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finite_state_machine.py
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finite_state_machine.py
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# -*- coding: utf-8 -*-
r"""
Finite State Machines, Automata, Transducers
This module adds support for finite state machines, automata and
transducers.
For creating automata and transducers you can use classes
- :class:`Automaton` and :class:`Transducer`
(or the more general class :class:`FiniteStateMachine`)
or the generators
- :class:`automata <sage.combinat.finite_state_machine_generators.AutomatonGenerators>` and
:class:`transducers <sage.combinat.finite_state_machine_generators.TransducerGenerators>`
which contain :doc:`preconstructed and commonly used automata and transducers
<finite_state_machine_generators>`. See also the
:ref:`examples <finite_state_machine_examples>` below.
Contents
========
:class:`FiniteStateMachine` and derived classes :class:`Transducer` and :class:`Automaton`
------------------------------------------------------------------------------------------
Accessing parts of a finite state machine
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. csv-table::
:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FiniteStateMachine.state` | Get a state by its label
:meth:`~FiniteStateMachine.states` | List of states
:meth:`~FiniteStateMachine.iter_states` | Iterator over the states
:meth:`~FiniteStateMachine.initial_states` | List of initial states
:meth:`~FiniteStateMachine.iter_initial_states` | Iterator over initial states
:meth:`~FiniteStateMachine.final_states` | List of final states
:meth:`~FiniteStateMachine.iter_final_states` | Iterator over final states
:meth:`~FiniteStateMachine.transition` | Get a transition by its states and labels
:meth:`~FiniteStateMachine.transitions` | List of transitions
:meth:`~FiniteStateMachine.iter_transitions` | Iterator over the transitions
:meth:`~FiniteStateMachine.predecessors` | List of predecessors of a state
:meth:`~FiniteStateMachine.induced_sub_finite_state_machine` | Induced sub-machine
:meth:`~FiniteStateMachine.accessible_components` | Accessible components
:meth:`~FiniteStateMachine.coaccessible_components` | Coaccessible components
:meth:`~FiniteStateMachine.final_components` | Final components (connected components which cannot be left again)
(Modified) Copies
^^^^^^^^^^^^^^^^^
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:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FiniteStateMachine.empty_copy` | Returns an empty deep copy
:meth:`~FiniteStateMachine.deepcopy` | Returns a deep copy
:meth:`~FiniteStateMachine.relabeled` | Returns a relabeled deep copy
:meth:`Automaton.with_output` | Extends an automaton to a transducer
Manipulation
^^^^^^^^^^^^
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:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FiniteStateMachine.add_state` | Add a state
:meth:`~FiniteStateMachine.add_states` | Add states
:meth:`~FiniteStateMachine.delete_state` | Delete a state
:meth:`~FiniteStateMachine.add_transition` | Add a transition
:meth:`~FiniteStateMachine.add_transitions_from_function` | Add transitions
:attr:`~FiniteStateMachine.input_alphabet` | Input alphabet
:attr:`~FiniteStateMachine.output_alphabet` | Output alphabet
:attr:`~FiniteStateMachine.on_duplicate_transition` | Hook for handling duplicate transitions
:meth:`~FiniteStateMachine.add_from_transition_function` | Add transitions by a transition function
:meth:`~FiniteStateMachine.delete_transition` | Delete a transition
:meth:`~FiniteStateMachine.remove_epsilon_transitions` | Remove epsilon transitions (not implemented)
:meth:`~FiniteStateMachine.split_transitions` | Split transitions with input words of length ``> 1``
:meth:`~FiniteStateMachine.determine_alphabets` | Determine input and output alphabets
:meth:`~FiniteStateMachine.determine_input_alphabet` | Determine input alphabet
:meth:`~FiniteStateMachine.determine_output_alphabet` | Determine output alphabet
:meth:`~FiniteStateMachine.construct_final_word_out` | Construct final output by implicitly reading trailing letters; cf. :meth:`~FiniteStateMachine.with_final_word_out`
Properties
