'\"
'\" Generated from file 'faop\&.man' by tcllib/doctools with format 'nroff'
'\" Copyright (c) 2004-2008 Andreas Kupries <andreas_kupries@users\&.sourceforge\&.net>
'\"
.TH "grammar::fa::op" n 0\&.4 tcllib "Finite automaton operations and usage"
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.BS
.SH NAME
grammar::fa::op \- Operations on finite automatons
.SH SYNOPSIS
package require \fBTcl  8\&.4\fR
.sp
package require \fBsnit \fR
.sp
package require \fBstruct::list \fR
.sp
package require \fBstruct::set \fR
.sp
package require \fBgrammar::fa::op  ?0\&.4\&.1?\fR
.sp
\fB::grammar::fa::op::constructor\fR \fIcmd\fR
.sp
\fB::grammar::fa::op::reverse\fR \fIfa\fR
.sp
\fB::grammar::fa::op::complete\fR \fIfa\fR ?\fIsink\fR?
.sp
\fB::grammar::fa::op::remove_eps\fR \fIfa\fR
.sp
\fB::grammar::fa::op::trim\fR \fIfa\fR ?\fIwhat\fR?
.sp
\fB::grammar::fa::op::determinize\fR \fIfa\fR ?\fImapvar\fR?
.sp
\fB::grammar::fa::op::minimize\fR \fIfa\fR ?\fImapvar\fR?
.sp
\fB::grammar::fa::op::complement\fR \fIfa\fR
.sp
\fB::grammar::fa::op::kleene\fR \fIfa\fR
.sp
\fB::grammar::fa::op::optional\fR \fIfa\fR
.sp
\fB::grammar::fa::op::union\fR \fIfa\fR \fIfb\fR ?\fImapvar\fR?
.sp
\fB::grammar::fa::op::intersect\fR \fIfa\fR \fIfb\fR ?\fImapvar\fR?
.sp
\fB::grammar::fa::op::difference\fR \fIfa\fR \fIfb\fR ?\fImapvar\fR?
.sp
\fB::grammar::fa::op::concatenate\fR \fIfa\fR \fIfb\fR ?\fImapvar\fR?
.sp
\fB::grammar::fa::op::fromRegex\fR \fIfa\fR \fIregex\fR ?\fIover\fR?
.sp
\fB::grammar::fa::op::toRegexp\fR \fIfa\fR
.sp
\fB::grammar::fa::op::toRegexp2\fR \fIfa\fR
.sp
\fB::grammar::fa::op::toTclRegexp\fR \fIregexp\fR \fIsymdict\fR
.sp
\fB::grammar::fa::op::simplifyRegexp\fR \fIregexp\fR
.sp
.BE
.SH DESCRIPTION
.PP
This package provides a number of complex operations on finite
automatons (Short: FA),
as provided by the package \fBgrammar::fa\fR\&.
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 \fBgrammar::fa\fR
and \fBgrammar::fa::interpreter\fR for that\&.
Another package related to this is \fBgrammar::fa::compiler\fR
which turns a FA into an executor class which has the definition of
the FA hardwired into it\&.
.PP
For more information about what a finite automaton is see section
\fIFINITE AUTOMATONS\fR in package \fBgrammar::fa\fR\&.
.SH 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\&.
\fIStructure operations\fR
.TP
\fB::grammar::fa::op::constructor\fR \fIcmd\fR
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\&.
.sp
Any container class using this package for complex operations should set
its own class command as the constructor\&. See package \fBgrammar::fa\fR
for an example\&.
.TP
\fB::grammar::fa::op::reverse\fR \fIfa\fR
Reverses the \fIfa\fR\&. This is done by reversing the direction of all
transitions and swapping the sets of \fIstart\fR and \fIfinal\fR
states\&. The language of \fIfa\fR changes unpredictably\&.
.TP
\fB::grammar::fa::op::complete\fR \fIfa\fR ?\fIsink\fR?
Completes the \fIfa\fR \fIcomplete\fR, but nothing is done if the
\fIfa\fR is already \fIcomplete\fR\&. This implies that only the first
in a series of multiple consecutive complete operations on \fIfa\fR
will perform anything\&. The remainder will be null operations\&.
