        ...
        TypeError: Composition with automaton is not possible.
        sage: A.composition(A, algorithm='direct')
        Traceback (most recent call last):
        ...
        TypeError: Composition with automaton is not possible.
        sage: A.composition(A, algorithm='explorative')
        Traceback (most recent call last):
        ...
        TypeError: Composition with automaton is not possible.
        sage: is_Automaton(A.composition(T, algorithm='direct'))
        True
        sage: is_Automaton(A.composition(T, algorithm='explorative'))
        True

  `concatenation`(other)

    TESTS:

        sage: F = FiniteStateMachine([('A', 'A')])
        sage: FiniteStateMachine().concatenation(F)
        Traceback (most recent call last):
        ...
        NotImplementedError

  `construct_final_word_out`(letters, allow_non_final=True)

    This is an inplace version of `with_final_word_out()`. See `with_final_word_out()` for documentation and examples.

    TESTS:

        sage: T = Transducer([(0, 1, 0, 0), (1, 0, 0, 1)],
        ....:                initial_states=[0],
        ....:                final_states=[0])
        sage: F = T.with_final_word_out(0)
        sage: T.construct_final_word_out(0)
        sage: T == F  # indirect doctest
        True
        sage: T = Transducer([(0, 1, 0, None)],
        ....:                final_states=[1])
        sage: F = T.with_final_word_out(0)
        sage: F.state(0).final_word_out
        []

  `copy`()

    Returns a (shallow) copy of the finite state machine.

    INPUT:

    Nothing.

    OUTPUT:

    A new finite state machine.

    TESTS:

        sage: copy(FiniteStateMachine())
        Traceback (most recent call last):
        ...
        NotImplementedError

  `deepcopy`(memo=None)

    Returns a deep copy of the finite state machine.

    INPUT:

    - `memo` – (default: `None`) a dictionary storing already processed elements.

    OUTPUT:

    A new finite state machine.

    EXAMPLES:

        sage: F = FiniteStateMachine([('A', 'A', 0, 1), ('A', 'A', 1, 0)])
        sage: deepcopy(F)
        Finite state machine with 1 states

    TESTS:

    Make sure that the links between transitions and states are still intact:

        sage: C = deepcopy(F)
        sage: C.transitions()[0].from_state is C.state('A')
        True
        sage: C.transitions()[0].to_state is C.state('A')
        True

  `default_format_letter`(x, combine_all=False)

    Return a `LatexExpr` built out of the argument `x`.

    INPUT:

    - `x` – a Sage object
    - `combine_all` – boolean (Default: `False`) If `combine_all` is `True` and the input is a tuple, then it does not return a tuple and instead returns a string with all the elements separated by a single space.

    OUTPUT:

    A `LatexExpr` built from `x`

    EXAMPLES:

        sage: latex(Integer(3))  # indirect doctest
        3
        sage: latex(1==0)
        \mathrm{False}
        sage: print latex([x,2])
        \left[x, 2\right]

    Check that trac ticket #11775 is fixed:

        sage: latex((x,2), combine_all=True)
        x 2

  `default_format_transition_label`(word)

    Default formatting of words in transition labels for LaTeX output.

    INPUT:

    `word` – list of letters

    OUTPUT:

    String representation of `word` suitable to be typeset in mathematical mode.

    - For a non-empty word: Concatenation of the letters, piped through `self.format_letter` and separated by blanks.
    - For an empty word: `sage.combinat.finite_state_machine.EmptyWordLaTeX`.

    There is also a variant `format_transition_label_reversed()` writing the words in reversed order.

    EXAMPLES:

    1. Example of a non-empty word:

           sage: T = Transducer()
           sage: print T.default_format_transition_label(
           ....:    ['a', 'alpha', 'a_1', '0', 0, (0, 1)])
           \text{\texttt{a}} \text{\texttt{alpha}}
           \text{\texttt{a{\char`\_}1}} 0 0 \left(0, 1\right)

    2. In the example above, `'a'` and `'alpha'` should perhaps be symbols:

           sage: var('a alpha a_1')
           (a, alpha, a_1)
           sage: print T.default_format_transition_label([a, alpha, a_1])
           a \alpha a_{1}

    3. Example of an empty word:

           sage: print T.default_format_transition_label([])
           \varepsilon

       We can change this by setting `sage.combinat.finite_state_machine.EmptyWordLaTeX`:

           sage: sage.combinat.finite_state_machine.EmptyWordLaTeX = ''
           sage: T.default_format_transition_label([])
           ''

       Finally, we restore the default value:

           sage: sage.combinat.finite_state_machine.EmptyWordLaTeX = r'\varepsilon'

    4. This method is the default value for `FiniteStateMachine.format_transition_label`. That can be changed to be any other function:

           sage: A = Automaton([(0, 1, 0)])
           sage: def custom_format_transition_label(word):
           ....:     return "t"
           sage: A.latex_options(format_transition_label=custom_format_transition_label)
           sage: print latex(A)
           \begin{tikzpicture}[auto, initial text=, >=latex]
           \node[state] (v0) at (3.000000, 0.000000) {$0$};
           \node[state] (v1) at (-3.000000, 0.000000) {$1$};
           \path[->] (v0) edge node[rotate=360.00, anchor=south] {$t$} (v1);
           \end{tikzpicture}

    TEST:

    > Check that #16357 is fixed:

    >     sage: T = Transducer()
    >     sage: T.default_format_transition_label([])
    >     '\\varepsilon'
    >     sage: T.default_format_transition_label(iter([]))
    >     '\\varepsilon'

  `delete_state`(s)

    Deletes a state and all transitions coming or going to this state.

    INPUT:

    - `s` – a label of a state or an `FSMState`.

    OUTPUT:

    Nothing.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMTransition
        sage: t1 = FSMTransition('A', 'B', 0)
        sage: t2 = FSMTransition('B', 'B', 1)
        sage: F = FiniteStateMachine([t1, t2])
        sage: F.delete_state('A')
        sage: F.transitions()
        [Transition from 'B' to 'B': 1|-]

    TESTS:

        sage: F._states_
        ['B']
        sage: F._states_dict_  # This shows that #16024 is fixed.
        {'B': 'B'}

  `delete_transition`(t)

    Deletes a transition by removing it from the list of transitions of the state, where the transition starts.

    INPUT:

    - `t` – a transition.

    OUTPUT:

    Nothing.

    EXAMPLES:

        sage: F = FiniteStateMachine([('A', 'B', 0), ('B', 'A', 1)])
        sage: F.delete_transition(('A', 'B', 0))
        sage: F.transitions()
        [Transition from 'B' to 'A': 1|-]

  `determine_alphabets`(reset=True)

    Determines the input and output alphabet according to the transitions in self.

    INPUT:

    - `reset` – If reset is `True`, then the existing input and output alphabets are erased, otherwise new letters are appended to the existing alphabets.

    OUTPUT:

    Nothing.

    After this operation the input alphabet and the output alphabet of self are a list of letters.

    Todo

    At the moment, the letters of the alphabets need to be hashable.

    EXAMPLES:

        sage: T = Transducer([(1, 1, 1, 0), (1, 2, 2, 1),
        ....:                 (2, 2, 1, 1), (2, 2, 0, 0)],
        ....:                final_states=[1],
        ....:                determine_alphabets=False)
        sage: T.state(1).final_word_out = [1, 4]
        sage: (T.input_alphabet, T.output_alphabet)
        (None, None)
        sage: T.determine_alphabets()
        sage: (T.input_alphabet, T.output_alphabet)
        ([0, 1, 2], [0, 1, 4])

  `digraph`(edge_labels='words_in_out')

    Returns the graph of the finite state machine with labeled vertices and labeled edges.

    INPUT:

    - `edge_label`: (default: `'words_in_out'`) can be
        - `'words_in_out'` (labels will be strings `'i|o'`)
        - a function with which takes as input a transition and outputs (returns) the label

    OUTPUT:

    A graph.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A')
        sage: T = Transducer()
        sage: T.graph()
        Digraph on 0 vertices
        sage: T.add_state(A)
        'A'
        sage: T.graph()
        Digraph on 1 vertex
        sage: T.add_transition(('A', 'A', 0, 1))
        Transition from 'A' to 'A': 0|1
        sage: T.graph()
        Looped digraph on 1 vertex

  `disjoint_union`(other)

    TESTS:

        sage: F = FiniteStateMachine([('A', 'A')])
        sage: FiniteStateMachine().disjoint_union(F)
        Traceback (most recent call last):
        ...
        NotImplementedError

  `empty_copy`(memo=None, new_class=None)

    Returns an empty deep copy of the finite state machine, i.e., `input_alphabet`, `output_alphabet`, `on_duplicate_transition` are preserved, but states and transitions are not.

    INPUT:

    - `memo` – a dictionary storing already processed elements.
    - `new_class` – a class for the copy. By default (`None`), the class of `self` is used.

    OUTPUT:

    A new finite state machine.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import duplicate_transition_raise_error
        sage: F = FiniteStateMachine([('A', 'A', 0, 2), ('A', 'A', 1, 3)],
        ....:                        input_alphabet=[0, 1],
        ....:                        output_alphabet=[2, 3],
        ....:                        on_duplicate_transition=duplicate_transition_raise_error)
        sage: FE = F.empty_copy(); FE
        Finite state machine with 0 states
        sage: FE.input_alphabet
        [0, 1]
        sage: FE.output_alphabet
        [2, 3]
        sage: FE.on_duplicate_transition == duplicate_transition_raise_error
        True

    TESTS:

        sage: T = Transducer()
        sage: type(T.empty_copy())
        <class 'sage.combinat.finite_state_machine.Transducer'>
        sage: type(T.empty_copy(new_class=Automaton))
        <class 'sage.combinat.finite_state_machine.Automaton'>

  `epsilon_successors`(state)

    Returns the dictionary with states reachable from `state` without reading anything from an input tape as keys. The values are lists of outputs.

    INPUT:

    - `state` – the state whose epsilon successors should be determined.

    OUTPUT:

    A dictionary mapping states to a list of output words.

    The states in the output are the epsilon successors of `state`. Each word of the list of output words is a word written when taking a path from `state` to the corresponding state.

    EXAMPLES:

        sage: T = Transducer([(0, 1, None, 'a'), (1, 2, None, 'b')])
        sage: T.epsilon_successors(0)
        {1: [['a']], 2: [['a', 'b']]}
        sage: T.epsilon_successors(1)
        {2: [['b']]}
        sage: T.epsilon_successors(2)
        {}

    If there is a cycle with only epsilon transitions, then this cycle is only processed once and there is no infinite loop:

        sage: S = Transducer([(0, 1, None, 'a'), (1, 0, None, 'b')])
        sage: S.epsilon_successors(0)
        {0: [['a', 'b']], 1: [['a']]}
        sage: S.epsilon_successors(1)
        {0: [['b']], 1: [['b', 'a']]}

  `equivalence_classes`()

    Returns a list of equivalence classes of states.

    INPUT:

    Nothing.

    OUTPUT:

    A list of equivalence classes of states.

    Two states \(a\) and \(b\) are equivalent if and only if there is a bijection \(\varphi\) between paths starting at \(a\) and paths starting at \(b\) with the following properties: Let \(p_a\) be a path from \(a\) to \(a'\) and \(p_b\) a path from \(b\) to \(b'\) such that \(\varphi(p_a)=p_b\), then

    - \(p_a.\mathit{word}_\mathit{in}=p_b.\mathit{word}_\mathit{in}\),
    - \(p_a.\mathit{word}_\mathit{out}=p_b.\mathit{word}_\mathit{out}\),
    - \(a'\) and \(b'\) have the same output label, and
    - \(a'\) and \(b'\) are both final or both non-final and have the same final output word.

    The function `equivalence_classes()` returns a list of the equivalence classes to this equivalence relation.

    This is one step of Moore’s minimization algorithm.

    See also

    `minimization()`

    EXAMPLES:

        sage: fsm = FiniteStateMachine([("A", "B", 0, 1), ("A", "B", 1, 0),
        ....:                           ("B", "C", 0, 0), ("B", "C", 1, 1),
        ....:                           ("C", "D", 0, 1), ("C", "D", 1, 0),
        ....:                           ("D", "A", 0, 0), ("D", "A", 1, 1)])
        sage: sorted(fsm.equivalence_classes())
        [['A', 'C'], ['B', 'D']]
        sage: fsm.state("A").is_final = True
        sage: sorted(fsm.equivalence_classes())
        [['A'], ['B'], ['C'], ['D']]
        sage: fsm.state("C").is_final = True
        sage: sorted(fsm.equivalence_classes())
        [['A', 'C'], ['B', 'D']]
        sage: fsm.state("A").final_word_out = 1
        sage: sorted(fsm.equivalence_classes())
        [['A'], ['B'], ['C'], ['D']]
        sage: fsm.state("C").final_word_out = 1
        sage: sorted(fsm.equivalence_classes())
        [['A', 'C'], ['B', 'D']]

  `final_components`()

    Returns the final components of a finite state machine as finite state machines.

    INPUT:

    Nothing.

    OUTPUT:

    A list of finite state machines, each representing a final component of `self`.

    A final component of a transducer `T` is a strongly connected component `C` such that there are no transitions of `T` leaving `C`.

    The final components are the only parts of a transducer which influence the main terms of the asympotic behaviour of the sum of output labels of a transducer, see [HKP2014] and [HKW2014].

    EXAMPLES:

        sage: T = Transducer([['A', 'B', 0, 0], ['B', 'C', 0, 1],
        ....:                 ['C', 'B', 0, 1], ['A', 'D', 1, 0],
        ....:                 ['D', 'D', 0, 0], ['D', 'B', 1, 0],
        ....:                 ['A', 'E', 2, 0], ['E', 'E', 0, 0]])
        sage: FC = T.final_components()
        sage: sorted(FC[0].transitions())
        [Transition from 'B' to 'C': 0|1,
         Transition from 'C' to 'B': 0|1]
        sage: FC[1].transitions()
        [Transition from 'E' to 'E': 0|0]

    Another example (cycle of length 2):

        sage: T = Automaton([[0, 1, 0], [1, 0, 0]])
        sage: len(T.final_components()) == 1
        True
        sage: T.final_components()[0].transitions()
        [Transition from 0 to 1: 0|-,
         Transition from 1 to 0: 0|-]

    REFERENCES:

    [HKP2014] | Clemens Heuberger, Sara Kropf, and Helmut Prodinger, Asymptotic analysis of the sum of the output of transducer, in preparation.

  `final_states`()

    Returns a list of all final states.

    INPUT:

    Nothing.

    OUTPUT:

    A list of all final states.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A', is_final=True)
        sage: B = FSMState('B', is_initial=True)
        sage: C = FSMState('C', is_final=True)
        sage: F = FiniteStateMachine([(A, B), (A, C)])
        sage: F.final_states()
        ['A', 'C']

  `format_letter`(x, combine_all=False)

    Return a `LatexExpr` built out of the argument `x`.

    INPUT:

    - `x` – a Sage object
    - `combine_all` – boolean (Default: `False`) If `combine_all` is `True` and the input is a tuple, then it does not return a tuple and instead returns a string with all the elements separated by a single space.

    OUTPUT:

    A `LatexExpr` built from `x`

    EXAMPLES:

        sage: latex(Integer(3))  # indirect doctest
        3
        sage: latex(1==0)
        \mathrm{False}
        sage: print latex([x,2])
        \left[x, 2\right]

    Check that trac ticket #11775 is fixed:

        sage: latex((x,2), combine_all=True)
        x 2

  `format_letter_negative`(letter)

    Format negative numbers as overlined numbers, everything else by standard LaTeX formatting.

    INPUT:

    `letter` – anything.

