FiniteStateMachines

This package provides three classes for different types of finite state machines.

1. FiniteStateMachine, for classical finite state machines over an alphabet in which all letters are weighted equally;
2. WeightedFiniteStateMachine, for finite state machines over an alphabet in which each transition has a weight that is a polynomial in x;
3. CombinatorialFSM, for finite state machines with no alphabet at all, in which transitions between a pair of states are just recorded with a weight that is an expression in x and possibly other variables.

It can be installed via pip with the command pip install finite_state_machines. Questions, comments, and improvements welcome!

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FiniteStateMachine

This is a Python class to perform basic operations on finite state machines, including union, intersection, and minimization.

>>> from finite_state_machines import FiniteStateMachine as FSM

>>> M = FSM.fsm_for_words_avoiding("000", alphabet=["0","1"])
>>> M.enumeration(10)
[1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504]

>>> N = FSM.fsm_for_words_avoiding("101", alphabet=["0","1"])
>>> N.enumeration(10)
[1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351]

>>> M.intersection(N).words_generated(3)
{'001', '010', '011', '100', '110', '111'}

>>> M.intersection(N).enumeration(10)
[1, 2, 4, 6, 9, 13, 19, 28, 41, 60, 88]

>>> M.union(N).enumeration(10)
[1, 2, 4, 8, 16, 32, 62, 118, 222, 414, 767]

WeightedFiniteStateMachine

In a weighted finite state machine, each transition is labeled with a weight that is a polynomial in x. This class has methods to generate words of a certain size ("length" is no longer the correct metric) and to count words of a given size without generating them with a dynamic programming algorithm. Functions are provided to convert back and forth between weighted and non-weighted FSMs. Intersection and union of such machines are not precisely defined and thus not implemented, and there is currently no minimization function.

>>> import sympy
>>> x = sympy.Symbol('x')
>>> from finite_state_machines import WeightedFiniteStateMachine as WFSM
>>> M = WFSM({"a", "b"}, 2, 0, {1}, {(0,"a"):(0,x),(0,"b"):(1,x),(1,"a"):(0,x**2),(1,"b"):(1,x**2)})
>>> [M.words_generated(i) for i in range(5)]
[set(), {'b'}, {'ab'}, {'aab', 'bb'}, {'aaab', 'abb', 'bab'}]

>>> M.enumeration(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

CombinatorialFSM

A combinatorial finite state machine (CFSM) has no alphabet. There are states (which in this implementation must be hashable), a single start state, a set of accepting states, and transitions between pairs of states that are weighted with sympy expressions that involve a main variable (x be default) and possibly other variables.

Construction of a CFSM is a bit different from the previous two classes. After creating the class, you feed it transitions and weights. The weight is added to any existing weight between these two same states.

The class comes with a minimization function and a function to write the transition matrix and solving routine to a Maple file.

>>> import sympy
>>> x, y, C = sympy.symbols('x y C')
>>> from finite_state_machines import CombinatorialFSM
>>> M = CombinatorialFSM()
>>> M.add_transition(0, 1, x*y)
>>> M.add_transition(0, 1, x**2)
>>> M.add_transition(0, 2, x*y**2/C)
>>> M.add_transition(0, 3, x*y**2/C)
>>> M.add_transition(1, 0, x*(1+y/2))
>>> M.add_transition(2, 1, x**2)
>>> M.add_transition(3, 1, x**2)
>>> M.set_start(0)
>>> M.set_accepting({0, 1})
>>> M.enumeration(5)
[1,
 y,
 y**2/2 + y + 1,
 y**3/2 + y**2 + y/2 + 1 + 2*y**2/C,
 y**4/4 + y**3 + 2*y**2 + 2*y + y**3/C + 2*y**2/C,
 y**5/4 + y**4 + 3*y**3/2 + 2*y**2 + 5*y/2 + 1 + 2*y**4/C + 4*y**3/C]

>>> len(M.states)
4

>>> minimized = M.minimize()
>>> len(minimized.states)
3

>>> minimized.enumeration(5)
[1,
 y,
 y**2/2 + y + 1,
 y**3/2 + y**2 + y/2 + 1 + 2*y**2/C,
 y**4/4 + y**3 + 2*y**2 + 2*y + y**3/C + 2*y**2/C,
 y**5/4 + y**4 + 3*y**3/2 + 2*y**2 + 5*y/2 + 1 + 2*y**4/C + 4*y**3/C]

>>> with open('CFSM_example.txt', 'w') as f:
>>>     minimized.write_to_maple_file(f)

The produced Maple file is:

start := 1:
accepting := [1, 2]:
M := Matrix(3,3, storage=sparse):
M[1,2] := -x**2 - x*y:
M[1,3] := -2*x*y**2/C:
M[2,1] := x*(-y/2 - 1):
M[3,2] := -x**2:
M[1,1] := 1 + M[1,1]:
M[2,2] := 1 + M[2,2]:
M[3,3] := 1 + M[3,3]:
V := Vector(LinearAlgebra[Dimensions](M)[1]):
for a in accepting do V[a] := 1: od:
infolevel[solve] := 5:
xvec := LinearAlgebra[LinearSolve](M, V):
f := xvec[1]:

---

If this code is useful to you in your work, please consider citing it.

Bibtex entry:

@misc{FiniteStateMachines,
	author = {Jay Pantone},
	howpublished = {\url{https://github.com/jaypantone/FiniteStateMachines}},
	month = {September},
	note = {DOI: \url{https://doi.org/10.5281/zenodo.4592555}},
	title = {Finite{S}tate{M}achines},
	year = {2021}
}

Biblatex entry:

@software{FiniteStateMachines,
	author = {Jay Pantone},
	date = {2021},
	doi = {10.5281/zenodo.4592555},
	month = {9},
	title = {FiniteStateMachines},
	url = {https://github.com/jaypantone/FiniteStateMachines}
}