^^^^^^^^^^
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:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FiniteStateMachine.has_state` | Checks for a state
:meth:`~FiniteStateMachine.has_initial_state` | Checks for an initial state
:meth:`~FiniteStateMachine.has_initial_states` | Checks for initial states
:meth:`~FiniteStateMachine.has_final_state` | Checks for an final state
:meth:`~FiniteStateMachine.has_final_states` | Checks for final states
:meth:`~FiniteStateMachine.has_transition` | Checks for a transition
:meth:`~FiniteStateMachine.is_deterministic` | Checks for a deterministic machine
:meth:`~FiniteStateMachine.is_complete` | Checks for a complete machine
:meth:`~FiniteStateMachine.is_connected` | Checks for a connected machine
:meth:`Automaton.is_equivalent` | Checks for equivalent automata
:meth:`~FiniteStateMachine.is_Markov_chain` | Checks for a Markov chain
:meth:`~FiniteStateMachine.is_monochromatic` | Checks whether the colors of all states are equal
:meth:`~FiniteStateMachine.number_of_words` | Determine the number of successful paths
:meth:`~FiniteStateMachine.asymptotic_moments` | Main terms of expectation and variance of sums of labels
:meth:`~FiniteStateMachine.moments_waiting_time` | Moments of the waiting time for first true output
:meth:`~FiniteStateMachine.epsilon_successors` | Epsilon successors of a state
:meth:`Automaton.shannon_parry_markov_chain` | Compute Markov chain with Parry measure
Operations
^^^^^^^^^^
.. csv-table::
:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FiniteStateMachine.disjoint_union` | Disjoint union
:meth:`~FiniteStateMachine.concatenation` | Concatenation
:meth:`~FiniteStateMachine.kleene_star` | Kleene star
:meth:`Automaton.complement` | Complement of an automaton
:meth:`Automaton.intersection` | Intersection of automata
:meth:`Transducer.intersection` | Intersection of transducers
:meth:`Transducer.cartesian_product` | Cartesian product of a transducer with another finite state machine
:meth:`~FiniteStateMachine.product_FiniteStateMachine` | Product of finite state machines
:meth:`~FiniteStateMachine.composition` | Composition (output of other is input of self)
:meth:`~FiniteStateMachine.__call__` | Composition with other finite state machine
:meth:`~FiniteStateMachine.input_projection` | Input projection (output is deleted)
:meth:`~FiniteStateMachine.output_projection` | Output projection (old output is new input)
:meth:`~FiniteStateMachine.projection` | Input or output projection
:meth:`~FiniteStateMachine.transposition` | Transposition (all transitions are reversed)
:meth:`~FiniteStateMachine.with_final_word_out` | Machine with final output constructed by implicitly reading trailing letters, cf. :meth:`~FiniteStateMachine.construct_final_word_out` for inplace version
:meth:`Automaton.determinisation` | Determinisation of an automaton
:meth:`~FiniteStateMachine.completion` | Completion of a finite state machine
:meth:`~FiniteStateMachine.process` | Process input
:meth:`~FiniteStateMachine.__call__` | Process input with shortened output
:meth:`Automaton.process` | Process input of an automaton (output differs from general case)
:meth:`Transducer.process` | Process input of a transducer (output differs from general case)
:meth:`~FiniteStateMachine.iter_process` | Return process iterator
:meth:`~FiniteStateMachine.language` | Return all possible output words
:meth:`Automaton.language` | Return all possible accepted words
Simplification
^^^^^^^^^^^^^^
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:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FiniteStateMachine.prepone_output` | Prepone output where possible
:meth:`~FiniteStateMachine.equivalence_classes` | List of equivalent states
:meth:`~FiniteStateMachine.quotient` | Quotient with respect to equivalence classes
:meth:`~FiniteStateMachine.merged_transitions` | Merge transitions while adding input
:meth:`~FiniteStateMachine.markov_chain_simplification` | Simplification of a Markov chain
:meth:`Automaton.minimization` | Minimization of an automaton
:meth:`Transducer.simplification` | Simplification of a transducer
Conversion
^^^^^^^^^^