    OUTPUT:

    Overlined absolute value if letter is a negative integer, `latex(letter)` otherwise.

    EXAMPLES:

        sage: A = Automaton([(0, 0, -1)])
        sage: map(A.format_letter_negative, [-1, 0, 1, 'a', None])
         ['\\overline{1}', 0, 1, \text{\texttt{a}}, \mbox{\rm None}]
        sage: A.latex_options(format_letter=A.format_letter_negative)
        sage: print(latex(A))
        \begin{tikzpicture}[auto, initial text=, >=latex]
        \node[state] (v0) at (3.000000, 0.000000) {$0$};
        \path[->] (v0) edge[loop above] node {$\overline{1}$} ();
        \end{tikzpicture}

  `format_transition_label`(word)

    Default formatting of words in transition labels for LaTeX output.

    INPUT:

    `word` – list of letters

    OUTPUT:

    String representation of `word` suitable to be typeset in mathematical mode.

    - For a non-empty word: Concatenation of the letters, piped through `self.format_letter` and separated by blanks.
    - For an empty word: `sage.combinat.finite_state_machine.EmptyWordLaTeX`.

    There is also a variant `format_transition_label_reversed()` writing the words in reversed order.

    EXAMPLES:

    1. Example of a non-empty word:

           sage: T = Transducer()
           sage: print T.default_format_transition_label(
           ....:    ['a', 'alpha', 'a_1', '0', 0, (0, 1)])
           \text{\texttt{a}} \text{\texttt{alpha}}
           \text{\texttt{a{\char`\_}1}} 0 0 \left(0, 1\right)

    2. In the example above, `'a'` and `'alpha'` should perhaps be symbols:

           sage: var('a alpha a_1')
           (a, alpha, a_1)
           sage: print T.default_format_transition_label([a, alpha, a_1])
           a \alpha a_{1}

    3. Example of an empty word:

           sage: print T.default_format_transition_label([])
           \varepsilon

       We can change this by setting `sage.combinat.finite_state_machine.EmptyWordLaTeX`:

           sage: sage.combinat.finite_state_machine.EmptyWordLaTeX = ''
           sage: T.default_format_transition_label([])
           ''

       Finally, we restore the default value:

           sage: sage.combinat.finite_state_machine.EmptyWordLaTeX = r'\varepsilon'

    4. This method is the default value for `FiniteStateMachine.format_transition_label`. That can be changed to be any other function:

           sage: A = Automaton([(0, 1, 0)])
           sage: def custom_format_transition_label(word):
           ....:     return "t"
           sage: A.latex_options(format_transition_label=custom_format_transition_label)
           sage: print latex(A)
           \begin{tikzpicture}[auto, initial text=, >=latex]
           \node[state] (v0) at (3.000000, 0.000000) {$0$};
           \node[state] (v1) at (-3.000000, 0.000000) {$1$};
           \path[->] (v0) edge node[rotate=360.00, anchor=south] {$t$} (v1);
           \end{tikzpicture}

    TEST:

    > Check that #16357 is fixed:

    >     sage: T = Transducer()
    >     sage: T.default_format_transition_label([])
    >     '\\varepsilon'
    >     sage: T.default_format_transition_label(iter([]))
    >     '\\varepsilon'

  `format_transition_label_reversed`(word)

    Format words in transition labels in reversed order.

    INPUT:

    `word` – list of letters.

    OUTPUT:

    String representation of `word` suitable to be typeset in mathematical mode, letters are written in reversed order.

    This is the reversed version of `default_format_transition_label()`.

    In digit expansions, digits are frequently processed from the least significant to the most significant position, but it is customary to write the least significant digit at the right-most position. Therefore, the labels have to be reversed.

    EXAMPLE:

        sage: T = Transducer([(0, 0, 0, [1, 2, 3])])
        sage: T.format_transition_label_reversed([1, 2, 3])
        '3 2 1'
        sage: T.latex_options(format_transition_label=T.format_transition_label_reversed)
        sage: print latex(T)
        \begin{tikzpicture}[auto, initial text=, >=latex]
        \node[state] (v0) at (3.000000, 0.000000) {$0$};
        \path[->] (v0) edge[loop above] node {$0\mid 3 2 1$} ();
        \end{tikzpicture}

    TEST:

    > Check that #16357 is fixed:

    >     sage: T = Transducer()
    >     sage: T.format_transition_label_reversed([])
    >     '\\varepsilon'

  `graph`(edge_labels='words_in_out')

    Returns the graph of the finite state machine with labeled vertices and labeled edges.

    INPUT:

    - `edge_label`: (default: `'words_in_out'`) can be
        - `'words_in_out'` (labels will be strings `'i|o'`)
        - a function with which takes as input a transition and outputs (returns) the label

    OUTPUT:

    A graph.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A')
        sage: T = Transducer()
        sage: T.graph()
        Digraph on 0 vertices
        sage: T.add_state(A)
        'A'
        sage: T.graph()
        Digraph on 1 vertex
        sage: T.add_transition(('A', 'A', 0, 1))
        Transition from 'A' to 'A': 0|1
        sage: T.graph()
        Looped digraph on 1 vertex

  `has_final_state`(state)

    Returns whether `state` is one of the final states of the finite state machine.

    INPUT:

    - `state` can be a `FSMState` or a label.

    OUTPUT:

    True or False.

    EXAMPLES:

        sage: FiniteStateMachine(final_states=['A']).has_final_state('A')
        True

  `has_final_states`()

    Returns whether the finite state machine has a final state.

    INPUT:

    Nothing.

    OUTPUT:

    True or False.

    EXAMPLES:

        sage: FiniteStateMachine().has_final_states()
        False

  `has_initial_state`(state)

    Returns whether `state` is one of the initial states of the finite state machine.

    INPUT:

    - `state` can be a `FSMState` or a label.

    OUTPUT:

    True or False.

    EXAMPLES:

        sage: F = FiniteStateMachine([('A', 'A')], initial_states=['A'])
        sage: F.has_initial_state('A')
        True

  `has_initial_states`()

    Returns whether the finite state machine has an initial state.

    INPUT:

    Nothing.

    OUTPUT:

    True or False.

    EXAMPLES:

        sage: FiniteStateMachine().has_initial_states()
        False

  `has_state`(state)

    Returns whether `state` is one of the states of the finite state machine.

    INPUT:

    - `state` can be a `FSMState` or a label of a state.

    OUTPUT:

    True or False.

    EXAMPLES:

        sage: FiniteStateMachine().has_state('A')
        False

  `has_transition`(transition)

    Returns whether `transition` is one of the transitions of the finite state machine.

    INPUT:

    - `transition` has to be a `FSMTransition`.

    OUTPUT:

    True or False.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMTransition
        sage: t = FSMTransition('A', 'A', 0, 1)
        sage: FiniteStateMachine().has_transition(t)
        False
        sage: FiniteStateMachine().has_transition(('A', 'A', 0, 1))
        Traceback (most recent call last):
        ...
        TypeError: Transition is not an instance of FSMTransition.

  `induced_sub_finite_state_machine`(states)

    Returns a sub-finite-state-machine of the finite state machine induced by the given states.

    INPUT:

    - `states` – a list (or an iterator) of states (either labels or instances of `FSMState`) of the sub-finite-state-machine.

    OUTPUT:

    A new finite state machine. It consists (of deep copies) of the given states and (deep copies) of all transitions of `self` between these states.

    EXAMPLE:

        sage: FSM = FiniteStateMachine([(0, 1, 0), (0, 2, 0),
        ....:                           (1, 2, 0), (2, 0, 0)])
        sage: sub_FSM = FSM.induced_sub_finite_state_machine([0, 1])
        sage: sub_FSM.states()
        [0, 1]
        sage: sub_FSM.transitions()
        [Transition from 0 to 1: 0|-]
        sage: FSM.induced_sub_finite_state_machine([3])
        Traceback (most recent call last):
        ...
        ValueError: 3 is not a state of this finite state machine.

    TESTS:

    Make sure that the links between transitions and states are still intact:

        sage: sub_FSM.transitions()[0].from_state is sub_FSM.state(0)
        True

  `initial_states`()

    Returns a list of all initial states.

    INPUT:

    Nothing.

    OUTPUT:

    A list of all initial states.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A', is_initial=True)
        sage: B = FSMState('B')
        sage: F = FiniteStateMachine([(A, B, 1, 0)])
        sage: F.initial_states()
        ['A']

  `input_projection`()

    Returns an automaton where the output of each transition of self is deleted.

    INPUT:

    Nothing

    OUTPUT:

    An automaton.

    EXAMPLES:

        sage: F = FiniteStateMachine([('A', 'B', 0, 1), ('A', 'A', 1, 1),
        ....:                         ('B', 'B', 1, 0)])
        sage: G = F.input_projection()
        sage: G.transitions()
        [Transition from 'A' to 'B': 0|-,
         Transition from 'A' to 'A': 1|-,
         Transition from 'B' to 'B': 1|-]

  `intersection`(other)

    TESTS:

        sage: FiniteStateMachine().intersection(FiniteStateMachine())
        Traceback (most recent call last):
        ...
        NotImplementedError

  `is_Markov_chain`(is_zero=None)

    Checks whether `self` is a Markov chain where the transition probabilities are modeled as input labels.

    INPUT:

    - `is_zero` – by default (`is_zero=None`), checking for zero is simply done by `is_zero()`. This parameter can be used to provide a more sophisticated check for zero, e.g. in the case of symbolic probabilities, see the examples below.

    OUTPUT:

    `True` or `False`.

    `on_duplicate_transition` must be `duplicate_transition_add_input()` and the sum of the input weights of the transitions leaving a state must add up to 1.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import duplicate_transition_add_input
        sage: F = Transducer([[0, 0, 1/4, 0], [0, 1, 3/4, 1],
        ....:                 [1, 0, 1/2, 0], [1, 1, 1/2, 1]],
        ....:                on_duplicate_transition=duplicate_transition_add_input)
        sage: F.is_Markov_chain()
        True

    `on_duplicate_transition` must be `duplicate_transition_add_input()`:

        sage: F = Transducer([[0, 0, 1/4, 0], [0, 1, 3/4, 1],
        ....:                 [1, 0, 1/2, 0], [1, 1, 1/2, 1]])
        sage: F.is_Markov_chain()
        False

    Sum of input labels of the transitions leaving states must be 1:

        sage: F = Transducer([[0, 0, 1/4, 0], [0, 1, 3/4, 1],
        ....:                 [1, 0, 1/2, 0]],
        ....:                on_duplicate_transition=duplicate_transition_add_input)
        sage: F.is_Markov_chain()
        False

    If the probabilities are variables in the symbolic ring, `assume()` will do the trick:

        sage: var('p q')
        (p, q)
        sage: F = Transducer([(0, 0, p, 1), (0, 0, q, 0)],
        ....:                on_duplicate_transition=duplicate_transition_add_input)
        sage: assume(p + q == 1)
        sage: (p + q - 1).is_zero()
        True
        sage: F.is_Markov_chain()
        True
        sage: forget()
        sage: del(p, q)

    If the probabilities are variables in some polynomial ring, the parameter `is_zero` can be used:

        sage: R.<p, q> = PolynomialRing(QQ)
        sage: def is_zero_polynomial(polynomial):
        ....:     return polynomial in (p + q - 1)*R
        sage: F = Transducer([(0, 0, p, 1), (0, 0, q, 0)],
        ....:                on_duplicate_transition=duplicate_transition_add_input)
        sage: F.is_Markov_chain()
        False
        sage: F.is_Markov_chain(is_zero_polynomial)
        True

  `is_complete`()

    Returns whether the finite state machine is complete.

    INPUT:

    Nothing.

    OUTPUT:

    `True` or `False`.

    A finite state machine is considered to be complete if each transition has an input label of length one and for each pair \((q, a)\) where \(q\) is a state and \(a\) is an element of the input alphabet, there is exactly one transition from \(q\) with input label \(a\).

    EXAMPLES:

        sage: fsm = FiniteStateMachine([(0, 0, 0, 0),
        ....:                           (0, 1, 1, 1),
        ....:                           (1, 1, 0, 0)],
        ....:                          determine_alphabets=False)
        sage: fsm.is_complete()
        Traceback (most recent call last):
        ...
        ValueError: No input alphabet is given. Try calling determine_alphabets().
        sage: fsm.input_alphabet = [0, 1]
        sage: fsm.is_complete()
        False
        sage: fsm.add_transition((1, 1, 1, 1))
        Transition from 1 to 1: 1|1
        sage: fsm.is_complete()
        True
        sage: fsm.add_transition((0, 0, 1, 0))
        Transition from 0 to 0: 1|0
        sage: fsm.is_complete()
        False

  `is_connected`()

    TESTS:

        sage: FiniteStateMachine().is_connected()
        Traceback (most recent call last):
        ...
        NotImplementedError

  `is_deterministic`()

    Returns whether the finite finite state machine is deterministic.

    INPUT:

    Nothing.

    OUTPUT:

    `True` or `False`.

    A finite state machine is considered to be deterministic if each transition has input label of length one and for each pair \((q,a)\) where \(q\) is a state and \(a\) is an element of the input alphabet, there is at most one transition from \(q\) with input label \(a\).

    TESTS:

        sage: fsm = FiniteStateMachine()
        sage: fsm.add_transition(('A', 'B', 0, []))
        Transition from 'A' to 'B': 0|-
        sage: fsm.is_deterministic()
        True
        sage: fsm.add_transition(('A', 'C', 0, []))
        Transition from 'A' to 'C': 0|-
        sage: fsm.is_deterministic()
        False
        sage: fsm.add_transition(('A', 'B', [0,1], []))
        Transition from 'A' to 'B': 0,1|-
        sage: fsm.is_deterministic()
        False

  `is_monochromatic`()

    Checks whether the colors of all states are equal.

    INPUT:

    Nothing.

    OUTPUT:

    `True` or `False`.

    EXAMPLES:

        sage: G = transducers.GrayCode()
        sage: [s.color for s in G.iter_states()]
        [None, None, None]
        sage: G.is_monochromatic()
        True
        sage: G.state(1).color = 'blue'
        sage: G.is_monochromatic()
        False

  `iter_final_states`()

    Returns an iterator of the final states.

    INPUT:

    Nothing.

    OUTPUT:

    An iterator over all initial states.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A', is_final=True)
        sage: B = FSMState('B', is_initial=True)
        sage: C = FSMState('C', is_final=True)
        sage: F = FiniteStateMachine([(A, B), (A, C)])
        sage: [s.label() for s in F.iter_final_states()]
        ['A', 'C']

  `iter_initial_states`()

    Returns an iterator of the initial states.

    INPUT:

    Nothing.

    OUTPUT:

    An iterator over all initial states.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A', is_initial=True)
        sage: B = FSMState('B')
        sage: F = FiniteStateMachine([(A, B, 1, 0)])
        sage: [s.label() for s in F.iter_initial_states()]
        ['A']

  `iter_process`(input_tape=None, initial_state=None, **kwargs)

    This function returns an instance of `FSMProcessIterator`. See `process()` (which runs this iterator until the end) for more information.

    EXAMPLES:

        sage: inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:     initial_states=['A'], final_states=['A'])
        sage: it = inverter.iter_process(input_tape=[0, 1, 1])
        sage: for current in it:
        ....:     print current
        process (1 branch)
        + at state 'A'
        +-- tape at 1, [[1]]
        process (1 branch)
        + at state 'A'
        +-- tape at 2, [[1, 0]]
        process (1 branch)
        + at state 'A'
        +-- tape at 3, [[1, 0, 0]]
        process (0 branches)
        sage: it.result()
        [(True, 'A', [1, 0, 0])]

    See also

    `FiniteStateMachine.process()`, `Automaton.process()`, `Transducer.process()`, `__call__()`, `FSMProcessIterator`.

