'\"
'\" Generated from file 'fa\&.man' by tcllib/doctools with format 'nroff'
'\" Copyright (c) 2004-2009 Andreas Kupries <andreas_kupries@users\&.sourceforge\&.net>
'\"
.TH "grammar::fa" n 0\&.6 tcllib "Finite automaton operations and usage"
.\" The -*- nroff -*- definitions below are for supplemental macros used
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\&\\$1 \\fI\\$2\\fP (\\$3)
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.\}
.el \}\
\h'-1.5n'\L'|\\n(^yu-1v'\h'\\n(^lu+3n'\L'\\n(^tu+1v-\\n(^yu'\l'|0u-1.5n\(ul'
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'ie '\\$1'' .ds So \\fBoptions\\fR
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.SH "STANDARD OPTIONS"
.LP
.nf
.ta 5.5c 11c
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.LP
See the \\*(So manual entry for details on the standard options.
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Command-Line Name:	\\fB\\$1\\fR
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Database Class:	\\fB\\$3\\fR
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.BS
.SH NAME
grammar::fa \- Create and manipulate finite automatons
.SH SYNOPSIS
package require \fBTcl 8\&.5 9\fR
.sp
package require \fBsnit 1\&.3\fR
.sp
package require \fBstruct::list\fR
.sp
package require \fBstruct::set\fR
.sp
package require \fBgrammar::fa::op ?0\&.3?\fR
.sp
package require \fBgrammar::fa ?0\&.6?\fR
.sp
\fB::grammar::fa\fR \fIfaName\fR ?\fB=\fR|\fB:=\fR|\fB<--\fR|\fBas\fR|\fBdeserialize\fR \fIsrc\fR|\fBfromRegex\fR \fIre\fR ?\fIover\fR??
.sp
\fBfaName\fR \fIoption\fR ?\fIarg arg \&.\&.\&.\fR?
.sp
\fIfaName\fR \fBdestroy\fR
.sp
\fIfaName\fR \fBclear\fR
.sp
\fIfaName\fR \fB=\fR \fIsrcFA\fR
.sp
\fIfaName\fR \fB-->\fR \fIdstFA\fR
.sp
\fIfaName\fR \fBserialize\fR
.sp
\fIfaName\fR \fBdeserialize\fR \fIserialization\fR
.sp
\fIfaName\fR \fBstates\fR
.sp
\fIfaName\fR \fBstate\fR \fBadd\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
.sp
\fIfaName\fR \fBstate\fR \fBdelete\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
.sp
\fIfaName\fR \fBstate\fR \fBexists\fR \fIs\fR
.sp
\fIfaName\fR \fBstate\fR \fBrename\fR \fIs\fR \fIsnew\fR
.sp
\fIfaName\fR \fBstartstates\fR
.sp
\fIfaName\fR \fBstart\fR \fBadd\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
.sp
\fIfaName\fR \fBstart\fR \fBremove\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
.sp
\fIfaName\fR \fBstart?\fR \fIs\fR
.sp
\fIfaName\fR \fBstart?set\fR \fIstateset\fR
.sp
\fIfaName\fR \fBfinalstates\fR
.sp
\fIfaName\fR \fBfinal\fR \fBadd\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
.sp
\fIfaName\fR \fBfinal\fR \fBremove\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
.sp
\fIfaName\fR \fBfinal?\fR \fIs\fR
.sp
\fIfaName\fR \fBfinal?set\fR \fIstateset\fR
.sp
\fIfaName\fR \fBsymbols\fR
.sp
\fIfaName\fR \fBsymbols@\fR \fIs\fR ?\fId\fR?
.sp
\fIfaName\fR \fBsymbols@set\fR \fIstateset\fR
.sp
\fIfaName\fR \fBsymbol\fR \fBadd\fR \fIsym1\fR ?\fIsym2\fR \&.\&.\&.?
.sp
\fIfaName\fR \fBsymbol\fR \fBdelete\fR \fIsym1\fR ?\fIsym2\fR \&.\&.\&.?