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:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FiniteStateMachine.adjacency_matrix` | (Weighted) adjacency :func:`matrix`
:meth:`~FiniteStateMachine.graph` | Underlying :class:`DiGraph`
:meth:`~FiniteStateMachine.plot` | Plot
LaTeX output
++++++++++++
.. csv-table::
:class: contentstable
:widths: 30, 70
:delim: |
:meth:`~FiniteStateMachine.latex_options` | Set options
:meth:`~FiniteStateMachine.set_coordinates` | Set coordinates of the states
:meth:`~FiniteStateMachine.default_format_transition_label` | Default formatting of words in transition labels
:meth:`~FiniteStateMachine.format_letter_negative` | Format negative numbers as overlined number
:meth:`~FiniteStateMachine.format_transition_label_reversed` | Format words in transition labels in reversed order
.. SEEALSO::
:ref:`finite_state_machine_LaTeX_output`
:class:`FSMState`
-----------------
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:class: contentstable
:widths: 30, 70
:delim: |
:attr:`~FSMState.final_word_out` | Final output of a state
:attr:`~FSMState.is_final` | Describes whether a state is final or not
:attr:`~FSMState.is_initial` | Describes whether a state is initial or not
:attr:`~FSMState.initial_probability` | Probability of starting in this state as part of a Markov chain
:meth:`~FSMState.label` | Label of a state
:meth:`~FSMState.relabeled` | Returns a relabeled deep copy of a state
:meth:`~FSMState.fully_equal` | Checks whether two states are fully equal (including all attributes)
:class:`FSMTransition`
----------------------
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:class: contentstable
:widths: 30, 70
:delim: |
:attr:`~FSMTransition.from_state` | State in which transition starts
:attr:`~FSMTransition.to_state` | State in which transition ends
:attr:`~FSMTransition.word_in` | Input word of the transition
:attr:`~FSMTransition.word_out` | Output word of the transition
:meth:`~FSMTransition.deepcopy` | Returns a deep copy of the transition
:class:`FSMProcessIterator`
---------------------------
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:class: contentstable
:widths: 30, 70
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:meth:`~FSMProcessIterator.next` | Makes one step in processing the input tape
:meth:`~FSMProcessIterator.preview_word` | Reads a word from the input tape
:meth:`~FSMProcessIterator.result` | Returns the finished branches during process
Helper Functions
----------------
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:class: contentstable
:widths: 30, 70
:delim: |
:func:`equal` | Checks whether all elements of ``iterator`` are equal
:func:`full_group_by` | Group iterable by values of some key
:func:`startswith` | Determine whether list starts with the given prefix
:func:`FSMLetterSymbol` | Returns a string associated to the input letter
:func:`FSMWordSymbol` | Returns a string associated to a word
:func:`is_FSMState` | Tests whether an object inherits from :class:`FSMState`
:func:`is_FSMTransition` | Tests whether an object inherits from :class:`FSMTransition`
:func:`is_FiniteStateMachine` | Tests whether an object inherits from :class:`FiniteStateMachine`
:func:`duplicate_transition_ignore` | Default function for handling duplicate transitions
:func:`duplicate_transition_raise_error` | Raise error when inserting a duplicate transition
:func:`duplicate_transition_add_input` | Add input when inserting a duplicate transition
.. _finite_state_machine_examples:
Examples
========
We start with a general :class:`FiniteStateMachine`. Later there will
be also an :class:`Automaton` and a :class:`Transducer`.
A simple finite state machine
-----------------------------
We can easily create a finite state machine by
::
sage: fsm = FiniteStateMachine()
sage: fsm
Empty finite state machine
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 :class:`directed graph <DiGraph>` by
::
sage: fsm.graph()
Looped multi-digraph on 2 vertices
To visualize a finite state machine, we can use
:func:`~sage.misc.latex.latex` and run the result through LaTeX,
see the section on :ref:`finite_state_machine_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 :class:`FiniteStateMachine` for a lot of possibilities to create
finite state machines.