  `iter_states`()

    Returns an iterator of the states.

    INPUT:

    Nothing.

    OUTPUT:

    An iterator of the states of the finite state machine.

    EXAMPLES:

        sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)])
        sage: [s.label() for s in FSM.iter_states()]
        ['1', '2']

  `iter_transitions`(from_state=None)

    Returns an iterator of all transitions.

    INPUT:

    - `from_state` – (default: `None`) If `from_state` is given, then a list of transitions starting there is given.

    OUTPUT:

    An iterator of all transitions.

    EXAMPLES:

        sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)])
        sage: [(t.from_state.label(), t.to_state.label())
        ....:     for t in FSM.iter_transitions('1')]
        [('1', '2')]
        sage: [(t.from_state.label(), t.to_state.label())
        ....:     for t in FSM.iter_transitions('2')]
        [('2', '2')]
        sage: [(t.from_state.label(), t.to_state.label())
        ....:     for t in FSM.iter_transitions()]
        [('1', '2'), ('2', '2')]

  `latex_options`(coordinates=None, format_state_label=None, format_letter=None, format_transition_label=None, loop_where=None, initial_where=None, accepting_style=None, accepting_distance=None, accepting_where=None, accepting_show_empty=None)

    Set options for LaTeX output via `latex()` and therefore `view()`.

    INPUT:

    - `coordinates` – a dictionary or a function mapping labels of states to pairs interpreted as coordinates. If no coordinates are given, states a placed equidistantly on a circle of radius \(3\). See also `set_coordinates()`.
    - `format_state_label` – a function mapping labels of states to a string suitable for typesetting in LaTeX’s mathematics mode. If not given, `latex()` is used.
    - `format_letter` – a function mapping letters of the input and output alphabets to a string suitable for typesetting in LaTeX’s mathematics mode. If not given, `default_format_transition_label()` uses `latex()`.
    - `format_transition_label` – a function mapping words over the input and output alphabets to a string suitable for typesetting in LaTeX’s mathematics mode. If not given, `default_format_transition_label()` is used.
    - `loop_where` – a dictionary or a function mapping labels of initial states to one of `'above'`, `'left'`, `'below'`, `'right'`. If not given, `'above'` is used.
    - `initial_where` – a dictionary or a function mapping labels of initial states to one of `'above'`, `'left'`, `'below'`, `'right'`. If not given, TikZ’ default (currently `'left'`) is used.
    - `accepting_style` – one of `'accepting by double'` and `'accepting by arrow'`. If not given, `'accepting by double'` is used unless there are non-empty final output words.
    - `accepting_distance` – a string giving a LaTeX length used for the length of the arrow leading from a final state. If not given, TikZ’ default (currently `'3ex'`) is used unless there are non-empty final output words, in which case `'7ex'` is used.
    - `accepting_where` – a dictionary or a function mapping labels of final states to one of `'above'`, `'left'`, `'below'`, `'right'`. If not given, TikZ’ default (currently `'right'`) is used. If the final state has a final output word, it is also possible to give an angle in degrees.
    - `accepting_show_empty` – if `True` the arrow of an empty final output word is labeled as well. Note that this implicitly implies `accepting_style='accepting by arrow'`. If not given, the default `False` is used.

    OUTPUT:

    Nothing.

    As TikZ (cf. the Wikipedia article PGF/TikZ) is used to typeset the graphics, the syntax is oriented on TikZ’ syntax.

    This is a convenience function collecting all options for LaTeX output. All of its functionality can also be achieved by directly setting the attributes

    - `coordinates`, `format_label`, `loop_where`, `initial_where`, and `accepting_where` of `FSMState` (here, `format_label` is a callable without arguments, everything else is a specific value);
    - `format_label` of `FSMTransition` (`format_label` is a callable without arguments);
    - `format_state_label`, `format_letter`, `format_transition_label`, `accepting_style`, `accepting_distance`, and `accepting_show_empty` of `FiniteStateMachine`.

    This function, however, also (somewhat) checks its input and serves to collect documentation on all these options.

    The function can be called several times, only those arguments which are not `None` are taken into account. By the same means, it can be combined with directly setting some attributes as outlined above.

    EXAMPLES:

    See also the section on LaTeX output in the introductory examples of this module.

        sage: T = Transducer(initial_states=[4],
        ....:     final_states=[0, 3])
        sage: for j in srange(4):
        ....:     T.add_transition(4, j, 0, [0, j])
        ....:     T.add_transition(j, 4, 0, [0, -j])
        ....:     T.add_transition(j, j, 0, 0)
        Transition from 4 to 0: 0|0,0
        Transition from 0 to 4: 0|0,0
        Transition from 0 to 0: 0|0
        Transition from 4 to 1: 0|0,1
        Transition from 1 to 4: 0|0,-1
        Transition from 1 to 1: 0|0
        Transition from 4 to 2: 0|0,2
        Transition from 2 to 4: 0|0,-2
        Transition from 2 to 2: 0|0
        Transition from 4 to 3: 0|0,3
        Transition from 3 to 4: 0|0,-3
        Transition from 3 to 3: 0|0
        sage: T.add_transition(4, 4, 0, 0)
        Transition from 4 to 4: 0|0
        sage: T.state(3).final_word_out = [0, 0]
        sage: T.latex_options(
        ....:     coordinates={4: (0, 0),
        ....:                  0: (-6, 3),
        ....:                  1: (-2, 3),
        ....:                  2: (2, 3),
        ....:                  3: (6, 3)},
        ....:     format_state_label=lambda x: r'\mathbf{%s}' % x,
        ....:     format_letter=lambda x: r'w_{%s}' % x,
        ....:     format_transition_label=lambda x:
        ....:         r"{\scriptstyle %s}" % T.default_format_transition_label(x),
        ....:     loop_where={4: 'below', 0: 'left', 1: 'above',
        ....:                 2: 'right', 3:'below'},
        ....:     initial_where=lambda x: 'above',
        ....:     accepting_style='accepting by double',
        ....:     accepting_distance='10ex',
        ....:     accepting_where={0: 'left', 3: 45}
        ....:     )
        sage: T.state(4).format_label=lambda: r'\mathcal{I}'
        sage: latex(T)
        \begin{tikzpicture}[auto, initial text=, >=latex]
        \node[state, initial, initial where=above] (v0) at (0.000000, 0.000000) {$\mathcal{I}$};
        \node[state, accepting, accepting where=left] (v1) at (-6.000000, 3.000000) {$\mathbf{0}$};
        \node[state, accepting, accepting where=45] (v2) at (6.000000, 3.000000) {$\mathbf{3}$};
        \path[->] (v2.45.00) edge node[rotate=45.00, anchor=south] {$$ \mid {\scriptstyle w_{0} w_{0}}$} ++(45.00:10ex);
        \node[state] (v3) at (-2.000000, 3.000000) {$\mathbf{1}$};
        \node[state] (v4) at (2.000000, 3.000000) {$\mathbf{2}$};
        \path[->] (v1) edge[loop left] node[rotate=90, anchor=south] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0}}$} ();
        \path[->] (v1.-21.57) edge node[rotate=-26.57, anchor=south] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0} w_{0}}$} (v0.148.43);
        \path[->] (v3) edge[loop above] node {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0}}$} ();
        \path[->] (v3.-51.31) edge node[rotate=-56.31, anchor=south] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0} w_{-1}}$} (v0.118.69);
        \path[->] (v4) edge[loop right] node[rotate=90, anchor=north] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0}}$} ();
        \path[->] (v4.-118.69) edge node[rotate=56.31, anchor=north] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0} w_{-2}}$} (v0.51.31);
        \path[->] (v2) edge[loop below] node {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0}}$} ();
        \path[->] (v2.-148.43) edge node[rotate=26.57, anchor=north] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0} w_{-3}}$} (v0.21.57);
        \path[->] (v0.158.43) edge node[rotate=333.43, anchor=north] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0} w_{0}}$} (v1.328.43);
        \path[->] (v0.128.69) edge node[rotate=303.69, anchor=north] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0} w_{1}}$} (v3.298.69);
        \path[->] (v0.61.31) edge node[rotate=56.31, anchor=south] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0} w_{2}}$} (v4.231.31);
        \path[->] (v0.31.57) edge node[rotate=26.57, anchor=south] {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0} w_{3}}$} (v2.201.57);
        \path[->] (v0) edge[loop below] node {${\scriptstyle w_{0}}\mid {\scriptstyle w_{0}}$} ();
        \end{tikzpicture}
        sage: view(T) # not tested

    To actually see this, use the live documentation in the Sage notebook and execute the cells.

    By changing some of the options, we get the following output:

        sage: T.latex_options(
        ....:     format_transition_label=T.default_format_transition_label,
        ....:     accepting_style='accepting by arrow',
        ....:     accepting_show_empty=True
        ....:     )
        sage: latex(T)
        \begin{tikzpicture}[auto, initial text=, >=latex, accepting text=, accepting/.style=accepting by arrow, accepting distance=10ex]
        \node[state, initial, initial where=above] (v0) at (0.000000, 0.000000) {$\mathcal{I}$};
        \node[state] (v1) at (-6.000000, 3.000000) {$\mathbf{0}$};
        \path[->] (v1.180.00) edge node[rotate=360.00, anchor=south] {$$ \mid \varepsilon$} ++(180.00:10ex);
        \node[state] (v2) at (6.000000, 3.000000) {$\mathbf{3}$};
        \path[->] (v2.45.00) edge node[rotate=45.00, anchor=south] {$$ \mid w_{0} w_{0}$} ++(45.00:10ex);
        \node[state] (v3) at (-2.000000, 3.000000) {$\mathbf{1}$};
        \node[state] (v4) at (2.000000, 3.000000) {$\mathbf{2}$};
        \path[->] (v1) edge[loop left] node[rotate=90, anchor=south] {$w_{0}\mid w_{0}$} ();
        \path[->] (v1.-21.57) edge node[rotate=-26.57, anchor=south] {$w_{0}\mid w_{0} w_{0}$} (v0.148.43);
        \path[->] (v3) edge[loop above] node {$w_{0}\mid w_{0}$} ();
        \path[->] (v3.-51.31) edge node[rotate=-56.31, anchor=south] {$w_{0}\mid w_{0} w_{-1}$} (v0.118.69);
        \path[->] (v4) edge[loop right] node[rotate=90, anchor=north] {$w_{0}\mid w_{0}$} ();
        \path[->] (v4.-118.69) edge node[rotate=56.31, anchor=north] {$w_{0}\mid w_{0} w_{-2}$} (v0.51.31);
        \path[->] (v2) edge[loop below] node {$w_{0}\mid w_{0}$} ();
        \path[->] (v2.-148.43) edge node[rotate=26.57, anchor=north] {$w_{0}\mid w_{0} w_{-3}$} (v0.21.57);
        \path[->] (v0.158.43) edge node[rotate=333.43, anchor=north] {$w_{0}\mid w_{0} w_{0}$} (v1.328.43);
        \path[->] (v0.128.69) edge node[rotate=303.69, anchor=north] {$w_{0}\mid w_{0} w_{1}$} (v3.298.69);
        \path[->] (v0.61.31) edge node[rotate=56.31, anchor=south] {$w_{0}\mid w_{0} w_{2}$} (v4.231.31);
        \path[->] (v0.31.57) edge node[rotate=26.57, anchor=south] {$w_{0}\mid w_{0} w_{3}$} (v2.201.57);
        \path[->] (v0) edge[loop below] node {$w_{0}\mid w_{0}$} ();
        \end{tikzpicture}
        sage: view(T) # not tested

    TESTS:

        sage: T.latex_options(format_state_label='Nothing')
        Traceback (most recent call last):
        ...
        TypeError: format_state_label must be callable.
        sage: T.latex_options(format_letter='')
        Traceback (most recent call last):
        ...
        TypeError: format_letter must be callable.
        sage: T.latex_options(format_transition_label='')
        Traceback (most recent call last):
        ...
        TypeError: format_transition_label must be callable.
        sage: T.latex_options(loop_where=37)
        Traceback (most recent call last):
        ...
        TypeError: loop_where must be a callable or a
        dictionary.
        sage: T.latex_options(loop_where=lambda x: 'top')
        Traceback (most recent call last):
        ...
        ValueError: loop_where for 4 must be in ['below',
        'right', 'above', 'left'].
        sage: T.latex_options(initial_where=90)
        Traceback (most recent call last):
        ...
        TypeError: initial_where must be a callable or a
        dictionary.
        sage: T.latex_options(initial_where=lambda x: 'top')
        Traceback (most recent call last):
        ...
        ValueError: initial_where for 4 must be in ['below',
        'right', 'above', 'left'].
        sage: T.latex_options(accepting_style='fancy')
        Traceback (most recent call last):
        ...
        ValueError: accepting_style must be in ['accepting by
        double', 'accepting by arrow'].
        sage: T.latex_options(accepting_where=90)
        Traceback (most recent call last):
        ...
        TypeError: accepting_where must be a callable or a
        dictionary.
        sage: T.latex_options(accepting_where=lambda x: 'top')
        Traceback (most recent call last):
        ...
        ValueError: accepting_where for 0 must be in ['below',
        'right', 'above', 'left'].
        sage: T.latex_options(accepting_where={0: 'above', 3: 'top'})
        Traceback (most recent call last):
        ...
        ValueError: accepting_where for 3 must be a real number or
        be in ['below', 'right', 'above', 'left'].

  `markov_chain_simplification`()

    Consider `self` as Markov chain with probabilities as input labels and simplify it.

    INPUT:

    Nothing.

    OUTPUT:

    Simplified version of `self`.

    EXAMPLE:

        sage: from sage.combinat.finite_state_machine import duplicate_transition_add_input
        sage: T = Transducer([[1, 2, 1/4, 0], [1, -2, 1/4, 0], [1, -2, 1/2, 0],
        ....:                 [2, 2, 1/4, 1], [2, -2, 1/4, 1], [-2, -2, 1/4, 1],
        ....:                 [-2, 2, 1/4, 1], [2, 3, 1/2, 2], [-2, 3, 1/2, 2]],
        ....:                initial_states=[1],
        ....:                final_states=[3],
        ....:                on_duplicate_transition=duplicate_transition_add_input)
        sage: T1 = T.markov_chain_simplification()
        sage: sorted(T1.transitions())
        [Transition from ((1,),) to ((2, -2),): 1|0,
         Transition from ((2, -2),) to ((2, -2),): 1/2|1,
         Transition from ((2, -2),) to ((3,),): 1/2|2]

  `merged_transitions`()

    Merges transitions which have the same `from_state`, `to_state` and `word_out` while adding their `word_in`.

    INPUT:

    Nothing.

    OUTPUT:

    A finite state machine with merged transitions. If no mergers occur, return `self`.

    EXAMPLE:

        sage: from sage.combinat.finite_state_machine import duplicate_transition_add_input
        sage: T = Transducer([[1, 2, 1/4, 1], [1, -2, 1/4, 1], [1, -2, 1/2, 1],
        ....:                 [2, 2, 1/4, 1], [2, -2, 1/4, 1], [-2, -2, 1/4, 1],
        ....:                 [-2, 2, 1/4, 1], [2, 3, 1/2, 1], [-2, 3, 1/2, 1]],
        ....:                on_duplicate_transition=duplicate_transition_add_input)
        sage: T1 = T.merged_transitions()
        sage: T1 is T
        False
        sage: sorted(T1.transitions())
        [Transition from -2 to -2: 1/4|1,
         Transition from -2 to 2: 1/4|1,
         Transition from -2 to 3: 1/2|1,
         Transition from 1 to 2: 1/4|1,
         Transition from 1 to -2: 3/4|1,
         Transition from 2 to -2: 1/4|1,
         Transition from 2 to 2: 1/4|1,
         Transition from 2 to 3: 1/2|1]

    Applying the function again does not change the result:

        sage: T2 = T1.merged_transitions()
        sage: T2 is T1
        True

  `on_duplicate_transition`(old_transition, new_transition)

    Which function to call when a duplicate transition is inserted. See the documentation of the parameter `on_duplicate_transition` of the class `FiniteStateMachine` for details.