.sp
\fIfaName\fR \fBsymbol\fR \fBrename\fR \fIsym\fR \fInewsym\fR
.sp
\fIfaName\fR \fBsymbol\fR \fBexists\fR \fIsym\fR
.sp
\fIfaName\fR \fBnext\fR \fIs\fR \fIsym\fR ?\fB-->\fR \fInext\fR?
.sp
\fIfaName\fR \fB!next\fR \fIs\fR \fIsym\fR ?\fB-->\fR \fInext\fR?
.sp
\fIfaName\fR \fBnextset\fR \fIstateset\fR \fIsym\fR
.sp
\fIfaName\fR \fBis\fR \fBdeterministic\fR
.sp
\fIfaName\fR \fBis\fR \fBcomplete\fR
.sp
\fIfaName\fR \fBis\fR \fBuseful\fR
.sp
\fIfaName\fR \fBis\fR \fBepsilon-free\fR
.sp
\fIfaName\fR \fBreachable_states\fR
.sp
\fIfaName\fR \fBunreachable_states\fR
.sp
\fIfaName\fR \fBreachable\fR \fIs\fR
.sp
\fIfaName\fR \fBuseful_states\fR
.sp
\fIfaName\fR \fBunuseful_states\fR
.sp
\fIfaName\fR \fBuseful\fR \fIs\fR
.sp
\fIfaName\fR \fBepsilon_closure\fR \fIs\fR
.sp
\fIfaName\fR \fBreverse\fR
.sp
\fIfaName\fR \fBcomplete\fR
.sp
\fIfaName\fR \fBremove_eps\fR
.sp
\fIfaName\fR \fBtrim\fR ?\fIwhat\fR?
.sp
\fIfaName\fR \fBdeterminize\fR ?\fImapvar\fR?
.sp
\fIfaName\fR \fBminimize\fR ?\fImapvar\fR?
.sp
\fIfaName\fR \fBcomplement\fR
.sp
\fIfaName\fR \fBkleene\fR
.sp
\fIfaName\fR \fBoptional\fR
.sp
\fIfaName\fR \fBunion\fR \fIfa\fR ?\fImapvar\fR?
.sp
\fIfaName\fR \fBintersect\fR \fIfa\fR ?\fImapvar\fR?
.sp
\fIfaName\fR \fBdifference\fR \fIfa\fR ?\fImapvar\fR?
.sp
\fIfaName\fR \fBconcatenate\fR \fIfa\fR ?\fImapvar\fR?
.sp
\fIfaName\fR \fBfromRegex\fR \fIregex\fR ?\fIover\fR?
.sp
.BE
.SH DESCRIPTION
.PP
This package provides a container class for
\fIfinite automatons\fR (Short: FA)\&.
It allows the incremental definition of the automaton, its
manipulation and querying of the definition\&.
While the package provides complex operations on the automaton
(via package \fBgrammar::fa::op\fR), it does not have the
ability to execute a definition for a stream of symbols\&.
Use the packages
\fBgrammar::fa::dacceptor\fR and
\fBgrammar::fa::dexec\fR for that\&.
Another package related to this is \fBgrammar::fa::compiler\fR\&. It
turns a FA into an executor class which has the definition of the FA
hardwired into it\&. The output of this package is configurable to suit
a large number of different implementation languages and paradigms\&.
.PP
For more information about what a finite automaton is see section
\fBFINITE AUTOMATONS\fR\&.
.SH API
The package exports the API described here\&.
.TP
\fB::grammar::fa\fR \fIfaName\fR ?\fB=\fR|\fB:=\fR|\fB<--\fR|\fBas\fR|\fBdeserialize\fR \fIsrc\fR|\fBfromRegex\fR \fIre\fR ?\fIover\fR??
Creates a new finite automaton with an associated global Tcl command
whose name is \fIfaName\fR\&. This command may be used to invoke various
operations on the automaton\&. It has the following general form:
.RS
.TP
\fBfaName\fR \fIoption\fR ?\fIarg arg \&.\&.\&.\fR?
\fIOption\fR and the \fIarg\fRs determine the exact behavior of the
command\&. See section \fBFA METHODS\fR for more explanations\&. The
new automaton will be empty if no \fIsrc\fR is specified\&. Otherwise
it will contain a copy of the definition contained in the \fIsrc\fR\&.