.sp
The language of \fIfa\fR is unchanged by this operation\&.
.sp
This is done by adding a single new state, the \fIsink\fR, 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\&.
.sp
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 \fInot\fR missing\&.
.sp
The argument \fIsink\fR provides the name for the new state and most
not be present in the \fIfa\fR if specified\&. If the name is not
specified the command will name the state "sink\fBn\fR", where \fBn\fR
is set so that there are no collisions with existing states\&.
.sp
Note that the sink state is \fInot useful\fR by definition\&.  In
other words, while the FA becomes complete, it is also
\fInot useful\fR in the strict sense as it has a state from which
no final state can be reached\&.
.TP
\fB::grammar::fa::op::remove_eps\fR \fIfa\fR
Removes all epsilon-transitions from the \fIfa\fR in such a manner the
the language of \fIfa\fR is unchanged\&. However nothing is done if the
\fIfa\fR is already \fIepsilon-free\fR\&.
This implies that only the first in a series of multiple consecutive
complete operations on \fIfa\fR will perform anything\&. The remainder
will be null operations\&.
.sp
\fINote:\fR This operation may cause states to become unreachable or
not useful\&. These states are not removed by this operation\&.
Use \fB::grammar::fa::op::trim\fR for that instead\&.
.TP
\fB::grammar::fa::op::trim\fR \fIfa\fR ?\fIwhat\fR?
Removes unwanted baggage from \fIfa\fR\&.
The legal values for \fIwhat\fR are listed below\&. The command defaults
to \fB!reachable|!useful\fR if no specific argument was given\&.
.RS
.TP
\fB!reachable\fR
Removes all states which are not reachable from a start state\&.
.TP
\fB!useful\fR
Removes all states which are unable to reach a final state\&.
.TP
\fB!reachable&!useful\fR
.TP
\fB!(reachable|useful)\fR
Removes all states which are not reachable from a start state and are
unable to reach a final state\&.
.TP
\fB!reachable|!useful\fR
.TP
\fB!(reachable&useful)\fR
Removes all states which are not reachable from a start state or are
unable to reach a final state\&.
.RE
.sp
.TP
\fB::grammar::fa::op::determinize\fR \fIfa\fR ?\fImapvar\fR?
Makes the \fIfa\fR deterministic without changing the language
accepted by the \fIfa\fR\&. However nothing is done if the \fIfa\fR is
already \fIdeterministic\fR\&. This implies that only the first in a
series of multiple consecutive complete operations on \fIfa\fR will
perform anything\&. The remainder will be null operations\&.
.sp
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 \fImapvar\fR, 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\&.
.sp
\fINote\fR: An empty dictionary signals that the command was able to
make the \fIfa\fR deterministic without performing a full subset
construction, just by removing states and shuffling transitions around
(As part of making the FA epsilon-free)\&.
.sp
\fINote\fR: 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
\fIdeterministic\fR\&.
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\&.
.TP
\fB::grammar::fa::op::minimize\fR \fIfa\fR ?\fImapvar\fR?
Creates a FA which accepts the same language as \fIfa\fR, but has a
minimal number of states\&. Uses Brzozowski's method to accomplish this\&.
.sp
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 \fImapvar\fR, 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\&.
.sp
\fINote\fR: An empty dictionary signals that the command was able to
minimize the \fIfa\fR without having to compute new states\&. This
should happen if and only if the input FA was already minimal\&.
.sp
\fINote\fR: 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\&.
.PP
\fILanguage operations\fR
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)\&.
.TP
\fB::grammar::fa::op::complement\fR \fIfa\fR
Complements \fIfa\fR\&. This is possible if and only if \fIfa\fR is
\fIcomplete\fR and \fIdeterministic\fR\&. The resulting FA accepts the
complementary language of \fIfa\fR\&. In other words, all inputs not
accepted by the input are accepted by the result, and vice versa\&.
.sp
The result will have all states and transitions of the input, and
different final states\&.