  `output_projection`()

    Returns a automaton where the input of each transition of self is deleted and the new input is the original output.

    INPUT:

    Nothing

    OUTPUT:

    An automaton.

    EXAMPLES:

        sage: F = FiniteStateMachine([('A', 'B', 0, 1), ('A', 'A', 1, 1),
        ....:                         ('B', 'B', 1, 0)])
        sage: G = F.output_projection()
        sage: G.transitions()
        [Transition from 'A' to 'B': 1|-,
         Transition from 'A' to 'A': 1|-,
         Transition from 'B' to 'B': 0|-]

    Final output words are also considered correctly:

        sage: H = Transducer([('A', 'B', 0, 1), ('A', 'A', 1, 1),
        ....:                 ('B', 'B', 1, 0), ('A', ('final', 0), 0, 0)],
        ....:                final_states=['A', 'B'])
        sage: H.state('B').final_word_out = 2
        sage: J = H.output_projection()
        sage: J.states()
        ['A', 'B', ('final', 0), ('final', 1)]
        sage: J.transitions()
        [Transition from 'A' to 'B': 1|-,
         Transition from 'A' to 'A': 1|-,
         Transition from 'A' to ('final', 0): 0|-,
         Transition from 'B' to 'B': 0|-,
         Transition from 'B' to ('final', 1): 2|-]
        sage: J.final_states()
        ['A', ('final', 1)]

  `plot`()

    Plots a graph of the finite state machine with labeled vertices and labeled edges.

    INPUT:

    Nothing.

    OUTPUT:

    A plot of the graph of the finite state machine.

    TESTS:

        sage: FiniteStateMachine([('A', 'A', 0)]).plot()
        Graphics object consisting of 3 graphics primitives

  `predecessors`(state, valid_input=None)

    Lists all predecessors of a state.

    INPUT:

    - `state` – the state from which the predecessors should be listed.
    - `valid_input` – If `valid_input` is a list, then we only consider transitions whose input labels are contained in `valid_input`. `state` has to be a `FSMState` (not a label of a state). If input labels of length larger than \(1\) are used, then `valid_input` has to be a list of lists.

    OUTPUT:

    A list of states.

    EXAMPLES:

        sage: A = Transducer([('I', 'A', 'a', 'b'), ('I', 'B', 'b', 'c'),
        ....:                 ('I', 'C', 'c', 'a'), ('A', 'F', 'b', 'a'),
        ....:                 ('B', 'F', ['c', 'b'], 'b'), ('C', 'F', 'a', 'c')],
        ....:                initial_states=['I'], final_states=['F'])
        sage: A.predecessors(A.state('A'))
        ['A', 'I']
        sage: A.predecessors(A.state('F'), valid_input=['b', 'a'])
        ['F', 'C', 'A', 'I']
        sage: A.predecessors(A.state('F'), valid_input=[['c', 'b'], 'a'])
        ['F', 'C', 'B']

  `prepone_output`()

    For all paths, shift the output of the path from one transition to the earliest possible preceeding transition of the path.

    INPUT:

    Nothing.

    OUTPUT:

    Nothing.

    Apply the following to each state \(s\) (except initial states) of the finite state machine as often as possible:

    If the letter \(a\) is a prefix of the output label of all transitions from \(s\) (including the final output of \(s\)), then remove it from all these labels and append it to all output labels of all transitions leading to \(s\).

    We assume that the states have no output labels, but final outputs are allowed.

    EXAMPLES:

        sage: A = Transducer([('A', 'B', 1, 1),
        ....:                 ('B', 'B', 0, 0),
        ....:                 ('B', 'C', 1, 0)],
        ....:                initial_states=['A'],
        ....:                final_states=['C'])
        sage: A.prepone_output()
        sage: A.transitions()
        [Transition from 'A' to 'B': 1|1,0,
         Transition from 'B' to 'B': 0|0,
         Transition from 'B' to 'C': 1|-]

        sage: B = Transducer([('A', 'B', 0, 1),
        ....:                 ('B', 'C', 1, [1, 1]),
        ....:                 ('B', 'C', 0, 1)],
        ....:                initial_states=['A'],
        ....:                final_states=['C'])
        sage: B.prepone_output()
        sage: B.transitions()
        [Transition from 'A' to 'B': 0|1,1,
         Transition from 'B' to 'C': 1|1,
         Transition from 'B' to 'C': 0|-]

    If initial states are not labeled as such, unexpected results may be obtained:

        sage: C = Transducer([(0,1,0,0)])
        sage: C.prepone_output()
        verbose 0 (...: finite_state_machine.py, prepone_output)
        All transitions leaving state 0 have an output label with
        prefix 0.  However, there is no inbound transition and it
        is not an initial state. This routine (possibly called by
        simplification) therefore erased this prefix from all
        outbound transitions.
        sage: C.transitions()
        [Transition from 0 to 1: 0|-]

    Also the final output of final states can be changed:

        sage: T = Transducer([('A', 'B', 0, 1),
        ....:                 ('B', 'C', 1, [1, 1]),
        ....:                 ('B', 'C', 0, 1)],
        ....:                initial_states=['A'],
        ....:                final_states=['B'])
        sage: T.state('B').final_word_out = [1]
        sage: T.prepone_output()
        sage: T.transitions()
        [Transition from 'A' to 'B': 0|1,1,
         Transition from 'B' to 'C': 1|1,
         Transition from 'B' to 'C': 0|-]
        sage: T.state('B').final_word_out
        []

        sage: S = Transducer([('A', 'B', 0, 1),
        ....:                 ('B', 'C', 1, [1, 1]),
        ....:                 ('B', 'C', 0, 1)],
        ....:                initial_states=['A'],
        ....:                final_states=['B'])
        sage: S.state('B').final_word_out = [0]
        sage: S.prepone_output()
        sage: S.transitions()
        [Transition from 'A' to 'B': 0|1,
         Transition from 'B' to 'C': 1|1,1,
         Transition from 'B' to 'C': 0|1]
        sage: S.state('B').final_word_out
        [0]

    Output labels do not have to be hashable:

        sage: C = Transducer([(0, 1, 0, []),
        ....:                 (1, 0, 0, [vector([0, 0]), 0]),
        ....:                 (1, 1, 1, [vector([0, 0]), 1]),
        ....:                 (0, 0, 1, 0)],
        ....:                 determine_alphabets=False,
        ....:                 initial_states=[0])
        sage: C.prepone_output()
        sage: sorted(C.transitions())
        [Transition from 0 to 1: 0|(0, 0),
         Transition from 0 to 0: 1|0,
         Transition from 1 to 0: 0|0,
         Transition from 1 to 1: 1|1,(0, 0)]

  `process`(*args, **kwargs)

    Returns whether the finite state machine accepts the input, the state where the computation stops and which output is generated.

    INPUT:

    - `input_tape` – the input tape can be a list or an iterable with entries from the input alphabet. If we are working with a multi-tape machine (see parameter `use_multitape_input` and notes below), then the tape is a list or tuple of tracks, each of which can be a list or an iterable with entries from the input alphabet.
    - `initial_state` or `initial_states` – the initial state(s) in which the machine starts. Either specify a single one with `initial_state` or a list of them with `initial_states`. If both are given, `initial_state` will be appended to `initial_states`. If neither is specified, the initial states of the finite state machine are taken.
    - `list_of_outputs` – (default: `None`) a boolean or `None`. If `True`, then the outputs are given in list form (even if we have no or only one single output). If `False`, then the result is never a list (an exception is raised if the result cannot be returned). If `list_of_outputs=None`, the method determines automatically what to do (e.g. if a non-deterministic machine returns more than one path, then the output is returned in list form).
    - `only_accepted` – (default: `False`) a boolean. If set, then the first argument in the output is guaranteed to be `True` (if the output is a list, then the first argument of each element will be `True`).
    - `always_include_output` – if set (not by default), always include the output. This is inconsequential for a `FiniteStateMachine`, but can be used in derived classes where the output is suppressed by default, cf. `Automaton.process()`.
    - `format_output` – a function that translates the written output (which is in form of a list) to something more readable. By default (`None`) identity is used here.
    - `check_epsilon_transitions` – (default: `True`) a boolean. If `False`, then epsilon transitions are not taken into consideration during process.
    - `write_final_word_out` – (default: `True`) a boolean specifying whether the final output words should be written or not.
    - `use_multitape_input` – (default: `False`) a boolean. If `True`, then the multi-tape mode of the process iterator is activated. See also the notes below for multi-tape machines.

    OUTPUT:

    A triple (or a list of triples, cf. parameter `list_of_outputs`), where

    - the first entry is `True` if the input string is accepted,
    - the second gives the reached state after processing the input tape (This is a state with label `None` if the input could not be processed, i.e., if at one point no transition to go on could be found.), and
    - the third gives a list of the output labels written during processing (in the case the finite state machine runs as transducer).

    Note that in the case the finite state machine is not deterministic, all possible paths are taken into account.

    This function uses an iterator which, in its simplest form, goes from one state to another in each step. To decide which way to go, it uses the input words of the outgoing transitions and compares them to the input tape. More precisely, in each step, the iterator takes an outgoing transition of the current state, whose input label equals the input letter of the tape. The output label of the transition, if present, is written on the output tape.

    If the choice of the outgoing transition is not unique (i.e., we have a non-deterministic finite state machine), all possibilites are followed. This is done by splitting the process into several branches, one for each of the possible outgoing transitions.

    The process (iteration) stops if all branches are finished, i.e., for no branch, there is any transition whose input word coincides with the processed input tape. This can simply happen when the entire tape was read.

    Also see `__call__()` for a version of `process()` with shortened output.

    Internally this function creates and works with an instance of `FSMProcessIterator`. This iterator can also be obtained with `iter_process()`.

    If working with multi-tape finite state machines, all input words of transitions are words of \(k\)-tuples of letters. Moreover, the input tape has to consist of \(k\) tracks, i.e., be a list or tuple of \(k\) iterators, one for each track.

    Warning

    Working with multi-tape finite state machines is still experimental and can lead to wrong outputs.

    EXAMPLES:

        sage: binary_inverter = FiniteStateMachine({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:                                      initial_states=['A'], final_states=['A'])
        sage: binary_inverter.process([0, 1, 0, 0, 1, 1])
        (True, 'A', [1, 0, 1, 1, 0, 0])

    Alternatively, we can invoke this function by:

        sage: binary_inverter([0, 1, 0, 0, 1, 1])
        (True, 'A', [1, 0, 1, 1, 0, 0])

    Below we construct a finite state machine which tests if an input is a non-adjacent form, i.e., no two neighboring letters are both nonzero (see also the example on non-adjacent forms in the documentation of the module `finite_state_machine`):

        sage: NAF = FiniteStateMachine(
        ....:     {'_': [('_', 0), (1, 1)], 1: [('_', 0)]},
        ....:     initial_states=['_'], final_states=['_', 1])
        sage: [NAF.process(w)[0] for w in [[0], [0, 1], [1, 1], [0, 1, 0, 1],
        ....:                           [0, 1, 1, 1, 0], [1, 0, 0, 1, 1]]]
        [True, True, False, True, False, False]

    Working only with the first component (i.e., returning whether accepted or not) usually corresponds to using the more specialized class `Automaton`.

    Non-deterministic finite state machines can be handeled as well.

        sage: T = Transducer([(0, 1, 0, 0), (0, 2, 0, 0)],
        ....:     initial_states=[0])
        sage: T.process([0])
        [(False, 1, [0]), (False, 2, [0])]

    Here is another non-deterministic finite state machine. Note that we use `format_output` (see `FSMProcessIterator`) to convert the written outputs (all characters) to strings.

        sage: T = Transducer([(0, 1, [0, 0], 'a'), (0, 2, [0, 0, 1], 'b'),
        ....:                 (0, 1, 1, 'c'), (1, 0, [], 'd'),
        ....:                 (1, 1, 1, 'e')],
        ....:                initial_states=[0], final_states=[0, 1])
        sage: T.process([0], format_output=lambda o: ''.join(o))
        (False, None, None)
        sage: T.process([0, 0], format_output=lambda o: ''.join(o))
        [(True, 0, 'ad'), (True, 1, 'a')]
        sage: T.process([1], format_output=lambda o: ''.join(o))
        [(True, 0, 'cd'), (True, 1, 'c')]
        sage: T.process([1, 1], format_output=lambda o: ''.join(o))
        [(True, 0, 'cdcd'), (True, 0, 'ced'),
         (True, 1, 'cdc'), (True, 1, 'ce')]
        sage: T.process([0, 0, 1], format_output=lambda o: ''.join(o))
        [(True, 0, 'adcd'), (True, 0, 'aed'),
         (True, 1, 'adc'), (True, 1, 'ae'), (False, 2, 'b')]
        sage: T.process([0, 0, 1], format_output=lambda o: ''.join(o),
        ....:           only_accepted=True)
        [(True, 0, 'adcd'), (True, 0, 'aed'),
         (True, 1, 'adc'), (True, 1, 'ae')]

    A simple example of a multi-tape finite state machine is the following: It writes the length of the first tape many letters `a` and then the length of the second tape many letters `b`:

        sage: M = FiniteStateMachine([(0, 0, (1, None), 'a'),
        ....:                         (0, 1, [], []),
        ....:                         (1, 1, (None, 1), 'b')],
        ....:                        initial_states=[0],
        ....:                        final_states=[1])
        sage: M.process(([1, 1], [1]), use_multitape_input=True)
        (True, 1, ['a', 'a', 'b'])

    See also

    `Automaton.process()`, `Transducer.process()`, `iter_process()`, `__call__()`, `FSMProcessIterator`.

