grammar::fa::op - Operations on finite automatons

package require Tcl 8.4 package require snit package require struct::list package require struct::set package require grammar::fa::op ?0.1? ::grammar::fa::op::reverse fa ::grammar::fa::op::complete fa ?sink? ::grammar::fa::op::remove_eps fa ::grammar::fa::op::trim fa ?what? ::grammar::fa::op::determinize fa ?mapvar? ::grammar::fa::op::minimize fa ?mapvar? ::grammar::fa::op::complement fa ::grammar::fa::op::kleene fa ::grammar::fa::op::optional fa ::grammar::fa::op::union fa fb ?mapvar? ::grammar::fa::op::intersect fa fb ?mapvar? ::grammar::fa::op::difference fa fb ?mapvar? ::grammar::fa::op::concatenate fa fb ?mapvar? ::grammar::fa::op::fromRegex fa regex ?over?

This package provides a number of complex operations on finite automatons (Short: FA), as provided by the package grammar::fa. The package does not provide the ability to create and/or manipulate such FAs, nor the ability to execute a FA for a stream of symbols. Use the packages grammar::fa and grammar::fa::interpreter for that. Another package related to this is grammar::fa::compiler which turns a FA into an executor class which has the definition of the FA hardwired into it.

For more information about what a finite automaton is see section FINITE AUTOMATONS in package grammar::fa.

The package exports the API described here. All commands modify their first argument. I.e. whatever FA they compute is stored back into it. Some of the operations will construct an automaton whose states are all new, but related to the states in the source automaton(s). These operations take variable names as optional arguments where they will store mappings which describe the relationship(s). The operations can be loosely partitioned into structural and language operations. The latter are defined in terms of the language the automaton(s) accept, whereas the former are defined in terms of the structural properties of the involved automaton(s). Some operations are both. Structure operations

::grammar::fa::op::reverse fa

Reverses the fa. This is done by reversing the direction of all transitions and swapping the sets of start and final states. The language of fa changes unpredictably.

::grammar::fa::op::complete fa ?sink?

Completes the fa complete, but nothing is done if the fa is already complete. This implies that only the first in a series of multiple consecutive complete operations on fa will perform anything. The remainder will be null operations. The language of fa is unchanged by this operation. This is done by adding a single new state, the sink, and transitions from all other states to that sink for all symbols they have no transitions for. The sink itself is made complete by adding loop transitions for all symbols. Note: When a FA has epsilon-transitions transitions over a symbol for a state S can be indirect, i.e. not attached directly to S, but to a state in the epsilon-closure of S. The symbols for such indirect transitions count when computing completeness of a state. In other words, these indirectly reached symbols are not missing. The argument sink provides the name for the new state and most not be present in the fa if specified. If the name is not specified the command will name the state "sinkn", where n is set so that there are no collisions with existing states. Note that the sink state is not useful by definition. In other words, while the FA becomes complete, it is also not useful in the strict sense as it has a state from which no final state can be reached.

::grammar::fa::op::remove_eps fa

Removes all epsilon-transitions from the fa in such a manner the the language of fa is unchanged. However nothing is done if the fa is already epsilon-free. This implies that only the first in a series of multiple consecutive complete operations on fa will perform anything. The remainder will be null operations. Note: This operation may cause states to become unreachable or not useful. These states are not removed by this operation. Use ::grammar::fa::op::trim for that instead.

::grammar::fa::op::trim fa ?what?

Removes unwanted baggage from fa. The legal values for what are listed below. The command defaults to !reachable|!useful if no specific argument was given.

!reachable

Removes all states which are not reachable from a start state.

!useful

Removes all states which are unable to reach a final state.

!reachable&!useful

!(reachable|useful)

Removes all states which are not reachable from a start state and are unable to reach a final state.

!reachable|!useful

!(reachable&useful)

Removes all states which are not reachable from a start state or are unable to reach a final state.

::grammar::fa::op::determinize fa ?mapvar?

Makes the fa deterministic without changing the language accepted by the fa. However nothing is done if the fa is already deterministic. This implies that only the first in a series of multiple consecutive complete operations on fa will perform anything. The remainder will be null operations. The command will store a dictionary describing the relationship between the new states of the resulting dfa and the states of the input nfa in mapvar, if it has been specified. Keys of the dictionary are the handles for the states of the resulting dfa, values are sets of states from the input nfa. Note: An empty dictionary signals that the command was able to make the fa deterministic without performing a full subset construction, just by removing states and shuffling transitions around (As part of making the FA epsilon-free). Note: The algorithm fails to make the FA deterministic in the technical sense if the FA has no start state(s), because determinism requires the FA to have exactly one start states. In that situation we make a best effort; and the missing start state will be the only condition preventing the generated result from being deterministic. It should also be noted that in this case the possibilities for trimming states from the FA are also severely reduced as we cannot declare states unreachable.

::grammar::fa::op::minimize fa ?mapvar?