.TP
\fB::grammar::fa::op::kleene\fR \fIfa\fR
Applies Kleene's closure to \fIfa\fR\&.
The resulting FA accepts all strings \fBS\fR for which we can find a
natural number \fBn\fR (0 inclusive) and strings \fBA1\fR \&.\&.\&. \fBAn\fR
in the language of \fIfa\fR such that \fBS\fR is the concatenation of
\fBA1\fR \&.\&.\&. \fBAn\fR\&.
In other words, the language of the result is the infinite union over
finite length concatenations over the language of \fIfa\fR\&.
.sp
The result will have all states and transitions of the input, and new
start and final states\&.
.TP
\fB::grammar::fa::op::optional\fR \fIfa\fR
Makes the \fIfa\fR optional\&. In other words it computes the FA which
accepts the language of \fIfa\fR and the empty the word (epsilon) as
well\&.
.sp
The result will have all states and transitions of the input, and new
start and final states\&.
.TP
\fB::grammar::fa::op::union\fR \fIfa\fR \fIfb\fR ?\fImapvar\fR?
Combines the FAs \fIfa\fR and \fIfb\fR such that the resulting FA
accepts the union of the languages of the two FAs\&.
.sp
The result will have all states and transitions of the two input FAs,
and new start and final states\&. All states of \fIfb\fR which exist in
\fIfa\fR as well will be renamed, and the \fImapvar\fR will contain a
mapping from the old states of \fIfb\fR to the new ones, if present\&.
.sp
It should be noted that the result will be non-deterministic, even if
the inputs are deterministic\&.
.TP
\fB::grammar::fa::op::intersect\fR \fIfa\fR \fIfb\fR ?\fImapvar\fR?
Combines the FAs \fIfa\fR and \fIfb\fR 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 \fIfa\fR and \fIfb\fR\&. The result will be useful, but
not necessarily deterministic or minimal\&.
.sp
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 \fImapvar\fR, 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 \fIfa\fR, the
second element will be drawn from \fIfb\fR\&.
.TP
\fB::grammar::fa::op::difference\fR \fIfa\fR \fIfb\fR ?\fImapvar\fR?
Combines the FAs \fIfa\fR and \fIfb\fR 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 \fIfa\fR, but not by \fIfb\fR\&. This can also be expressed
as the intersection of \fIfa\fR with the complement of \fIfb\fR\&. The
result will be useful, but not necessarily deterministic or minimal\&.
.sp
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 \fImapvar\fR, 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 \fIfa\fR, the
second element will be drawn from \fIfb\fR\&.
.TP
\fB::grammar::fa::op::concatenate\fR \fIfa\fR \fIfb\fR ?\fImapvar\fR?
Combines the FAs \fIfa\fR and \fIfb\fR 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 \fIfa\fR, and \fIfb\fR resp\&. and W is the concatenation of
A and B\&.
.sp
The result FA will be non-deterministic\&.
.TP
\fB::grammar::fa::op::fromRegex\fR \fIfa\fR \fIregex\fR ?\fIover\fR?
Generates a non-deterministic FA which accepts the same language as
the regular expression \fIregex\fR\&. If the \fIover\fR 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 \fIregex\fR when \fIover\fR was not
specified\&. This set is important if and only if the complement
operator "!" is used in \fIregex\fR as the complementary language of
an FA is quite different for different sets of symbols\&.
.sp
The regular expression is represented by a nested list, which forms
a syntax tree\&. The following structures are legal:
.RS
.TP
{S x}
Atomic regular expression\&. Everything else is constructed from
these\&. Accepts the \fBS\fRymbol "x"\&.
.TP
{\&. A1 A2 \&.\&.\&.}
Concatenation operator\&. Accepts the concatenation of the regular
expressions \fBA1\fR, \fBA2\fR, etc\&.
.sp
\fINote\fR that this operator accepts zero or more arguments\&. With zero
arguments the represented language is \fIepsilon\fR, the empty word\&.