    TESTS:

        sage: T = Transducer([(0, 1, [0, 0], 0), (0, 2, [0, 0, 1], 0),
        ....:                 (0, 1, 1, 2), (1, 0, [], 1), (1, 1, 1, 3)],
        ....:     initial_states=[0], final_states=[0, 1])
        sage: T.process([0])
        (False, None, None)
        sage: T.process([0, 0])
        [(True, 0, [0, 1]), (True, 1, [0])]
        sage: T.process([1])
        [(True, 0, [2, 1]), (True, 1, [2])]
        sage: T.process([1, 1])
        [(True, 0, [2, 1, 2, 1]), (True, 0, [2, 3, 1]),
         (True, 1, [2, 1, 2]), (True, 1, [2, 3])]

        sage: F = FiniteStateMachine([(0, 0, 0, 0)],
        ....:                        initial_states=[0])
        sage: F.process([0], only_accepted=True)
        []
        sage: F.process([0], only_accepted=True, list_of_outputs=False)
        Traceback (most recent call last):
        ...
        ValueError: No accepting output was found but according to the
        given options, an accepting output should be returned. Change
        only_accepted and/or list_of_outputs options.
        sage: F.process([0], only_accepted=True, list_of_outputs=True)
        []
        sage: F.process([0], only_accepted=False)
        (False, 0, [0])
        sage: F.process([0], only_accepted=False, list_of_outputs=False)
        (False, 0, [0])
        sage: F.process([0], only_accepted=False, list_of_outputs=True)
        [(False, 0, [0])]
        sage: F.process([1], only_accepted=True)
        []
        sage: F.process([1], only_accepted=True, list_of_outputs=False)
        Traceback (most recent call last):
        ...
        ValueError: No accepting output was found but according to the
        given options, an accepting output should be returned. Change
        only_accepted and/or list_of_outputs options.
        sage: F.process([1], only_accepted=True, list_of_outputs=True)
        []
        sage: F.process([1], only_accepted=False)
        (False, None, None)
        sage: F.process([1], only_accepted=False, list_of_outputs=False)
        (False, None, None)
        sage: F.process([1], only_accepted=False, list_of_outputs=True)
        []

        sage: F = FiniteStateMachine([(0, 1, 1, 'a'), (0, 2, 2, 'b')],
        ....:                        initial_states=[0],
        ....:                        final_states=[1])
        sage: A = Automaton([(0, 1, 1), (0, 2, 2)],
        ....:               initial_states=[0],
        ....:               final_states=[1])
        sage: T = Transducer([(0, 1, 1, 'a'), (0, 2, 2, 'b')],
        ....:                initial_states=[0],
        ....:                final_states=[1])
        sage: F.process([1])
        (True, 1, ['a'])
        sage: A.process([1])
        (True, 1)
        sage: T.process([1])
        (True, 1, ['a'])
        sage: F.process([2])
        (False, 2, ['b'])
        sage: A.process([2])
        (False, 2)
        sage: T.process([2])
        (False, 2, ['b'])
        sage: F.process([3])
        (False, None, None)
        sage: A.process([3])
        (False, None)
        sage: T.process([3])
        (False, None, None)

  `product_FiniteStateMachine`(other, function, new_input_alphabet=None, only_accessible_components=True, final_function=None, new_class=None)

    Returns a new finite state machine whose states are \(d\)-tuples of states of the original finite state machines.

    INPUT:

    - `other` – a finite state machine (for \(d=2\)) or a list (or iterable) of \(d-1\) finite state machines.
    - `function` has to accept \(d\) transitions from \(A_j\) to \(B_j\) for \(j\in\{1, \ldots, d\}\) and returns a pair `(word_in, word_out)` which is the label of the transition \(A=(A_1, \ldots, A_d)\) to \(B=(B_1, \ldots, B_d)\). If there is no transition from \(A\) to \(B\), then `function` should raise a `LookupError`.
    - `new_input_alphabet` (optional) – the new input alphabet as a list.
    - `only_accessible_components` – If `True` (default), then the result is piped through `accessible_components()`. If no `new_input_alphabet` is given, it is determined by `determine_alphabets()`.
    - `final_function` – A function mapping \(d\) final states of the original finite state machines to the final output of the corresponding state in the new finite state machine. By default, the final output is the empty word if both final outputs of the constituent states are empty; otherwise, a `ValueError` is raised.
    - `new_class` – Class of the new finite state machine. By default (`None`), the class of `self` is used.

    OUTPUT:

    A finite state machine whose states are \(d\)-tuples of states of the original finite state machines. A state is initial or final if all constituent states are initial or final, respectively.

    The labels of the transitions are defined by `function`.

    The final output of a final state is determined by calling `final_function` on the constituent states.

    The color of a new state is the tuple of colors of the constituent states of `self` and `other`.

    EXAMPLES:

        sage: F = Automaton([('A', 'B', 1), ('A', 'A', 0), ('B', 'A', 2)],
        ....:               initial_states=['A'], final_states=['B'],
        ....:               determine_alphabets=True)
        sage: G = Automaton([(1, 1, 1)], initial_states=[1], final_states=[1])
        sage: def addition(transition1, transition2):
        ....:     return (transition1.word_in[0] + transition2.word_in[0],
        ....:             None)
        sage: H = F.product_FiniteStateMachine(G, addition, [0, 1, 2, 3], only_accessible_components=False)
        sage: H.transitions()
        [Transition from ('A', 1) to ('B', 1): 2|-,
         Transition from ('A', 1) to ('A', 1): 1|-,
         Transition from ('B', 1) to ('A', 1): 3|-]
        sage: H1 = F.product_FiniteStateMachine(G, addition, [0, 1, 2, 3], only_accessible_components=False)
        sage: H1.states()[0].label()[0] is F.states()[0]
        True
        sage: H1.states()[0].label()[1] is G.states()[0]
        True

        sage: F = Automaton([(0,1,1/4), (0,0,3/4), (1,1,3/4), (1,0,1/4)],
        ....:                initial_states=[0] )
        sage: G = Automaton([(0,0,1), (1,1,3/4), (1,0,1/4)],
        ....:                initial_states=[0] )
        sage: H = F.product_FiniteStateMachine(
        ....:         G, lambda t1,t2: (t1.word_in[0]*t2.word_in[0], None))
        sage: H.states()
        [(0, 0), (1, 0)]

        sage: F = Automaton([(0,1,1/4), (0,0,3/4), (1,1,3/4), (1,0,1/4)],
        ....:                initial_states=[0] )
        sage: G = Automaton([(0,0,1), (1,1,3/4), (1,0,1/4)],
        ....:                initial_states=[0] )
        sage: H = F.product_FiniteStateMachine(G,
        ....:                                  lambda t1,t2: (t1.word_in[0]*t2.word_in[0], None),
        ....:                                  only_accessible_components=False)
        sage: H.states()
        [(0, 0), (1, 0), (0, 1), (1, 1)]

    Also final output words are considered according to the function `final_function`:

        sage: F = Transducer([(0, 1, 0, 1), (1, 1, 1, 1), (1, 1, 0, 1)],
        ....:                final_states=[1])
        sage: F.state(1).final_word_out = 1
        sage: G = Transducer([(0, 0, 0, 1), (0, 0, 1, 0)], final_states=[0])
        sage: G.state(0).final_word_out = 1
        sage: def minus(t1, t2):
        ....:     return (t1.word_in[0] - t2.word_in[0],
        ....:                t1.word_out[0] - t2.word_out[0])
        sage: H = F.product_FiniteStateMachine(G, minus)
        Traceback (most recent call last):
        ...
        ValueError: A final function must be given.
        sage: def plus(s1, s2):
        ....:     return s1.final_word_out[0] + s2.final_word_out[0]
        sage: H = F.product_FiniteStateMachine(G, minus,
        ....:                                  final_function=plus)
        sage: H.final_states()
        [(1, 0)]
        sage: H.final_states()[0].final_word_out
        [2]

    Products of more than two finite state machines are also possible:

        sage: def plus(s1, s2, s3):
        ....:     if s1.word_in == s2.word_in == s3.word_in:
        ....:          return (s1.word_in,
        ....:                  sum(s.word_out[0] for s in (s1, s2, s3)))
        ....:     else:
        ....:         raise LookupError
        sage: T0 = transducers.CountSubblockOccurrences([0, 0], [0, 1, 2])
        sage: T1 = transducers.CountSubblockOccurrences([1, 1], [0, 1, 2])
        sage: T2 = transducers.CountSubblockOccurrences([2, 2], [0, 1, 2])
        sage: T = T0.product_FiniteStateMachine([T1, T2], plus)
        sage: T.transitions()
        [Transition from ((), (), ()) to ((0,), (), ()): 0|0,
         Transition from ((), (), ()) to ((), (1,), ()): 1|0,
         Transition from ((), (), ()) to ((), (), (2,)): 2|0,
         Transition from ((0,), (), ()) to ((0,), (), ()): 0|1,
         Transition from ((0,), (), ()) to ((), (1,), ()): 1|0,
         Transition from ((0,), (), ()) to ((), (), (2,)): 2|0,
         Transition from ((), (1,), ()) to ((0,), (), ()): 0|0,
         Transition from ((), (1,), ()) to ((), (1,), ()): 1|1,
         Transition from ((), (1,), ()) to ((), (), (2,)): 2|0,
         Transition from ((), (), (2,)) to ((0,), (), ()): 0|0,
         Transition from ((), (), (2,)) to ((), (1,), ()): 1|0,
         Transition from ((), (), (2,)) to ((), (), (2,)): 2|1]
        sage: T([0, 0, 1, 1, 2, 2, 0, 1, 2, 2])
        [0, 1, 0, 1, 0, 1, 0, 0, 0, 1]

    `other` can also be an iterable:

        sage: T == T0.product_FiniteStateMachine(iter([T1, T2]), plus)
        True

    TESTS:

    Check that colors are correctly dealt with. In particular, the new colors have to be hashable such that `Automaton.determinisation()` does not fail:

        sage: A = Automaton([[0, 0, 0]], initial_states=[0])
        sage: B = A.product_FiniteStateMachine(A,
        ....:                                  lambda t1, t2: (0, None))
        sage: B.states()[0].color
        (None, None)
        sage: B.determinisation()
        Automaton with 1 states

    Check handling of the parameter `other`:

        sage: A.product_FiniteStateMachine(None, plus)
        Traceback (most recent call last):
        ...
        ValueError: other must be a finite state machine or a list
        of finite state machines.
        sage: A.product_FiniteStateMachine([None], plus)
        Traceback (most recent call last):
        ...
        ValueError: other must be a finite state machine or a list
        of finite state machines.

    Test whether `new_class` works:

        sage: T = Transducer()
        sage: type(T.product_FiniteStateMachine(T, None))
        <class 'sage.combinat.finite_state_machine.Transducer'>
        sage: type(T.product_FiniteStateMachine(T, None,
        ....:      new_class=Automaton))
        <class 'sage.combinat.finite_state_machine.Automaton'>

    Check that isolated vertices are kept (trac ticket #16762):

        sage: F = Transducer(initial_states=[0])
        sage: F.add_state(1)
        1
        sage: G = Transducer(initial_states=['A'])
        sage: F.product_FiniteStateMachine(G, None).states()
        [(0, 'A')]
        sage: F.product_FiniteStateMachine(
        ....:     G, None, only_accessible_components=False).states()
        [(0, 'A'), (1, 'A')]

  `projection`(what='input')

    Returns an Automaton which transition labels are the projection of the transition labels of the input.

    INPUT:

    - `what` – (default: `input`) either `input` or `output`.

    OUTPUT:

    An automaton.

    EXAMPLES:

        sage: F = FiniteStateMachine([('A', 'B', 0, 1), ('A', 'A', 1, 1),
        ....:                         ('B', 'B', 1, 0)])
        sage: G = F.projection(what='output')
        sage: G.transitions()
        [Transition from 'A' to 'B': 1|-,
         Transition from 'A' to 'A': 1|-,
         Transition from 'B' to 'B': 0|-]

  `quotient`(classes)

    Constructs the quotient with respect to the equivalence classes.

    INPUT:

    - `classes` is a list of equivalence classes of states.

    OUTPUT:

    A finite state machine.

    The labels of the new states are tuples of states of the `self`, corresponding to `classes`.

    Assume that \(c\) is a class, and \(a\) and \(b\) are states in \(c\). Then there is a bijection \(\varphi\) between the transitions from \(a\) and the transitions from \(b\) with the following properties: if \(\varphi(t_a)=t_b\), then

    - \(t_a.\mathit{word}_\mathit{in}=t_b.\mathit{word}_\mathit{in}\),
    - \(t_a.\mathit{word}_\mathit{out}=t_b.\mathit{word}_\mathit{out}\), and
    - \(t_a\) and \(t_b\) lead to some equivalent states \(a'\) and \(b'\).

    Non-initial states may be merged with initial states, the resulting state is an initial state.

    All states in a class must have the same `is_final`, `final_word_out` and `word_out` values.

    EXAMPLES:

        sage: fsm = FiniteStateMachine([("A", "B", 0, 1), ("A", "B", 1, 0),
        ....:                           ("B", "C", 0, 0), ("B", "C", 1, 1),
        ....:                           ("C", "D", 0, 1), ("C", "D", 1, 0),
        ....:                           ("D", "A", 0, 0), ("D", "A", 1, 1)])
        sage: fsmq = fsm.quotient([[fsm.state("A"), fsm.state("C")],
        ....:                      [fsm.state("B"), fsm.state("D")]])
        sage: fsmq.transitions()
        [Transition from ('A', 'C')
                      to ('B', 'D'): 0|1,
         Transition from ('A', 'C')
                      to ('B', 'D'): 1|0,
         Transition from ('B', 'D')
                      to ('A', 'C'): 0|0,
         Transition from ('B', 'D')
                      to ('A', 'C'): 1|1]
        sage: fsmq.relabeled().transitions()
        [Transition from 0 to 1: 0|1,
         Transition from 0 to 1: 1|0,
         Transition from 1 to 0: 0|0,
         Transition from 1 to 0: 1|1]
        sage: fsmq1 = fsm.quotient(fsm.equivalence_classes())
        sage: fsmq1 == fsmq
        True
        sage: fsm.quotient([[fsm.state("A"), fsm.state("B"), fsm.state("C"), fsm.state("D")]])
        Traceback (most recent call last):
            ...
        AssertionError: Transitions of state 'A' and 'B' are incompatible.

    TESTS:

        sage: fsm = FiniteStateMachine([("A", "B", 0, 1), ("A", "B", 1, 0),
        ....:                           ("B", "C", 0, 0), ("B", "C", 1, 1),
        ....:                           ("C", "D", 0, 1), ("C", "D", 1, 0),
        ....:                           ("D", "A", 0, 0), ("D", "A", 1, 1)],
        ....:                           final_states=["A", "C"])
        sage: fsm.state("A").final_word_out = 1
        sage: fsm.state("C").final_word_out = 2
        sage: fsmq = fsm.quotient([[fsm.state("A"), fsm.state("C")],
        ....:                      [fsm.state("B"), fsm.state("D")]])
        Traceback (most recent call last):
            ...
        AssertionError: Class ['A', 'C'] mixes
        final states with different final output words.

  `relabeled`(memo=None, labels=None)

    Returns a deep copy of the finite state machine, but the states are relabeled.

    INPUT:

    - `memo` – (default: `None`) a dictionary storing already processed elements.
    - `labels` – (default: `None`) a dictionary or callable mapping old labels to new labels. If `None`, then the new labels are integers starting with 0.

    OUTPUT:

    A new finite state machine.

    EXAMPLES:

        sage: FSM1 = FiniteStateMachine([('A', 'B'), ('B', 'C'), ('C', 'A')])
        sage: FSM1.states()
        ['A', 'B', 'C']
        sage: FSM2 = FSM1.relabeled()
        sage: FSM2.states()
        [0, 1, 2]
        sage: FSM3 = FSM1.relabeled(labels={'A': 'a', 'B': 'b', 'C': 'c'})
        sage: FSM3.states()
        ['a', 'b', 'c']
        sage: FSM4 = FSM2.relabeled(labels=lambda x: 2*x)
        sage: FSM4.states()
        [0, 2, 4]

    TESTS:

        sage: FSM2.relabeled(labels=1)
        Traceback (most recent call last):
        ...
        TypeError: labels must be None, a callable or a dictionary.

  `remove_epsilon_transitions`()

    TESTS:

        sage: FiniteStateMachine().remove_epsilon_transitions()
        Traceback (most recent call last):
        ...
        NotImplementedError

  `set_coordinates`(coordinates, default=True)

    Set coordinates of the states for the LaTeX representation by a dictionary or a function mapping labels to coordinates.

    INPUT:

    - `coordinates` – a dictionary or a function mapping labels of states to pairs interpreted as coordinates.
    - `default` – If `True`, then states not given by `coordinates` get a default position on a circle of radius 3.

    OUTPUT:

    Nothing.

