FSA

This module defines an FSA class, for representing and operating on finite-state automata (FSAs). FSAs can be used to represent regular expressions and to test sequences for membership in the languages described by regular expressions.

FSAs can be deterministic or nondeterministic, and they can contain epsilon transitions. Methods to determinize an automaton (also eliminating its epsilon transitions), and to minimize an automaton, are provided.

The transition labels for an FSA can be symbols from an alphabet, as in the standard formal definition of an FSA, but they can also be instances which represent predicates. If these instances implement instance.matches(), then the FSA nextState() function and accepts() predicate can be used. If they implement instance.complement() and instance.intersection(), the FSA can be be determinized and minimized, to find a minimal deterministic FSA that accepts an equivalent language.

Quick Start

Instances of FSA can be created out of labels (for instance, strings) by the singleton() function, and combined to create more complex FSAs through the complement(), closure(), concatenation(), union(), and other constructors. For example, concatenation(singleton('a'), union(singleton('b'), closure(singleton('c')))) creates an FSA that accepts the strings 'a', 'ab', 'ac', 'acc', 'accc', and so on.

Instances of FSA can also be created with the compileRE() function, which compiles a simple regular expression (using only '*', '?', '+', '|', '(', and ')' as metacharacters) into an FSA. For example, compileRE('a(b|c*)') returns an FSA equivalent to the example in the previous paragraph.

FSAs can be determinized, to create equivalent FSAs (FSAs accepting the same language) with unique successor states for each input, and minimized, to create an equivalent deterministic FSA with the smallest number of states. FSAs can also be complemented, intersected, unioned, and so forth as described under 'FSA Functions' below.

FSA Methods

The class FSA defines the following methods.

Acceptance

fsa.nextStates(state, input): returns a list of states
fsa.nextState(state, input): returns None or a single state if nextStates <= 1, otherwise it raises an exception
fsa.nextStateSet(states, input): returns a list of states
fsa.accepts(sequence): returns true or false

Accessors and predicates

isEmpty(): returns true iff the language accepted by the FSA is the empty language
labels(): returns a list of labels that are used in any transition
nextAvailableState(): returns an integer n such that no states in the FSA are numeric values >= n

Reductions

sorted(initial=0): returns an equivalent FSA whose states are numbered upwards from 0
determinized(): returns an equivalent deterministic FSA
minimized(): returns an equivalent minimal FSA
trimmed(): returns an equivalent FSA that contains no unreachable or dead states

Presentation

toDotString(): returns a string suitable as *.dot file for the 'dot' program from AT&T GraphViz
view(): views the FSA with a gs viewer, if gs and dot are installed

FSA Functions

Construction from FSAs

complement(a): returns an fsa that accepts exactly those sequences that its argument does not
closure(a): returns an fsa that accepts sequences composed of zero or more concatenations of sequences accepted by the argument
concatenation(a, b): returns an fsa that accepts sequences composed of a sequence accepted by a, followed by a sequence accepted by b
containment(a, occurrences=1): returns an fsa that accepts sequences that contain at least occurrences occurrences of a subsequence recognized by the argument.
difference(a, b): returns an fsa that accepts those sequences accepted by a but not b
intersection(a, b): returns an fsa that accepts sequences accepted by both a and b
iteration(a, min=1, max=None): returns an fsa that accepts sequences consisting of from min to max (or any number, if max is None) of sequences accepted by its first argument
option(a): equivalent to union(a, EMPTY_STRING_FSA)
reverse(a): returns an fsa that accepts strings whose reversal is accepted by the argument
union(a, b): returns an fsa that accepts sequences accepted by both a and b

Predicates

equivalent(a, b): returns true iff a and b accept the same language

Reductions (these equivalent to the similarly-named methods)

determinize(fsa): returns an equivalent deterministic FSA
minimize(fsa): returns an equivalent minimal FSA
sort(fsa, initial=0): returns an equivalent FSA whose states are numbered from initial
trim(fsa): returns an equivalent FSA that contains no dead or unreachable states

Construction from labels

compileRE(string): returns an FSA that accepts the language described by string, where string is a list of symbols and '*', '+', '?', and '|' operators, with '(' and ')' to control precedence.
sequence(sequence): returns an fsa that accepts sequences that are matched by the elements of the argument. For example, sequence('abc') returns an fsa that accepts 'abc' and ['a', 'b', 'c'].
singleton(label): returns an fsa that accepts singletons whose elements are matched by label. For example, singleton('a') returns an fsa that accepts only the string 'a'.

FSA Constants

EMPTY_STRING_FSA is an FSA that accepts the language consisting only of the empty string.

NULL_FSA is an FSA that accepts the null language.

UNIVERSAL_FSA is an FSA that accepts S*, where S is any object.

FSA instance creation

FSA is initialized with a list of states, an alphabet, a list of transition, an initial state, and a list of final states. If fsa is an FSA, fsa.tuple() returns these values in that order, i.e. (states, alphabet, transitions, initialState, finalStates). They're also available as fields of fsa with those names.

Each element of transition is a tuple of a start state, an end state, and a label: (startState, endSTate, label).

If the list of states is None, it's computed from initialState, finalStates, and the states in transitions.

If alphabet is None, an open alphabet is used: labels are assumed to be objects that implements label.matches(input), label.complement(), and label.intersection() as follows:

label.matches(input) returns true iff label matches input

label.complement() returnseither a label or a list of labels which,

together with the receiver, partition the input alphabet

label.intersection(other) returns either None (if label and other don't

both match any symbol), or a label that matches the set of symbols that both label and other match

As a special case, strings can be used as labels. If a strings 'a' and 'b' are used as a label and there's no alphabet, '~a' and '~b' are their respective complements, and '~a&~b' is the intersection of '~a' and '~b'. (The intersections of 'a' and 'b', 'a' and '~b', and '~a' and 'b' are, respectively, None, 'a', and 'b'.)

Goals

Design Goals:

easy to use
easy to read (simple implementation, direct expression of algorithms)
extensible

Non-Goals:

efficiency