.TP
{| A1 A2 \&.\&.\&.}
Choice operator, also called "Alternative"\&. Accepts all input accepted
by at least one of the regular expressions \fBA1\fR, \fBA2\fR, etc\&. In
other words, the union of \fBA1\fR, \fBA2\fR\&.
.sp
\fINote\fR that this operator accepts zero or more arguments\&. With zero
arguments the represented language is the \fIempty\fR language,
the language without words\&.
.TP
{& A1 A2 \&.\&.\&.}
Intersection operator, logical and\&. Accepts all input accepted which
is accepted by all of the regular expressions \fBA1\fR, \fBA2\fR,
etc\&. In other words, the intersection of \fBA1\fR, \fBA2\fR\&.
.TP
{? A}
Optionality operator\&. Accepts the empty word and anything from the
regular expression \fBA\fR\&.
.TP
{* A}
Kleene closure\&. Accepts the empty word and any finite concatenation of
words accepted by the regular expression \fBA\fR\&.
.TP
{+ A}
Positive Kleene closure\&. Accepts any finite concatenation of words
accepted by the regular expression \fBA\fR, but not the empty word\&.
.TP
{! A}
Complement operator\&. Accepts any word not accepted by the regular
expression \fBA\fR\&. Note that the complement depends on the set of
symbol the result should run over\&. See the discussion of the argument
\fIover\fR before\&.
.RE
.TP
\fB::grammar::fa::op::toRegexp\fR \fIfa\fR
This command generates and returns a regular expression which accepts
the same language as the finite automaton \fIfa\fR\&. The regular
expression is in the format as described above, for
\fB::grammar::fa::op::fromRegex\fR\&.
.TP
\fB::grammar::fa::op::toRegexp2\fR \fIfa\fR
This command has the same functionality as \fB::grammar::fa::op::toRegexp\fR,
but uses a different algorithm to simplify the generated regular expressions\&.
.TP
\fB::grammar::fa::op::toTclRegexp\fR \fIregexp\fR \fIsymdict\fR
This command generates and returns a regular expression in Tcl syntax for the
regular expression \fIregexp\fR, if that is possible\&. \fIregexp\fR is in the
same format as expected by \fB::grammar::fa::op::fromRegex\fR\&.
.sp
The command will fail and throw an error if \fIregexp\fR contains
complementation and intersection operations\&.
.sp
The argument \fIsymdict\fR is a dictionary mapping symbol names to
pairs of \fIsyntactic type\fR and Tcl-regexp\&. If a symbol
occurring in the \fIregexp\fR 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\&.
.TP
\fB::grammar::fa::op::simplifyRegexp\fR \fIregexp\fR
This command simplifies a regular expression by applying the following
algorithm first to the main expression and then recursively to all
sub-expressions:
.RS
.IP [1]
Convert the expression into a finite automaton\&.
.IP [2]
Minimize the automaton\&.
.IP [3]
Convert the automaton back to a regular expression\&.
.IP [4]
Choose the shorter of original expression and expression from
the previous step\&.
.RE
.PP
.PP
.SH EXAMPLES
.SH "BUGS, IDEAS, FEEDBACK"
This document, and the package it describes, will undoubtedly contain
bugs and other problems\&.
Please report such in the category \fIgrammar_fa\fR of the
\fITcllib Trackers\fR [http://core\&.tcl\&.tk/tcllib/reportlist]\&.
Please also report any ideas for enhancements you may have for either
package and/or documentation\&.
.PP
When proposing code changes, please provide \fIunified diffs\fR,
i\&.e the output of \fBdiff -u\fR\&.
.PP
Note further that \fIattachments\fR are strongly preferred over
inlined patches\&. Attachments can be made by going to the \fBEdit\fR
form of the ticket immediately after its creation, and then using the
left-most button in the secondary navigation bar\&.
.SH KEYWORDS
automaton, finite automaton, grammar, parsing, regular expression, regular grammar, regular languages, state, transducer
.SH CATEGORY
Grammars and finite automata
.SH COPYRIGHT
.nf
Copyright (c) 2004-2008 Andreas Kupries <andreas_kupries@users\&.sourceforge\&.net>

.fi