    EXAMPLES:

        sage: F = Automaton([[0, 1, 1], [1, 2, 2], [2, 0, 0]])
        sage: F.set_coordinates({0: (0, 0), 1: (2, 0), 2: (1, 1)})
        sage: F.state(0).coordinates
        (0, 0)

    We can also use a function to determine the coordinates:

        sage: F = Automaton([[0, 1, 1], [1, 2, 2], [2, 0, 0]])
        sage: F.set_coordinates(lambda l: (l, 3/(l+1)))
        sage: F.state(2).coordinates
        (2, 1)

  `split_transitions`()

    Returns a new transducer, where all transitions in self with input labels consisting of more than one letter are replaced by a path of the corresponding length.

    INPUT:

    Nothing.

    OUTPUT:

    A new transducer.

    EXAMPLES:

        sage: A = Transducer([('A', 'B', [1, 2, 3], 0)],
        ....:                initial_states=['A'], final_states=['B'])
        sage: A.split_transitions().states()
        [('A', ()), ('B', ()),
         ('A', (1,)), ('A', (1, 2))]

  `state`(state)

    Returns the state of the finite state machine.

    INPUT:

    - `state` – If `state` is not an instance of `FSMState`, then it is assumed that it is the label of a state.

    OUTPUT:

    Returns the state of the finite state machine corresponding to `state`.

    If no state is found, then a `LookupError` is thrown.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMState
        sage: A = FSMState('A')
        sage: FSM = FiniteStateMachine([(A, 'B'), ('C', A)])
        sage: FSM.state('A') == A
        True
        sage: FSM.state('xyz')
        Traceback (most recent call last):
        ...
        LookupError: No state with label xyz found.

  `states`()

    Returns the states of the finite state machine.

    INPUT:

    Nothing.

    OUTPUT:

    The states of the finite state machine as list.

    EXAMPLES:

        sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)])
        sage: FSM.states()
        ['1', '2']

  `transition`(transition)

    Returns the transition of the finite state machine.

    INPUT:

    - `transition` – If `transition` is not an instance of `FSMTransition`, then it is assumed that it is a tuple `(from_state, to_state, word_in, word_out)`.

    OUTPUT:

    Returns the transition of the finite state machine corresponding to `transition`.

    If no transition is found, then a `LookupError` is thrown.

    EXAMPLES:

        sage: from sage.combinat.finite_state_machine import FSMTransition
        sage: t = FSMTransition('A', 'B', 0)
        sage: F = FiniteStateMachine([t])
        sage: F.transition(('A', 'B', 0))
        Transition from 'A' to 'B': 0|-
        sage: id(t) == id(F.transition(('A', 'B', 0)))
        True

  `transitions`(from_state=None)

    Returns a list of all transitions.

    INPUT:

    - `from_state` – (default: `None`) If `from_state` is given, then a list of transitions starting there is given.

    OUTPUT:

    A list of all transitions.

    EXAMPLES:

        sage: FSM = Automaton([('1', '2', 1), ('2', '2', 0)])
        sage: FSM.transitions()
        [Transition from '1' to '2': 1|-,
         Transition from '2' to '2': 0|-]

  `transposition`()

    Returns a new finite state machine, where all transitions of the input finite state machine are reversed.

    INPUT:

    Nothing.

    OUTPUT:

    A new finite state machine.

    EXAMPLES:

        sage: aut = Automaton([('A', 'A', 0), ('A', 'A', 1), ('A', 'B', 0)],
        ....:                 initial_states=['A'], final_states=['B'])
        sage: aut.transposition().transitions('B')
        [Transition from 'B' to 'A': 0|-]

        sage: aut = Automaton([('1', '1', 1), ('1', '2', 0), ('2', '2', 0)],
        ....:                 initial_states=['1'], final_states=['1', '2'])
        sage: aut.transposition().initial_states()
        ['1', '2']

    TESTS:

    If a final state of `self` has a non-empty final output word, transposition is not implemented:

        sage: T = Transducer([('1', '1', 1, 0), ('1', '2', 0, 1),
        ....:                 ('2', '2', 0, 2)],
        ....:                 initial_states=['1'],
        ....:                 final_states=['1', '2'])
        sage: T.state('1').final_word_out = [2, 5]
        sage: T.transposition()
        Traceback (most recent call last):
        ...
        NotImplementedError: Transposition for transducers with
        final output words is not implemented.

  `with_final_word_out`(letters, allow_non_final=True)

    Constructs a new finite state machine with final output words for all states by implicitly reading trailing letters until a final state is reached.

    INPUT:

    - `letters` – either an element of the input alphabet or a list of such elements. This is repeated cyclically when needed.
    - `allow_non_final` – a boolean (default: `True`) which indicates whether we allow that some states may be non-final in the resulting finite state machine. I.e., if `False` then each state has to have a path to a final state with input label matching `letters`.

    OUTPUT:

    A finite state machine.

    The inplace version of this function is `construct_final_word_out()`.

    Suppose for the moment a single element `letter` as input for `letters`. This is equivalent to `letters = [letter]`. We will discuss the general case below.

    Let `word_in` be a word over the input alphabet and assume that the original finite state machine transforms `word_in` to `word_out` reaching a possibly non-final state `s`. Let further \(k\) be the minimum number of letters `letter` such that there is a path from `s` to some final state `f` whose input label consists of \(k\) copies of `letter` and whose output label is `path_word_out`. Then the state `s` of the resulting finite state machine is a final state with final output `path_word_out + f.final_word_out`. Therefore, the new finite state machine transforms `word_in` to `word_out + path_word_out + f.final_word_out`.

    This is e.g. useful for finite state machines operating on digit expansions: there, it is sometimes required to read a sufficient number of trailing zeros (at the most significant positions) in order to reach a final state and to flush all carries. In this case, this method constructs an essentially equivalent finite state machine in the sense that it not longer requires adding sufficiently many trailing zeros. However, it is the responsibility of the user to make sure that if adding trailing zeros to the input anyway, the output is equivalent.

    If `letters` consists of more than one letter, then it is assumed that (not necessarily complete) cycles of `letters` are appended as trailing input.

    See also

    example on Gray code

    EXAMPLES:

    > 1. A simple transducer transforming \(00\) blocks to \(01\) blocks:

    >        sage: T = Transducer([(0, 1, 0, 0), (1, 0, 0, 1)],
    >        ....:                initial_states=[0],
    >        ....:                final_states=[0])
    >        sage: T.process([0, 0, 0])
    >        (False, 1, [0, 1, 0])
    >        sage: T.process([0, 0, 0, 0])
    >        (True, 0, [0, 1, 0, 1])
    >        sage: F = T.with_final_word_out(0)
    >        sage: for f in F.iter_final_states():
    >        ....:     print f, f.final_word_out
    >        0 []
    >        1 [1]
    >        sage: F.process([0, 0, 0])
    >        (True, 1, [0, 1, 0, 1])
    >        sage: F.process([0, 0, 0, 0])
    >        (True, 0, [0, 1, 0, 1])

    > 2. A more realistic example: Addition of \(1\) in binary. We construct a transition function transforming the input to its binary expansion:

    >        sage: def binary_transition(carry, input):
    >        ....:     value = carry + input
    >        ....:     if value.mod(2) == 0:
    >        ....:         return (value/2, 0)
    >        ....:     else:
    >        ....:         return ((value-1)/2, 1)

    >    Now, we only have to start with a carry of \(1\) to get the required transducer:

    >        sage: T = Transducer(binary_transition,
    >        ....:                input_alphabet=[0, 1],
    >        ....:                initial_states=[1],
    >        ....:                final_states=[0])

    >    We test this for the binary expansion of \(7\):

    >        sage: T.process([1, 1, 1])
    >        (False, 1, [0, 0, 0])

    >    The final carry \(1\) has not be flushed yet, we have to add a trailing zero:

    >        sage: T.process([1, 1, 1, 0])
    >        (True, 0, [0, 0, 0, 1])

    >    We check that with this trailing zero, the transducer performs as advertised:

    >        sage: all(ZZ(T(k.bits()+[0]), base=2) == k + 1
    >        ....:     for k in srange(16))
    >        True

    >    However, most of the time, we produce superfluous trailing zeros:

    >        sage: T(11.bits()+[0])
    >        [0, 0, 1, 1, 0]

    >    We now use this method:

    >        sage: F = T.with_final_word_out(0)
    >        sage: for f in F.iter_final_states():
    >        ....:     print f, f.final_word_out
    >        1 [1]
    >        0 []

    >    The same tests as above, but we do not have to pad with trailing zeros anymore:

    >        sage: F.process([1, 1, 1])
    >        (True, 1, [0, 0, 0, 1])
    >        sage: all(ZZ(F(k.bits()), base=2) == k + 1
    >        ....:     for k in srange(16))
    >        True

    >    No more trailing zero in the output:

    >        sage: F(11.bits())
    >        [0, 0, 1, 1]
    >        sage: all(F(k.bits())[-1] == 1
    >        ....:     for k in srange(16))
    >        True

    > 3. Here is an example, where we allow trailing repeated \(10\):

    >        sage: T = Transducer([(0, 1, 0, 'a'),
    >        ....:                 (1, 2, 1, 'b'),
    >        ....:                 (2, 0, 0, 'c')],
    >        ....:                initial_states=[0],
    >        ....:                final_states=[0])
    >        sage: F = T.with_final_word_out([1, 0])
    >        sage: for f in F.iter_final_states():
    >        ....:     print f, ''.join(f.final_word_out)
    >        0
    >        1 bc

    >    Trying this with trailing repeated \(01\) does not produce a `final_word_out` for state `1`, but for state `2`:

    >        sage: F = T.with_final_word_out([0, 1])
    >        sage: for f in F.iter_final_states():
    >        ....:     print f, ''.join(f.final_word_out)
    >        0
    >        2 c

    > 4. Here another example with a more-letter trailing input:

    >        sage: T = Transducer([(0, 1, 0, 'a'),
    >        ....:                 (1, 2, 0, 'b'), (1, 2, 1, 'b'),
    >        ....:                 (2, 3, 0, 'c'), (2, 0, 1, 'e'),
    >        ....:                 (3, 1, 0, 'd'), (3, 1, 1, 'd')],
    >        ....:                initial_states=[0],
    >        ....:                final_states=[0],
    >        ....:                with_final_word_out=[0, 0, 1, 1])
    >        sage: for f in T.iter_final_states():
    >        ....:     print f, ''.join(f.final_word_out)
    >        0
    >        1 bcdbcdbe
    >        2 cdbe
    >        3 dbe

    TESTS:

    > 1. Reading copies of `letter` may result in a cycle. In this simple example, we have no final state at all:

    >        sage: T = Transducer([(0, 1, 0, 0), (1, 0, 0, 0)],
    >        ....:                initial_states=[0])
    >        sage: T.with_final_word_out(0)
    >        Traceback (most recent call last):
    >        ...
    >        ValueError: The finite state machine contains
    >        a cycle starting at state 0 with input label 0
    >        and no final state.

    > 2. A unique transition with input word `letter` is required:

    >        sage: T = Transducer([(0, 1, 0, 0), (0, 2, 0, 0)])
    >        sage: T.with_final_word_out(0)
    >        Traceback (most recent call last):
    >        ...
    >        ValueError: No unique transition leaving state 0
    >        with input label 0.

    >    It is not a problem if there is no transition starting at state `1` with input word `letter`:

    >        sage: T = Transducer([(0, 1, 0, 0)])
    >        sage: F = T.with_final_word_out(0)
    >        sage: for f in F.iter_final_states():
    >        ....:     print f, f.final_word_out

    >    Anyhow, you can override this by:

    >        sage: T = Transducer([(0, 1, 0, 0)])
    >        sage: T.with_final_word_out(0, allow_non_final=False)
    >        Traceback (most recent call last):
    >        ...
    >        ValueError: No unique transition leaving state 1
    >        with input label 0.

    > 3. All transitions must have input labels of length \(1\):

    >        sage: T = Transducer([(0, 0, [], 0)])
    >        sage: T.with_final_word_out(0)
    >        Traceback (most recent call last):
    >        ...
    >        NotImplementedError: All transitions must have input
    >        labels of length 1. Consider calling split_transitions().
    >        sage: T = Transducer([(0, 0, [0, 1], 0)])
    >        sage: T.with_final_word_out(0)
    >        Traceback (most recent call last):
    >        ...
    >        NotImplementedError: All transitions must have input
    >        labels of length 1. Consider calling split_transitions().

    > 4. An empty list as input is not allowed:

    >        sage: T = Transducer([(0, 0, [], 0)])
    >        sage: T.with_final_word_out([])
    >        Traceback (most recent call last):
    >        ...
    >        ValueError: letters is not allowed to be an empty list.

class `sage.combinat.finite_state_machine.``Transducer`(data=None, initial_states=None, final_states=None, input_alphabet=None, output_alphabet=None, determine_alphabets=None, with_final_word_out=None, store_states_dict=True, on_duplicate_transition=None)

  Bases: `sage.combinat.finite_state_machine.FiniteStateMachine`

  This creates a transducer, which is a finite state machine, whose transitions have input and output labels.

  An transducer has additional features like creating a simplified transducer.

  See class `FiniteStateMachine` for more information.

  EXAMPLES:

  We can create a transducer performing the addition of 1 (for numbers given in binary and read from right to left) in the following way:

      sage: T = Transducer([('C', 'C', 1, 0), ('C', 'N', 0, 1),
      ....:                 ('N', 'N', 0, 0), ('N', 'N', 1, 1)],
      ....:                initial_states=['C'], final_states=['N'])
      sage: T
      Transducer with 2 states
      sage: T([0])
      [1]
      sage: T([1,1,0])
      [0, 0, 1]
      sage: ZZ(T(15.digits(base=2)+[0]), base=2)
      16

  Note that we have padded the binary input sequence by a \(0\) so that the transducer can reach its final state.

  TESTS:

      sage: Transducer()
      Transducer with 0 states

  `cartesian_product`(other, only_accessible_components=True)

    Warning

    The default output of this method is scheduled to change. This docstring describes the new default behaviour, which can already be achieved by setting `FSMOldCodeTransducerCartesianProduct` to `False`.

    Return a new transducer which can simultaneously process an input with the machines `self` and `other` where the output labels are \(d\)-tuples of the original output labels.

    INPUT:

    - `other` - a finite state machine (if \(d=2\)) or a list (or other iterable) of \(d-1\) finite state machines
    - `only_accessible_components` – If `True` (default), then the result is piped through `accessible_components()`. If no `new_input_alphabet` is given, it is determined by `determine_alphabets()`.

    OUTPUT:

    A transducer which can simultaneously process an input with `self` and the machine(s) in `other`.

    The set of states of the new transducer is the cartesian product of the set of states of `self` and `other`.

    Let \((A_j, B_j, a_j, b_j)\) for \(j\in\{1, \ldots, d\}\) be transitions in the machines `self` and in `other`. Then there is a transition \(((A_1, \ldots, A_d), (B_1, \ldots, B_d), a, (b_1, \ldots, b_d))\) in the new transducer if \(a_1 = \cdots = a_d =: a\).

