DFA

A simple python implementation of a DFA.

Table of Contents

• Installation
• Usage
  • Membership Queries
  • Transitions and Traces
  • Non-boolean output alphabets
  • Moore Machines
  • DFA <-> Dictionary
  • Computing Reachable States
  • Sampling Paths
  • Running interactively (Co-Routine API)
  • Visualizing DFAs

Features:

1. State can be any Hashable object.
2. Alphabet can be any finite sequence of Hashable objects.
3. Designed to be immutable and hashable (assuming components are immutable and hashable).
4. Design choice to allow transition map and accepting set to be given as functions rather than an explicit dict or set.

Installation

If you just need to use dfa, you can just run:

$ pip install dfa

For developers, note that this project uses the poetry python package/dependency management tool. Please familarize yourself with it and then run:

$ poetry install

Usage

The dfa api is centered around the DFA object.

By default, the DFA object models a Deterministic Finite Acceptor, e.g., a recognizer of a Regular Language.

Example Usage:

from dfa import DFA

dfa1 = DFA(
    start=0,
    inputs={0, 1},
    label=lambda s: (s % 4) == 3,
    transition=lambda s, c: (s + c) % 4,
)

dfa2 = DFA(
    start="left",
    inputs={"move right", "move left"},
    label=lambda s: s == "left",
    transition=lambda s, c: "left" if c == "move left" else "right",
)

Membership Queries

assert dfa1.label([1, 1, 1, 1])
assert not dfa1.label([1, 0])

assert dfa2.label(["move right"]*100 + ["move left"])
assert not dfa2.label(["move left", "move right"])

Transitions and Traces

assert dfa1.transition([1, 1, 1]) == 3
assert list(dfa1.trace([1, 1, 1])) == [0, 1, 2, 3]

Non-boolean output alphabets

Sometimes, it is useful to model an automata which can label a word using a non-Boolean alphabet. For example, {True, False, UNSURE}.

The DFA object supports this by specifying the output alphabet.

UNSURE = None

def my_labeler(s):
    if s % 4 == 2:
       return None
    return (s % 4) == 3


dfa3 = DFA(
    start=0,
    inputs={0, 1},
    label=my_labeler,
    transition=lambda s, c: (s + c) % 4,
    outputs={True, False, UNSURE},
)

Note: If outputs is set to None, then no checks are done that the outputs are within the output alphabet.

dfa3 = DFA(
    start=0,
    inputs={0, 1},
    label=my_labeler,
    transition=lambda s, c: (s + c) % 4,
    outputs=None,
)

Moore Machines

Finally, by reinterpreting the structure of the DFA object, one can model a Moore Machine. For example, in 3 state counter, dfa1, the Moore Machine can output the current count.

assert dfa1.transduce(()) == ()
assert dfa1.transduce((1,)) == (False,)
assert dfa1.transduce((1, 1, 1, 1)) == (False, False, False, True)

Language Queries

Utility functions are available for testing if a language:

1. Is empty: utils.find_word
2. Is equivilent to another language: utils.find_equiv_counterexample
3. Is a subset of a another language: utils.find_subset_counterexample

These operate by returning None if the property holds, i.e., lang(dfa1) = ∅, lang(dfa1) ≡ lang(dfa2), lang(dfa1) ⊆ lang(dfa2), and returning a counterexample Word otherwise.

DFA <-> Dictionary

Note that dfa provides helper functions for going from a dictionary based representation of a deterministic transition system to a DFA object and back.

from dfa import dfa2dict, dict2dfa

# DFA encoded a nested dictionaries with the following
# signature.
#     <state>: (<label>, {<action>: <next state>})

dfa_dict = {
    0: (False, {0: 0, 1: 1}),
    1: (False, {0: 1, 1: 2}),
    2: (False, {0: 2, 1: 3}), 
    3: (True, {0: 3, 1: 0})
}

# Dictionary -> DFA
dfa = dict2dfa(dfa_dict, start=0)

# DFA -> Dictionary
dfa_dict2, start = dfa2dict(dfa)

assert (dfa_dict, 0) == (dfa_dict2, start)

Computing Reachable States

# Perform a depth first traversal to collect all reachable states.
assert dfa1.states() == {0, 1, 2, 3}

Finding Words and Access strings

To generate accepting strings (words) in a DFA (breadth first using string length) one can use the dfa.utils.words function:

from dfa.utils.import dfa2dict, words, find_words

dfa_dict = {
    0: (False, {0: 0, 1: 1}),
    1: (False, {0: 1, 1: 2}),
    2: (False, {0: 2, 1: 3}),
    3: (True, {0: 3, 1: 0})
}
lang = dict2dfa(dfa_dict, start=0)

xs = set(fn.take(5, words(lang)))
assert len(xs) == 5
assert all(lang.label(x) for x in xs)

To get a single word, a helper function is provided in dfa.utils.find_word which returns None if the language of the DFA is empty:

# ... Same as above ...

x = find_word(lang)
assert x is not None
assert lang.label(x)

Often times, it is useful to sample a path between two states, say a and b. dfa supports this using dfa.utils.paths. This function returns a generator of words, w, such that dfa.transition(w, start=b) == a. For example:

from dfa.utils import paths

access_strings = paths(
    dfa1, 
    start=0,
    end=1,  # Optional. If not specified returns all paths
            # starting at `start`.
    max_length=7,  #  Defaults to float('inf')
    randomize=True,  #  Randomize the order. Shorter paths still found first.
)

for word in access_strings:
    assert dfa1.transition(word, start=0) == 1

DFA minimization

DFAs can be minimized using the minimize method.

my_dfa = my_dfa.minimize()

DFA advancement (progression)

One can create the DFA starting at the state indexed by a given word by using the advance method.

my_dfa = my_dfa.advance(word)

Running interactively (Co-Routine API)

dfa supports interactively stepping through a DFA object via co-routines. This is particularly useful when using DFA in a control loop. For example, the following code counts how many 1's it takes to advance dfa1's state back to the start state.

machine = dfa1.run()

next(machine)
state = None

count = 0
while state != dfa1.start:
    count += 1
    state = machine.send(1)

Visualizing DFAs

dfa optionally supports visualizing DFAs using graphviz. To use this functionality be sure to install dfa using with the draw option:

pip install dfa[draw]

or

poetry install -E draw

Then one can simply use dfa.draw.write_dot to write a .dot file representing the DFA. This .dot file can be rendered using any graphviz supporting tool.

from dfa.draw import write_dot

write_dot(dfa1, "path/to/dfa1.dot")

Using the dot command in linux results in the following rendering of dfa1.

$ dot -Tsvg path/to/dfa1.dot > dfa1.svg