.. _finite_state_machine_recognizing_NAFs_example:
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: 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.
We can also let an automaton act on a :doc:`word <words/words>`::
sage: W = Words([-1, 0, 1]); W
Finite and infinite words over {-1, 0, 1}
sage: w = W([1, 0, 1, 0, -1]); w
word: 1,0,1,0,-1
sage: NAF(w)
True
Recognizing NAFs via Automata Operations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Alternatively, we can use automata operations to recognize NAFs; for
simplicity, we only use the input alphabet ``[0, 1]``. On the one
hand, we can construct such an automaton by forbidding the word
``11``::
sage: forbidden = automata.ContainsWord([1, 1], input_alphabet=[0, 1])
sage: NAF_negative = forbidden.complement()
sage: NAF_negative([1, 1, 0, 1])
False
sage: NAF_negative([1, 0, 1, 0, 1])
True
On the other hand, we can write this as a regular expression and
translate that into automata operations::
sage: zero = automata.Word([0])
sage: one = automata.Word([1])
sage: epsilon = automata.EmptyWord(input_alphabet=[0, 1])
sage: NAF_positive = (zero + one*zero).kleene_star() * (epsilon + one)
We check that the two approaches are equivalent::
sage: NAF_negative.is_equivalent(NAF_positive)
True
.. SEEALSO::
:meth:`~sage.combinat.finite_state_machine_generators.AutomatonGenerators.ContainsWord`,
:meth:`~sage.combinat.finite_state_machine_generators.AutomatonGenerators.Word`,
:meth:`~Automaton.complement`,
:meth:`~FiniteStateMachine.kleene_star`,
:meth:`~sage.combinat.finite_state_machine_generators.AutomatonGenerators.EmptyWord`,
:meth:`~Automaton.is_equivalent`.
.. _finite_state_machine_LaTeX_output:
LaTeX output
------------
We can visualize a finite state machine by converting it to LaTeX by
using the usual function :func:`~sage.misc.latex.latex`. Within LaTeX,
TikZ is used for typesetting the graphics, see the
:wikipedia:`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
:meth:`~FiniteStateMachine.latex_options` for details. In particular,
we use :meth:`~FiniteStateMachine.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
To use the output of :func:`~sage.misc.latex.latex` in your own
`\LaTeX` file, you have to include
.. code-block:: latex
\usepackage{tikz}
\usetikzlibrary{automata}
into the preamble of your file.
A simple transducer (binary inverter)
-------------------------------------
Let's build a simple transducer, which rewrites a binary word by
inverting 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.
.. _finite_state_machine_division_by_3_example:
Transducers and (in)finite Words
--------------------------------
A transducer can also act on everything iterable, in particular, on
Sage's :doc:`words <words/words>`.
::
sage: W = Words([0, 1]); W
Finite and infinite words over {0, 1}
Let us take the inverter from the previous section and feed some
finite word into it::
sage: w = W([1, 1, 0, 1]); w
word: 1101
sage: inverter(w)
word: 0010
We see that the output is again a word (this is a consequence of
calling :meth:`~Transducer.process` with ``automatic_output_type``).
We can even input something infinite like an infinite word::
sage: tm = words.ThueMorseWord(); tm
word: 0110100110010110100101100110100110010110...
sage: inverter(tm)
word: 1001011001101001011010011001011001101001...
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`.
.. _finite_state_machine_gray_code_example:
Gray Code
---------
The Gray code is a binary :wikipedia:`numeral system <Numeral_system>`
where two successive values differ in only one bit, cf. the
:wikipedia:`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: 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
:meth:`Cartesian product <Transducer.cartesian_product>` between the
shifted version and the original input (represented here by the
``shift_right_transducer`` and the :meth:`identity transducer
<sage.combinat.finite_state_machine_generators.TransducerGenerators.Identity>`,
respectively). This Cartesian product is then fed into the
``xor_transducer`` as a :meth:`composition
<FiniteStateMachine.composition>` of transducers.