The \fIsrc\fR has to be a FA object reference for all operators except
\fBdeserialize\fR and \fBfromRegex\fR\&. The \fBdeserialize\fR
operator requires \fIsrc\fR to be the serialization of a FA instead,
and \fBfromRegex\fR takes a regular expression in the form a of a
syntax tree\&. See \fB::grammar::fa::op::fromRegex\fR for more detail on
that\&.
.RE
.PP
.SH "FA METHODS"
.PP
All automatons provide the following methods for their manipulation:
.TP
\fIfaName\fR \fBdestroy\fR
Destroys the automaton, including its storage space and associated
command\&.
.TP
\fIfaName\fR \fBclear\fR
Clears out the definition of the automaton contained in \fIfaName\fR,
but does \fInot\fR destroy the object\&.
.TP
\fIfaName\fR \fB=\fR \fIsrcFA\fR
Assigns the contents of the automaton contained
in \fIsrcFA\fR to \fIfaName\fR, overwriting any
existing definition\&.
This is the assignment operator for automatons\&. It copies the
automaton contained in the FA object \fIsrcFA\fR over the automaton
definition in \fIfaName\fR\&. The old contents of \fIfaName\fR are
deleted by this operation\&.
.sp
This operation is in effect equivalent to
.sp
.CS


    \fIfaName\fR \fBdeserialize\fR [\fIsrcFA\fR \fBserialize\fR]

.CE
.TP
\fIfaName\fR \fB-->\fR \fIdstFA\fR
This is the reverse assignment operator for automatons\&. It copies the
automation contained in the object \fIfaName\fR over the automaton
definition in the object \fIdstFA\fR\&.
The old contents of \fIdstFA\fR are deleted by this operation\&.
.sp
This operation is in effect equivalent to
.sp
.CS


    \fIdstFA\fR \fBdeserialize\fR [\fIfaName\fR \fBserialize\fR]

.CE
.TP
\fIfaName\fR \fBserialize\fR
This method serializes the automaton stored in \fIfaName\fR\&. In other
words it returns a tcl \fIvalue\fR completely describing that
automaton\&.
This allows, for example, the transfer of automatons over arbitrary
channels, persistence, etc\&.
This method is also the basis for both the copy constructor and the
assignment operator\&.
.sp
The result of this method has to be semantically identical over all
implementations of the \fBgrammar::fa\fR interface\&. This is what
will enable us to copy automatons between different implementations of
the same interface\&.
.sp
The result is a list of three elements with the following structure:
.RS
.IP [1]
The constant string \fBgrammar::fa\fR\&.
.IP [2]
A list containing the names of all known input symbols\&. The order of
elements in this list is not relevant\&.
.IP [3]
The last item in the list is a dictionary, however the order of the
keys is important as well\&. The keys are the states of the serialized
FA, and their order is the order in which to create the states when
deserializing\&. This is relevant to preserve the order relationship
between states\&.
.sp
The value of each dictionary entry is a list of three elements
describing the state in more detail\&.
.RS
.IP [1]
A boolean flag\&. If its value is \fBtrue\fR then the state is a
start state, otherwise it is not\&.
.IP [2]
A boolean flag\&. If its value is \fBtrue\fR then the state is a
final state, otherwise it is not\&.
.IP [3]
The last element is a dictionary describing the transitions for the
state\&. The keys are symbols (or the empty string), and the values are
sets of successor states\&.
.RE
.RE
.sp
Assuming the following FA (which describes the life of a truck driver
in a very simple way :)
.sp
.CS


    Drive -- yellow --> Brake -- red --> (Stop) -- red/yellow --> Attention -- green --> Drive
    (\&.\&.\&.) is the start state\&.

