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

NAME

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

SYNOPSIS

       package require Tcl  8.4

       package require snit

       package require struct::list

       package require struct::set

       package require grammar::fa::op  ?0.4.1?

       ::grammar::fa::op::constructor cmd

       ::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?

       ::grammar::fa::op::toRegexp fa

       ::grammar::fa::op::toRegexp2 fa

       ::grammar::fa::op::toTclRegexp regexp symdict

       ::grammar::fa::op::simplifyRegexp regexp

_________________________________________________________________________________________________

DESCRIPTION

       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.

API

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::constructor cmd
This command has to be called by the user of the package before any other
operations is performed, to establish a command which can be used to construct a FA
container object. If this is not done several operations will fail as they are
unable to construct internal and transient containers to hold state and/or partial
results.

Any container class using this package for complex operations should set its own
class command as the constructor. See package grammar::fa for an example.

::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.

::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.

Note that this operator accepts zero or more arguments. With zero arguments
the represented language is epsilon, the empty word.

{| 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.

Note that this operator accepts zero or more arguments. With zero arguments
the represented language is the empty language, the language without words.

{& 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.

::grammar::fa::op::toRegexp fa
This command generates and returns a regular expression which accepts the same
language as the finite automaton fa. The regular expression is in the format as
described above, for ::grammar::fa::op::fromRegex.

::grammar::fa::op::toRegexp2 fa
This command has the same functionality as ::grammar::fa::op::toRegexp, but uses a
different algorithm to simplify the generated regular expressions.

::grammar::fa::op::toTclRegexp regexp symdict
This command generates and returns a regular expression in Tcl syntax for the
regular expression regexp, if that is possible. regexp is in the same format as
expected by ::grammar::fa::op::fromRegex.

The command will fail and throw an error if regexp contains complementation and
intersection operations.

The argument symdict is a dictionary mapping symbol names to pairs of syntactic
type and Tcl-regexp. If a symbol occurring in the regexp is not listed in this
dictionary then single-character symbols are considered to designate themselves
whereas multiple-character symbols are considered to be a character class name.

::grammar::fa::op::simplifyRegexp regexp
This command simplifies a regular expression by applying the following algorithm
first to the main expression and then recursively to all sub-expressions:

[1] Convert the expression into a finite automaton.

[2] Minimize the automaton.

[3] Convert the automaton back to a regular expression.

[4] Choose the shorter of original expression and expression from the previous
step.

EXAMPLES

BUGS, IDEAS, FEEDBACK

       This document, and the package it describes,  will  undoubtedly  contain  bugs  and  other
       problems.   Please  report  such  in  the  category  grammar_fa  of  the  Tcllib  Trackers
       [http://core.tcl.tk/tcllib/reportlist].  Please also report any ideas for enhancements you
       may have for either package and/or documentation.

KEYWORDS

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

COPYRIGHT

       Copyright (c) 2004-2008 Andreas Kupries <andreas_kupries@users.sourceforge.net>