    EXAMPLES:

    Originally a different output was constructed by `Transducer.cartesian_product()`. This output is now produced by `Transducer.intersection()`.

        sage: transducer1 = Transducer([('A', 'A', 0, 0),
        ....:                           ('A', 'A', 1, 1)],
        ....:                          initial_states=['A'],
        ....:                          final_states=['A'],
        ....:                          determine_alphabets=True)
        sage: transducer2 = Transducer([(0, 1, 0, ['b', 'c']),
        ....:                           (0, 0, 1, 'b'),
        ....:                           (1, 1, 0, 'a')],
        ....:                          initial_states=[0],
        ....:                          final_states=[1],
        ....:                          determine_alphabets=True)
        sage: result = transducer1.cartesian_product(transducer2)
        doctest:...: DeprecationWarning: The output of
        Transducer.cartesian_product will change.
        Please use Transducer.intersection for the original output.
        See http://trac.sagemath.org/16061 for details.
        sage: result
        Transducer with 1 states

    By setting `FSMOldCodeTransducerCartesianProduct` to `False` the new desired output is produced.

        sage: sage.combinat.finite_state_machine.FSMOldCodeTransducerCartesianProduct = False
        sage: result = transducer1.cartesian_product(transducer2)
        sage: result
        Transducer with 2 states
        sage: result.transitions()
        [Transition from ('A', 0) to ('A', 1): 0|(0, 'b'),(None, 'c'),
         Transition from ('A', 0) to ('A', 0): 1|(1, 'b'),
         Transition from ('A', 1) to ('A', 1): 0|(0, 'a')]
        sage: result([1, 0, 0])
        [(1, 'b'), (0, 'b'), (None, 'c'),  (0, 'a')]
        sage: (transducer1([1, 0, 0]), transducer2([1, 0, 0]))
        ([1, 0, 0], ['b', 'b', 'c', 'a'])

    Also final output words are correctly processed:

        sage: transducer1.state('A').final_word_out = 2
        sage: result = transducer1.cartesian_product(transducer2)
        sage: result.final_states()[0].final_word_out
        [(2, None)]

    The following transducer counts the number of 11 blocks minus the number of 10 blocks over the alphabet `[0, 1]`.

        sage: count_11 = transducers.CountSubblockOccurrences(
        ....:     [1, 1],
        ....:     input_alphabet=[0, 1])
        sage: count_10 = transducers.CountSubblockOccurrences(
        ....:     [1, 0],
        ....:     input_alphabet=[0, 1])
        sage: count_11x10 = count_11.cartesian_product(count_10)
        sage: difference = transducers.sub([0, 1])(count_11x10)
        sage: T = difference.simplification().relabeled()
        sage: T.initial_states()
        [1]
        sage: sorted(T.transitions())
        [Transition from 0 to 1: 0|-1,
         Transition from 0 to 0: 1|1,
         Transition from 1 to 1: 0|0,
         Transition from 1 to 0: 1|0]
        sage: input =  [0, 1, 1,  0, 1,  0, 0, 0, 1, 1, 1,  0]
        sage: output = [0, 0, 1, -1, 0, -1, 0, 0, 0, 1, 1, -1]
        sage: T(input) == output
        True

    If `other` is an automaton, then `cartesian_product()` returns `self` where the input is restricted to the input accepted by `other`.

    For example, if the transducer transforms the standard binary expansion into the non-adjacent form and the automaton recognizes the binary expansion without adjacent ones, then the cartesian product of these two is a transducer which does not change the input (except for changing `a` to `(a, None)` and ignoring a leading \(0\)).

        sage: NAF = Transducer([(0, 1, 0, None),
        ....:                   (0, 2, 1, None),
        ....:                   (1, 1, 0, 0),
        ....:                   (1, 2, 1, 0),
        ....:                   (2, 1, 0, 1),
        ....:                   (2, 3, 1, -1),
        ....:                   (3, 2, 0, 0),
        ....:                   (3, 3, 1, 0)],
        ....:                  initial_states=[0],
        ....:                  final_states=[1],
        ....:                  determine_alphabets=True)
        sage: aut11 = Automaton([(0, 0, 0), (0, 1, 1), (1, 0, 0)],
        ....:                   initial_states=[0],
        ....:                   final_states=[0, 1],
        ....:                   determine_alphabets=True)
        sage: res = NAF.cartesian_product(aut11)
        sage: res([1, 0, 0, 1, 0, 1, 0])
        [(1, None), (0, None), (0, None), (1, None), (0, None), (1, None)]

    This is obvious because if the standard binary expansion does not have adjacent ones, then it is the same as the non-adjacent form.

    Be aware that `cartesian_product()` is not commutative.

        sage: aut11.cartesian_product(NAF)
        Traceback (most recent call last):
        ...
        TypeError: Only an automaton can be intersected with an automaton.

    The cartesian product of more than two finite state machines can also be computed:

        sage: T0 = transducers.CountSubblockOccurrences([0, 0], [0, 1, 2])
        sage: T1 = transducers.CountSubblockOccurrences([1, 1], [0, 1, 2])
        sage: T2 = transducers.CountSubblockOccurrences([2, 2], [0, 1, 2])
        sage: T = T0.cartesian_product([T1, T2])
        sage: T.transitions()
        [Transition from ((), (), ()) to ((0,), (), ()): 0|(0, 0, 0),
         Transition from ((), (), ()) to ((), (1,), ()): 1|(0, 0, 0),
         Transition from ((), (), ()) to ((), (), (2,)): 2|(0, 0, 0),
         Transition from ((0,), (), ()) to ((0,), (), ()): 0|(1, 0, 0),
         Transition from ((0,), (), ()) to ((), (1,), ()): 1|(0, 0, 0),
         Transition from ((0,), (), ()) to ((), (), (2,)): 2|(0, 0, 0),
         Transition from ((), (1,), ()) to ((0,), (), ()): 0|(0, 0, 0),
         Transition from ((), (1,), ()) to ((), (1,), ()): 1|(0, 1, 0),
         Transition from ((), (1,), ()) to ((), (), (2,)): 2|(0, 0, 0),
         Transition from ((), (), (2,)) to ((0,), (), ()): 0|(0, 0, 0),
         Transition from ((), (), (2,)) to ((), (1,), ()): 1|(0, 0, 0),
         Transition from ((), (), (2,)) to ((), (), (2,)): 2|(0, 0, 1)]
        sage: T([0, 0, 1, 1, 2, 2, 0, 1, 2, 2])
        [(0, 0, 0),
         (1, 0, 0),
         (0, 0, 0),
         (0, 1, 0),
         (0, 0, 0),
         (0, 0, 1),
         (0, 0, 0),
         (0, 0, 0),
         (0, 0, 0),
         (0, 0, 1)]

  `intersection`(other, only_accessible_components=True)

    Returns a new transducer which accepts an input if it is accepted by both given finite state machines producing the same output.

    INPUT:

    - `other` – a transducer
    - `only_accessible_components` – If `True` (default), then the result is piped through `accessible_components()`. If no `new_input_alphabet` is given, it is determined by `determine_alphabets()`.

    OUTPUT:

    A new transducer which computes the intersection (see below) of the languages of `self` and `other`.

    The set of states of the transducer is the cartesian product of the set of states of both given transducer. There is a transition \(((A, B), (C, D), a, b)\) in the new transducer if there are transitions \((A, C, a, b)\) and \((B, D, a, b)\) in the old transducers.

    EXAMPLES:

        sage: transducer1 = Transducer([('1', '2', 1, 0),
        ....:                           ('2', '2', 1, 0),
        ....:                           ('2', '2', 0, 1)],
        ....:                          initial_states=['1'],
        ....:                          final_states=['2'])
        sage: transducer2 = Transducer([('A', 'A', 1, 0),
        ....:                           ('A', 'B', 0, 0),
        ....:                           ('B', 'B', 0, 1),
        ....:                           ('B', 'A', 1, 1)],
        ....:                          initial_states=['A'],
        ....:                          final_states=['B'])
        sage: res = transducer1.intersection(transducer2)
        sage: res.transitions()
        [Transition from ('1', 'A') to ('2', 'A'): 1|0,
         Transition from ('2', 'A') to ('2', 'A'): 1|0]

    In general, transducers are not closed under intersection. But for transducer which do not have epsilon-transitions, the intersection is well defined (cf. [BaWo2012]). However, in the next example the intersection of the two transducers is not well defined. The intersection of the languages consists of \((a^n, b^n c^n)\). This set is not recognizable by a finite transducer.

        sage: t1 = Transducer([(0, 0, 'a', 'b'),
        ....:                  (0, 1, None, 'c'),
        ....:                  (1, 1, None, 'c')],
        ....:                 initial_states=[0],
        ....:                 final_states=[0, 1])
        sage: t2 = Transducer([('A', 'A', None, 'b'),
        ....:                  ('A', 'B', 'a', 'c'),
        ....:                  ('B', 'B', 'a', 'c')],
        ....:                 initial_states=['A'],
        ....:                 final_states=['A', 'B'])
        sage: t2.intersection(t1)
        Traceback (most recent call last):
        ...
        ValueError: An epsilon-transition (with empty input or output)
        was found.

    TESTS:

        sage: transducer1 = Transducer([('1', '2', 1, 0)],
        ....:                          initial_states=['1'],
        ....:                          final_states=['2'])
        sage: transducer2 = Transducer([('A', 'B', 1, 0)],
        ....:                          initial_states=['A'],
        ....:                          final_states=['B'])
        sage: res = transducer1.intersection(transducer2)
        sage: res.final_states()
        [('2', 'B')]
        sage: transducer1.state('2').final_word_out = 1
        sage: transducer2.state('B').final_word_out = 2
        sage: res = transducer1.intersection(transducer2)
        sage: res.final_states()
        []

    REFERENCES:

    [BaWo2012] | Javier Baliosian and Dina Wonsever, Finite State Transducers, chapter in Handbook of Finite State Based Models and Applications, edited by Jiacun Wang, Chapman and Hall/CRC, 2012.

  `process`(*args, **kwargs)

    Warning

    The default output of this method is scheduled to change. This docstring describes the new default behaviour, which can already be achieved by setting `FSMOldProcessOutput` to `False`.

    Returns whether the transducer accepts the input, the state where the computation stops and which output is generated.

    INPUT:

    - `input_tape` – the input tape can be a list or an iterable with entries from the input alphabet. If we are working with a multi-tape machine (see parameter `use_multitape_input` and notes below), then the tape is a list or tuple of tracks, each of which can be a list or an iterable with entries from the input alphabet.
    - `initial_state` or `initial_states` – the initial state(s) in which the machine starts. Either specify a single one with `initial_state` or a list of them with `initial_states`. If both are given, `initial_state` will be appended to `initial_states`. If neither is specified, the initial states of the finite state machine are taken.
    - `list_of_outputs` – (default: `None`) a boolean or `None`. If `True`, then the outputs are given in list form (even if we have no or only one single output). If `False`, then the result is never a list (an exception is raised if the result cannot be returned). If `list_of_outputs=None` the method determines automatically what to do (e.g. if a non-deterministic machine returns more than one path, then the output is returned in list form).
    - `only_accepted` – (default: `False`) a boolean. If set, then the first argument in the output is guaranteed to be `True` (if the output is a list, then the first argument of each element will be `True`).
    - `full_output` – (default: `True`) a boolean. If set, then the full output is given, otherwise only the generated output (the third entry below only). If the input is not accepted, a `ValueError` is raised.
    - `always_include_output` – if set (not by default), always include the output. This is inconsequential for a `Transducer`, but can be used in other classes derived from `FiniteStateMachine` where the output is suppressed by default, cf. `Automaton.process()`.
    - `format_output` – a function that translates the written output (which is in form of a list) to something more readable. By default (`None`) identity is used here.
    - `check_epsilon_transitions` – (default: `True`) a boolean. If `False`, then epsilon transitions are not taken into consideration during process.
    - `write_final_word_out` – (default: `True`) a boolean specifying whether the final output words should be written or not.
    - `use_multitape_input` – (default: `False`) a boolean. If `True`, then the multi-tape mode of the process iterator is activated. See also the notes below for multi-tape machines.

    OUTPUT:

    The full output is a triple (or a list of triples, cf. parameter `list_of_outputs`), where

    - the first entry is `True` if the input string is accepted,
    - the second gives the reached state after processing the input tape (This is a state with label `None` if the input could not be processed, i.e., if at one point no transition to go on could be found.), and
    - the third gives a list of the output labels written during processing.

    If `full_output` is `False`, then only the third entry is returned.

    Note that in the case the transducer is not deterministic, all possible paths are taken into account.

    By setting `FSMOldProcessOutput` to `False` the new desired output is produced.

    This function uses an iterator which, in its simplest form, goes from one state to another in each step. To decide which way to go, it uses the input words of the outgoing transitions and compares them to the input tape. More precisely, in each step, the iterator takes an outgoing transition of the current state, whose input label equals the input letter of the tape. The output label of the transition, if present, is written on the output tape.

    If the choice of the outgoing transition is not unique (i.e., we have a non-deterministic finite state machine), all possibilites are followed. This is done by splitting the process into several branches, one for each of the possible outgoing transitions.

    The process (iteration) stops if all branches are finished, i.e., for no branch, there is any transition whose input word coincides with the processed input tape. This can simply happen when the entire tape was read.

    Also see `__call__()` for a version of `process()` with shortened output.

    Internally this function creates and works with an instance of `FSMProcessIterator`. This iterator can also be obtained with `iter_process()`.

    If working with multi-tape finite state machines, all input words of transitions are words of \(k\)-tuples of letters. Moreover, the input tape has to consist of \(k\) tracks, i.e., be a list or tuple of \(k\) iterators, one for each track.

    Warning

    Working with multi-tape finite state machines is still experimental and can lead to wrong outputs.

    EXAMPLES:

        sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False  # activate new output behavior
        sage: binary_inverter = Transducer({'A': [('A', 0, 1), ('A', 1, 0)]},
        ....:                              initial_states=['A'], final_states=['A'])
        sage: binary_inverter.process([0, 1, 0, 0, 1, 1])
        (True, 'A', [1, 0, 1, 1, 0, 0])

    If we are only interested in the output, we can also use:

        sage: binary_inverter([0, 1, 0, 0, 1, 1])
        [1, 0, 1, 1, 0, 0]

    The following transducer transforms \(0^n 1\) to \(1^n 2\):

        sage: T = Transducer([(0, 0, 0, 1), (0, 1, 1, 2)])
        sage: T.state(0).is_initial = True
        sage: T.state(1).is_final = True

    We can see the different possibilites of the output by:

        sage: [T.process(w) for w in [[1], [0, 1], [0, 0, 1], [0, 1, 1],
        ....:                         [0], [0, 0], [2, 0], [0, 1, 2]]]
        [(True, 1, [2]), (True, 1, [1, 2]),
         (True, 1, [1, 1, 2]), (False, None, None),
         (False, 0, [1]), (False, 0, [1, 1]),
         (False, None, None), (False, None, None)]

    If we just want a condensed output, we use:

        sage: [T.process(w, full_output=False)
        ....:      for w in [[1], [0, 1], [0, 0, 1]]]
        [[2], [1, 2], [1, 1, 2]]
        sage: T.process([0], full_output=False)
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.
        sage: T.process([0, 1, 2], full_output=False)
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.

    It is equivalent to:

        sage: [T(w) for w in [[1], [0, 1], [0, 0, 1]]]
        [[2], [1, 2], [1, 1, 2]]
        sage: T([0])
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.
        sage: T([0, 1, 2])
        Traceback (most recent call last):
        ...
        ValueError: Invalid input sequence.

    A cycle with empty input and empty output is correctly processed:

        sage: T = Transducer([(0, 1, None, None), (1, 0, None, None)],
        ....:                initial_states=[0], final_states=[1])
        sage: T.process([])
        [(False, 0, []), (True, 1, [])]
        sage: _ = T.add_transition(-1, 0, 0, 'r')
        sage: T.state(-1).is_initial = True
        sage: T.state(0).is_initial = False
        sage: T.process([0])
        [(False, 0, ['r']), (True, 1, ['r'])]

    If there is a cycle with empty input but non-empty output, the possible outputs would be an infinite set:

        sage: T = Transducer([(0, 1, None, 'z'), (1, 0, None, None)],
        ....:                initial_states=[0], final_states=[1])
        sage: T.process([])
        Traceback (most recent call last):
        ...
        RuntimeError: State 0 is in an epsilon cycle (no input),
        but output is written.