.CE
.sp
a possible serialization is
.sp
.CS


    grammar::fa \\
    {yellow red green red/yellow} \\
    {Drive     {0 0 {yellow     Brake}} \\
     Brake     {0 0 {red        Stop}} \\
     Stop      {1 0 {red/yellow Attention}} \\
     Attention {0 0 {green      Drive}}}

.CE
.sp
A possible one, because I did not care about creation order here
.TP
\fIfaName\fR \fBdeserialize\fR \fIserialization\fR
This is the complement to \fBserialize\fR\&. It replaces the
automaton definition in \fIfaName\fR with the automaton described by
the \fIserialization\fR value\&. The old contents of \fIfaName\fR are
deleted by this operation\&.
.TP
\fIfaName\fR \fBstates\fR
Returns the set of all states known to \fIfaName\fR\&.
.TP
\fIfaName\fR \fBstate\fR \fBadd\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
Adds the states \fIs1\fR, \fIs2\fR, et cetera to the FA definition in
\fIfaName\fR\&. The operation will fail any of the new states is already
declared\&.
.TP
\fIfaName\fR \fBstate\fR \fBdelete\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
Deletes the state \fIs1\fR, \fIs2\fR, et cetera, and all associated
information from the FA definition in \fIfaName\fR\&. The latter means
that the information about in- or outbound transitions is deleted as
well\&. If the deleted state was a start or final state then this
information is invalidated as well\&. The operation will fail if the
state \fIs\fR is not known to the FA\&.
.TP
\fIfaName\fR \fBstate\fR \fBexists\fR \fIs\fR
A predicate\&. It tests whether the state \fIs\fR is known to the FA in
\fIfaName\fR\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
state \fIs\fR is known, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBstate\fR \fBrename\fR \fIs\fR \fIsnew\fR
Renames the state \fIs\fR to \fIsnew\fR\&. Fails if \fIs\fR is not a
known state\&. Also fails if \fIsnew\fR is already known as a state\&.
.TP
\fIfaName\fR \fBstartstates\fR
Returns the set of states which are marked as \fIstart\fR states,
also known as \fIinitial\fR states\&.
See \fBFINITE AUTOMATONS\fR for explanations what this means\&.
.TP
\fIfaName\fR \fBstart\fR \fBadd\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
Mark the states \fIs1\fR, \fIs2\fR, et cetera in the FA \fIfaName\fR
as \fIstart\fR (aka \fIinitial\fR)\&.
.TP
\fIfaName\fR \fBstart\fR \fBremove\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
Mark the states \fIs1\fR, \fIs2\fR, et cetera in the FA \fIfaName\fR
as \fInot start\fR (aka \fInot accepting\fR)\&.
.TP
\fIfaName\fR \fBstart?\fR \fIs\fR
A predicate\&. It tests if the state \fIs\fR in the FA \fIfaName\fR is
\fIstart\fR or not\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
state \fIs\fR is \fIstart\fR, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBstart?set\fR \fIstateset\fR
A predicate\&. It tests if the set of states \fIstateset\fR contains at
least one start state\&. They operation will fail if the set contains an
element which is not a known state\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if a
start state is present in \fIstateset\fR, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBfinalstates\fR
Returns the set of states which are marked as \fIfinal\fR states,
also known as \fIaccepting\fR states\&.
See \fBFINITE AUTOMATONS\fR for explanations what this means\&.
.TP
\fIfaName\fR \fBfinal\fR \fBadd\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
Mark the states \fIs1\fR, \fIs2\fR, et cetera in the FA \fIfaName\fR
as \fIfinal\fR (aka \fIaccepting\fR)\&.
.TP
\fIfaName\fR \fBfinal\fR \fBremove\fR \fIs1\fR ?\fIs2\fR \&.\&.\&.?
Mark the states \fIs1\fR, \fIs2\fR, et cetera in the FA \fIfaName\fR
as \fInot final\fR (aka \fInot accepting\fR)\&.
.TP
\fIfaName\fR \fBfinal?\fR \fIs\fR
A predicate\&. It tests if the state \fIs\fR in the FA \fIfaName\fR is
\fIfinal\fR or not\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
state \fIs\fR is \fIfinal\fR, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBfinal?set\fR \fIstateset\fR
A predicate\&. It tests if the set of states \fIstateset\fR contains at
least one final state\&. They operation will fail if the set contains an
element which is not a known state\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if a
final state is present in \fIstateset\fR, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBsymbols\fR
Returns the set of all symbols known to the FA \fIfaName\fR\&.