::
sage: product_transducer = shift_right_transducer.cartesian_product(transducers.Identity([0, 1]))
sage: Gray_transducer = xor_transducer(product_transducer)
We use :meth:`~FiniteStateMachine.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 :meth:`prepackaged transducer
<sage.combinat.finite_state_machine_generators.TransducerGenerators.GrayCode>`
for Gray code, let's see whether they agree. We have to use
:meth:`~FiniteStateMachine.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 :ref:`previous example "division by
3" <finite_state_machine_division_by_3_example>` 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:`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 :meth:`Automaton.process` and
:meth:`Transducer.process` (or even
:meth:`FiniteStateMachine.process`), the explanation of the parameter
``hook`` and the examples in :class:`FSMState` and
:class:`FSMTransition`, and the description and examples in
:class:`FSMProcessIterator` for more information on processing and
hooks.
REFERENCES:
.. [HKP2015] Clemens Heuberger, Sara Kropf, and Helmut Prodinger,
*Output sum of transducers: Limiting distribution and periodic
fluctuation*,
`Electron. J. Combin. 22 (2015), #P2.19 <http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p19>`_.
.. [HKW2015] Clemens Heuberger, Sara Kropf and Stephan Wagner,
*Variances and Covariances in the Central Limit Theorem for the Output
of a Transducer*, European J. Combin. 49 (2015), 167-187,
:doi:`10.1016/j.ejc.2015.03.004`.
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
:trac:`15841`, :trac:`15847`, :trac:`15848`, :trac:`15849`, :trac:`15850`, :trac:`15922`, :trac:`15923`, :trac:`15924`,
:trac:`15925`, :trac:`15928`, :trac:`15960`, :trac:`15961`, :trac:`15962`, :trac:`15963`, :trac:`15975`, :trac:`16016`,
:trac:`16024`, :trac:`16061`, :trac:`16128`, :trac:`16132`, :trac:`16138`, :trac:`16139`, :trac:`16140`, :trac:`16143`,
:trac:`16144`, :trac:`16145`, :trac:`16146`, :trac:`16191`, :trac:`16200`, :trac:`16205`, :trac:`16206`, :trac:`16207`,
:trac:`16229`, :trac:`16253`, :trac:`16254`, :trac:`16255`, :trac:`16266`, :trac:`16355`, :trac:`16357`, :trac:`16387`,
:trac:`16425`, :trac:`16539`, :trac:`16555`, :trac:`16557`, :trac:`16588`, :trac:`16589`, :trac:`16666`, :trac:`16668`,
:trac:`16674`, :trac:`16675`, :trac:`16677`.
- Daniel Krenn (2015-09-14): cleanup :trac:`18227`
ACKNOWLEDGEMENT:
- Clemens Heuberger, Daniel Krenn and Sara Kropf are supported by the
Austrian Science Fund (FWF): P 24644-N26.
Methods
=======
"""
#*****************************************************************************
# Copyright (C) 2012--2015 Clemens Heuberger <clemens.heuberger@aau.at>
# 2012--2015 Daniel Krenn <dev@danielkrenn.at>
# 2012--2015 Sara Kropf <sara.kropf@aau.at>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# http://www.gnu.org/licenses/
#*****************************************************************************
from __future__ import print_function
from builtins import zip
import six
from six.moves import range
from six import itervalues
from six.moves import zip_longest
import collections
import itertools
import sage
def full_group_by(l, key=lambda x: x):
"""
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 `<https://stackoverflow.com/a/15250161>`_).
However, it is required
- that distinct keys have distinct ``str``-representations.
The implementation is inspired by
`<https://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("{} {}".format(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.
"""
elements = collections.defaultdict(list)