Creates a FA which accepts the same language as fa, but has a minimal number of states. Uses Brzozowski's method to accomplish this. The command will store a dictionary describing the relationship between the new states of the resulting minimal fa and the states of the input fa in mapvar, if it has been specified. Keys of the dictionary are the handles for the states of the resulting minimal fa, values are sets of states from the input fa. Note: An empty dictionary signals that the command was able to minimize the fa without having to compute new states. This should happen if and only if the input FA was already minimal. Note: If the algorithm has no start or final states to work with then the result might be technically minimal, but have a very unexpected structure. It should also be noted that in this case the possibilities for trimming states from the FA are also severely reduced as we cannot declare states unreachable. Language operations All operations in this section require that all input FAs have at least one start and at least one final state. Otherwise the language of the FAs will not be defined, making the operation senseless (as it operates on the languages of the FAs in a defined manner).

::grammar::fa::op::complement fa

Complements fa. This is possible if and only if fa is complete and deterministic. The resulting FA accepts the complementary language of fa. In other words, all inputs not accepted by the input are accepted by the result, and vice versa. The result will have all states and transitions of the input, and different final states.

::grammar::fa::op::kleene fa

Applies Kleene's closure to fa. The resulting FA accepts all strings S for which we can find a natural number n (0 inclusive) and strings A1 ... An in the language of fa such that S is the concatenation of A1 ... An. In other words, the language of the result is the infinite union over finite length concatenations over the language of fa. The result will have all states and transitions of the input, and new start and final states.

::grammar::fa::op::optional fa

Makes the fa optional. In other words it computes the FA which accepts the language of fa and the empty the word (epsilon) as well. The result will have all states and transitions of the input, and new start and final states.

::grammar::fa::op::union fa fb ?mapvar?

Combines the FAs fa and fb such that the resulting FA accepts the union of the languages of the two FAs. The result will have all states and transitions of the two input FAs, and new start and final states. All states of fb which exist in fa as well will be renamed, and the mapvar will contain a mapping from the old states of fb to the new ones, if present. It should be noted that the result will be non-deterministic, even if the inputs are deterministic.

::grammar::fa::op::intersect fa fb ?mapvar?

Combines the FAs fa and fb such that the resulting FA accepts the intersection of the languages of the two FAs. In other words, the result will accept a word if and only if the word is accepted by both fa and fb. The result will be useful, but not necessarily deterministic or minimal. The command will store a dictionary describing the relationship between the new states of the resulting fa and the pairs of states of the input FAs in mapvar, if it has been specified. Keys of the dictionary are the handles for the states of the resulting fa, values are pairs of states from the input FAs. Pairs are represented by lists. The first element in each pair will be a state in fa, the second element will be drawn from fb.

::grammar::fa::op::difference fa fb ?mapvar?

Combines the FAs fa and fb such that the resulting FA accepts the difference of the languages of the two FAs. In other words, the result will accept a word if and only if the word is accepted by fa, but not by fb. This can also be expressed as the intersection of fa with the complement of fb. The result will be useful, but not necessarily deterministic or minimal. The command will store a dictionary describing the relationship between the new states of the resulting fa and the pairs of states of the input FAs in mapvar, if it has been specified. Keys of the dictionary are the handles for the states of the resulting fa, values are pairs of states from the input FAs. Pairs are represented by lists. The first element in each pair will be a state in fa, the second element will be drawn from fb.

::grammar::fa::op::concatenate fa fb ?mapvar?

Combines the FAs fa and fb such that the resulting FA accepts the cross-product of the languages of the two FAs. I.e. a word W will be accepted by the result if there are two words A and B accepted by fa, and fb resp. and W is the concatenation of A and B. The result FA will be non-deterministic.

::grammar::fa::op::fromRegex fa regex ?over?

Generates a non-deterministic FA which accepts the same language as the regular expression regex. If the over is specified it is treated as the set of symbols the regular expression and the automaton are defined over. The command will compute the set from the "S" constructors in regex when over was not specified. This set is important if and only if the complement operator "!" is used in regex as the complementary language of an FA is quite different for different sets of symbols. The regular expression is represented by a nested list, which forms a syntax tree. The following structures are legal:

{S x}

Atomic regular expression. Everything else is constructed from these. Accepts the Symbol "x".

{. A1 A2 ...}

Concatenation operator. Accepts the concatenation of the regular expressions A1, A2, etc.

{| A1 A2 ...}

Choice operator, also called "Alternative". Accepts all input accepted by at least one of the regular expressions A1, A2, etc. In other words, the union of A1, A2.

{& A1 A2 ...}

Intersection operator, logical and. Accepts all input accepted which is accepted by all of the regular expressions A1, A2, etc. In other words, the intersection of A1, A2.

{? A}

Optionality operator. Accepts the empty word and anything from the regular expression A.

{* A}

Kleene closure. Accepts the empty word and any finite concatenation of words accepted by the regular expression A.

{+ A}

Positive Kleene closure. Accepts any finite concatenation of words accepted by the regular expression A, but not the empty word.

{! A}

Complement operator. Accepts any word not accepted by the regular expression A. Note that the complement depends on the set of symbol the result should run over. See the discussion of the argument over before.

automaton, finite automaton, grammar, parsing, regular expression, regular grammar, regular languages, state, transducer

Copyright (c) 2004 Andreas Kupries <andreas_kupries@users.sourceforge.net>