    But if this cycle with empty input and non-empty output is not reached, the correct output is produced:

        sage: _ = T.add_transition(-1, 0, 0, 'r')
        sage: T.state(-1).is_initial = True
        sage: T.state(0).is_initial = False
        sage: T.process([])
        (False, -1, [])
        sage: T.process([0])
        Traceback (most recent call last):
        ...
        RuntimeError: State 0 is in an epsilon cycle (no input),
        but output is written.

    If we set `check_epsilon_transitions=False`, then no transitions with empty input are considered anymore. Thus cycles with empty input are no problem anymore:

        sage: T.process([0], check_epsilon_transitions=False)
        (False, 0, ['r'])

    A simple example of a multi-tape transducer is the following: It writes the length of the first tape many letters `a` and then the length of the second tape many letters `b`:

        sage: M = Transducer([(0, 0, (1, None), 'a'),
        ....:                 (0, 1, [], []),
        ....:                 (1, 1, (None, 1), 'b')],
        ....:                initial_states=[0],
        ....:                final_states=[1])
        sage: M.process(([1, 1], [1]), use_multitape_input=True)
        (True, 1, ['a', 'a', 'b'])

    See also

    `FiniteStateMachine.process()`, `Automaton.process()`, `iter_process()`, `__call__()`, `FSMProcessIterator`.

    TESTS:

        sage: T = Transducer([(0, 1, 1, 'a'), (1, 0, 1, 'b')],
        ....:                initial_states=[0, 1], final_states=[1])
        sage: T.process([1, 1])
        [(False, 0, ['a', 'b']), (True, 1, ['b', 'a'])]
        sage: T.process([1, 1], T.state(0))
        (False, 0, ['a', 'b'])
        sage: T.state(1).final_word_out = 'c'
        sage: T.process([1, 1], T.state(1))
        (True, 1, ['b', 'a', 'c'])
        sage: T.process([1, 1], T.state(1), write_final_word_out=False)
        (True, 1, ['b', 'a'])

    The parameter `input_tape` is required:

        sage: T.process()
        Traceback (most recent call last):
        ...
        TypeError: No input tape given.

  `simplification`()

    Returns a simplified transducer.

    INPUT:

    Nothing.

    OUTPUT:

    A new transducer.

    This function simplifies a transducer by Moore’s algorithm, first moving common output labels of transitions leaving a state to output labels of transitions entering the state (cf. `prepone_output()`).

    The resulting transducer implements the same function as the original transducer.

    EXAMPLES:

        sage: fsm = Transducer([("A", "B", 0, 1), ("A", "B", 1, 0),
        ....:                           ("B", "C", 0, 0), ("B", "C", 1, 1),
        ....:                           ("C", "D", 0, 1), ("C", "D", 1, 0),
        ....:                           ("D", "A", 0, 0), ("D", "A", 1, 1)])
        sage: fsms = fsm.simplification()
        sage: fsms
        Transducer with 2 states
        sage: fsms.transitions()
        [Transition from ('A', 'C')
                      to ('B', 'D'): 0|1,
         Transition from ('A', 'C')
                      to ('B', 'D'): 1|0,
         Transition from ('B', 'D')
                      to ('A', 'C'): 0|0,
         Transition from ('B', 'D')
                      to ('A', 'C'): 1|1]
        sage: fsms.relabeled().transitions()
        [Transition from 0 to 1: 0|1,
         Transition from 0 to 1: 1|0,
         Transition from 1 to 0: 0|0,
         Transition from 1 to 0: 1|1]

        sage: fsm = Transducer([("A", "A", 0, 0),
        ....:                   ("A", "B", 1, 1),
        ....:                   ("A", "C", 1, -1),
        ....:                   ("B", "A", 2, 0),
        ....:                   ("C", "A", 2, 0)])
        sage: fsm_simplified = fsm.simplification()
        sage: fsm_simplified
        Transducer with 2 states
        sage: fsm_simplified.transitions()
        [Transition from ('A',) to ('A',): 0|0,
         Transition from ('A',) to ('B', 'C'): 1|1,0,
         Transition from ('A',) to ('B', 'C'): 1|-1,0,
         Transition from ('B', 'C') to ('A',): 2|-]

        sage: from sage.combinat.finite_state_machine import duplicate_transition_add_input
        sage: T = Transducer([('A', 'A', 1/2, 0),
        ....:                 ('A', 'B', 1/4, 1),
        ....:                 ('A', 'C', 1/4, 1),
        ....:                 ('B', 'A', 1, 0),
        ....:                 ('C', 'A', 1, 0)],
        ....:                initial_states=[0],
        ....:                final_states=['A', 'B', 'C'],
        ....:                on_duplicate_transition=duplicate_transition_add_input)
        sage: sorted(T.simplification().transitions())
        [Transition from ('A',) to ('A',): 1/2|0,
         Transition from ('A',) to ('B', 'C'): 1/2|1,
         Transition from ('B', 'C') to ('A',): 1|0]

    Illustrating the use of colors in order to avoid identification of states:

        sage: T = Transducer( [[0,0,0,0], [0,1,1,1],
        ....:                  [1,0,0,0], [1,1,1,1]],
        ....:                 initial_states=[0],
        ....:                 final_states=[0,1])
        sage: sorted(T.simplification().transitions())
        [Transition from (0, 1) to (0, 1): 0|0,
         Transition from (0, 1) to (0, 1): 1|1]
        sage: T.state(0).color = 0
        sage: T.state(0).color = 1
        sage: sorted(T.simplification().transitions())
        [Transition from (0,) to (0,): 0|0,
         Transition from (0,) to (1,): 1|1,
         Transition from (1,) to (0,): 0|0,
         Transition from (1,) to (1,): 1|1]

`sage.combinat.finite_state_machine.``duplicate_transition_add_input`(old_transition, new_transition)

  Alternative function for handling duplicate transitions in finite state machines. This implementation adds the input label of the new transition to the input label of the old transition. This is intended for the case where a Markov chain is modelled by a finite state machine using the input labels as transition probabilities.

  See the documentation of the `on_duplicate_transition` parameter of `FiniteStateMachine`.

  INPUT:

  - `old_transition` – A transition in a finite state machine.
  - `new_transition` – A transition, identical to `old_transition`, which is to be inserted into the finite state machine.

  OUTPUT:

  A transition whose input weight is the sum of the input weights of `old_transition` and `new_transition`.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import duplicate_transition_add_input
      sage: from sage.combinat.finite_state_machine import FSMTransition
      sage: duplicate_transition_add_input(FSMTransition('a', 'a', 1/2),
      ....:                                FSMTransition('a', 'a', 1/2))
      Transition from 'a' to 'a': 1|-

  Input labels must be lists of length 1:

      sage: duplicate_transition_add_input(FSMTransition('a', 'a', [1, 1]),
      ....:                                FSMTransition('a', 'a', [1, 1]))
      Traceback (most recent call last):
      ...
      TypeError: Trying to use duplicate_transition_add_input on
      "Transition from 'a' to 'a': 1,1|-" and
      "Transition from 'a' to 'a': 1,1|-",
      but input words are assumed to be lists of length 1

`sage.combinat.finite_state_machine.``duplicate_transition_ignore`(old_transition, new_transition)

  Default function for handling duplicate transitions in finite state machines. This implementation ignores the occurrence.

  See the documentation of the `on_duplicate_transition` parameter of `FiniteStateMachine`.

  INPUT:

  - `old_transition` – A transition in a finite state machine.
  - `new_transition` – A transition, identical to `old_transition`, which is to be inserted into the finite state machine.

  OUTPUT:

  The same transition, unchanged.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import duplicate_transition_ignore
      sage: from sage.combinat.finite_state_machine import FSMTransition
      sage: duplicate_transition_ignore(FSMTransition(0, 0, 1),
      ....:                             FSMTransition(0, 0, 1))
      Transition from 0 to 0: 1|-

`sage.combinat.finite_state_machine.``duplicate_transition_raise_error`(old_transition, new_transition)

  Alternative function for handling duplicate transitions in finite state machines. This implementation raises a `ValueError`.

  See the documentation of the `on_duplicate_transition` parameter of `FiniteStateMachine`.

  INPUT:

  - `old_transition` – A transition in a finite state machine.
  - `new_transition` – A transition, identical to `old_transition`, which is to be inserted into the finite state machine.

  OUTPUT:

  Nothing. A `ValueError` is raised.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import duplicate_transition_raise_error
      sage: from sage.combinat.finite_state_machine import FSMTransition
      sage: duplicate_transition_raise_error(FSMTransition(0, 0, 1),
      ....:                                  FSMTransition(0, 0, 1))
      Traceback (most recent call last):
      ...
      ValueError: Attempting to re-insert transition Transition from 0 to 0: 1|-

`sage.combinat.finite_state_machine.``equal`(iterator)

  Checks whether all elements of `iterator` are equal.

  INPUT:

  - `iterator` – an iterator of the elements to check

  OUTPUT:

  `True` or `False`.

  This implements http://stackoverflow.com/a/3844832/1052778.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import equal
      sage: equal([0, 0, 0])
      True
      sage: equal([0, 1, 0])
      False
      sage: equal([])
      True
      sage: equal(iter([None, None]))
      True

  We can test other properties of the elements than the elements themselves. In the following example, we check whether all tuples have the same lengths:

      sage: equal(len(x) for x in [(1, 2), (2, 3), (3, 1)])
      True
      sage: equal(len(x) for x in [(1, 2), (1, 2, 3), (3, 1)])
      False

`sage.combinat.finite_state_machine.``full_group_by`(l, key=<function <lambda> at 0xe4d2e8ec>)

  Group iterable `l` by values of `key`.

  INPUT:

  - iterable `l`
  - key function `key`

  OUTPUT:

  A list of pairs `(k, elements)` such that `key(e)=k` for all `e` in `elements`.

  This is similar to `itertools.groupby` except that lists are returned instead of iterables and no prior sorting is required.

  We do not require

  - that the keys are sortable (in contrast to the approach via `sorted` and `itertools.groupby`) and
  - that the keys are hashable (in contrast to the implementation proposed in http://stackoverflow.com/a/15250161).

  However, it is required

  - that distinct keys have distinct `str`-representations.

  The implementation is inspired by http://stackoverflow.com/a/15250161, but non-hashable keys are allowed.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import full_group_by
      sage: t = [2/x, 1/x, 2/x]
      sage: r = full_group_by([0, 1, 2], key=lambda i:t[i])
      sage: sorted(r, key=lambda p:p[1])
      [(2/x, [0, 2]), (1/x, [1])]
      sage: from itertools import groupby
      sage: for k, elements in groupby(sorted([0, 1, 2],
      ....:                            key=lambda i:t[i]),
      ....:                            key=lambda i:t[i]):
      ....:     print k, list(elements)
      2/x [0]
      1/x [1]
      2/x [2]

  Note that the behavior is different from `itertools.groupby` because neither \(1/x<2/x\) nor \(2/x<1/x\) does hold.

  Here, the result `r` has been sorted in order to guarantee a consistent order for the doctest suite.

`sage.combinat.finite_state_machine.``is_Automaton`(FSM)

  Tests whether or not `FSM` inherits from `Automaton`.

  TESTS:

      sage: from sage.combinat.finite_state_machine import is_FiniteStateMachine, is_Automaton
      sage: is_Automaton(FiniteStateMachine())
      False
      sage: is_Automaton(Automaton())
      True
      sage: is_FiniteStateMachine(Automaton())
      True

`sage.combinat.finite_state_machine.``is_FSMProcessIterator`(PI)

  Tests whether or not `PI` inherits from `FSMProcessIterator`.

  TESTS:

      sage: from sage.combinat.finite_state_machine import is_FSMProcessIterator, FSMProcessIterator
      sage: is_FSMProcessIterator(FSMProcessIterator(FiniteStateMachine([[0, 0, 0, 0]], initial_states=[0]), []))
      True

`sage.combinat.finite_state_machine.``is_FSMState`(S)

  Tests whether or not `S` inherits from `FSMState`.

  TESTS:

      sage: from sage.combinat.finite_state_machine import is_FSMState, FSMState
      sage: is_FSMState(FSMState('A'))
      True

`sage.combinat.finite_state_machine.``is_FSMTransition`(T)

  Tests whether or not `T` inherits from `FSMTransition`.

  TESTS:

      sage: from sage.combinat.finite_state_machine import is_FSMTransition, FSMTransition
      sage: is_FSMTransition(FSMTransition('A', 'B'))
      True

`sage.combinat.finite_state_machine.``is_FiniteStateMachine`(FSM)

  Tests whether or not `FSM` inherits from `FiniteStateMachine`.

  TESTS:

      sage: from sage.combinat.finite_state_machine import is_FiniteStateMachine
      sage: is_FiniteStateMachine(FiniteStateMachine())
      True
      sage: is_FiniteStateMachine(Automaton())
      True
      sage: is_FiniteStateMachine(Transducer())
      True

`sage.combinat.finite_state_machine.``is_Transducer`(FSM)

  Tests whether or not `FSM` inherits from `Transducer`.

  TESTS:

      sage: from sage.combinat.finite_state_machine import is_FiniteStateMachine, is_Transducer
      sage: is_Transducer(FiniteStateMachine())
      False
      sage: is_Transducer(Transducer())
      True
      sage: is_FiniteStateMachine(Transducer())
      True

`sage.combinat.finite_state_machine.``setup_latex_preamble`()

  This function adds the package `tikz` with support for automata to the preamble of Latex so that the finite state machines can be drawn nicely.

  INPUT:

  Nothing.

  OUTPUT:

  Nothing.

  See the section on LaTeX output in the introductory examples of this module.

  TESTS:

      sage: from sage.combinat.finite_state_machine import setup_latex_preamble
      sage: setup_latex_preamble()
      sage: ("\usepackage{tikz}" in latex.extra_preamble()) == latex.has_file("tikz.sty")
      True

`sage.combinat.finite_state_machine.``startswith`(list, prefix)

  Determine whether list starts with the given prefix.

  INPUT:

  - `list` – list
  - `prefix` – list representing the prefix

  OUTPUT:

  `True` or `False`.

  Similar to `str.startswith()`.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import startswith
      sage: startswith([1, 2, 3], [1, 2])
      True
      sage: startswith([1], [1, 2])
      False
      sage: startswith([1, 3, 2], [1, 2])
      False

`sage.combinat.finite_state_machine.``tupleofwords_to_wordoftuples`(tupleofwords)

  Transposes a tuple of words over the alphabet to a word of tuples.

  INPUT:

  - `tupleofwords` – a tuple of a list of letters.

  OUTPUT:

  A list of tuples.

  Missing letters in the words are padded with the letter `None` (from the empty word).

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import (
      ....:     tupleofwords_to_wordoftuples)
      sage: tupleofwords_to_wordoftuples(
      ....:     ([1, 2], [3, 4, 5, 6], [7]))
      [(1, 3, 7), (2, 4, None), (None, 5, None), (None, 6, None)]

`sage.combinat.finite_state_machine.``wordoftuples_to_tupleofwords`(wordoftuples)

  Transposes a word of tuples to a tuple of words over the alphabet.

  INPUT:

  - `wordoftuples` – a list of tuples of letters.

  OUTPUT:

  A tuple of lists.

  Letters `None` (empty word) are removed from each word in the output.

  EXAMPLES:

      sage: from sage.combinat.finite_state_machine import (
      ....:     wordoftuples_to_tupleofwords)
      sage: wordoftuples_to_tupleofwords(
      ....:     [(1, 2), (1, None), (1, None), (1, 2), (None, 2)])
      ([1, 1, 1, 1], [2, 2, 2])

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