.TP
\fIfaName\fR \fBsymbols@\fR \fIs\fR ?\fId\fR?
Returns the set of all symbols for which the state \fIs\fR has transitions\&.
If the empty symbol is present then \fIs\fR has epsilon transitions\&. If two
states are specified the result is the set of symbols which have transitions
from \fIs\fR to \fIt\fR\&. This set may be empty if there are no transitions
between the two specified states\&.
.TP
\fIfaName\fR \fBsymbols@set\fR \fIstateset\fR
Returns the set of all symbols for which at least one state in the set
of states \fIstateset\fR has transitions\&.
In other words, the union of [\fIfaName\fR \fBsymbols@\fR \fBs\fR]
for all states \fBs\fR in \fIstateset\fR\&.
If the empty symbol is present then at least one state contained in
\fIstateset\fR has epsilon transitions\&.
.TP
\fIfaName\fR \fBsymbol\fR \fBadd\fR \fIsym1\fR ?\fIsym2\fR \&.\&.\&.?
Adds the symbols \fIsym1\fR, \fIsym2\fR, et cetera to the FA
definition in \fIfaName\fR\&. The operation will fail any of the symbols
is already declared\&. The empty string is not allowed as a value for the symbols\&.
.TP
\fIfaName\fR \fBsymbol\fR \fBdelete\fR \fIsym1\fR ?\fIsym2\fR \&.\&.\&.?
Deletes the symbols \fIsym1\fR, \fIsym2\fR et cetera, and all
associated information from the FA definition in \fIfaName\fR\&. The
latter means that all transitions using the symbols are deleted as
well\&. The operation will fail if any of the symbols is not known to
the FA\&.
.TP
\fIfaName\fR \fBsymbol\fR \fBrename\fR \fIsym\fR \fInewsym\fR
Renames the symbol \fIsym\fR to \fInewsym\fR\&. Fails if \fIsym\fR is
not a known symbol\&. Also fails if \fInewsym\fR is already known as a
symbol\&.
.TP
\fIfaName\fR \fBsymbol\fR \fBexists\fR \fIsym\fR
A predicate\&. It tests whether the symbol \fIsym\fR is known to the FA
in \fIfaName\fR\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
symbol \fIsym\fR is known, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBnext\fR \fIs\fR \fIsym\fR ?\fB-->\fR \fInext\fR?
Define or query transition information\&.
.sp
If \fInext\fR is specified, then the method will add a transition from
the state \fIs\fR to the \fIsuccessor\fR state \fInext\fR labeled with
the symbol \fIsym\fR to the FA contained in \fIfaName\fR\&. The
operation will fail if \fIs\fR, or \fInext\fR are not known states, or
if \fIsym\fR is not a known symbol\&. An exception to the latter is that
\fIsym\fR is allowed to be the empty string\&. In that case the new
transition is an \fIepsilon transition\fR which will not consume
input when traversed\&. The operation will also fail if the combination
of (\fIs\fR, \fIsym\fR, and \fInext\fR) is already present in the FA\&.
.sp
If \fInext\fR was not specified, then the method will return
the set of states which can be reached from \fIs\fR through
a single transition labeled with symbol \fIsym\fR\&.
.TP
\fIfaName\fR \fB!next\fR \fIs\fR \fIsym\fR ?\fB-->\fR \fInext\fR?
Remove one or more transitions from the Fa in \fIfaName\fR\&.
.sp
If \fInext\fR was specified then the single transition from the state
\fIs\fR to the state \fInext\fR labeled with the symbol \fIsym\fR is
removed from the FA\&. Otherwise \fIall\fR transitions originating in
state \fIs\fR and labeled with the symbol \fIsym\fR will be removed\&.
.sp
The operation will fail if \fIs\fR and/or \fInext\fR are not known as
states\&. It will also fail if a non-empty \fIsym\fR is not known as
symbol\&. The empty string is acceptable, and allows the removal of
epsilon transitions\&.
.TP
\fIfaName\fR \fBnextset\fR \fIstateset\fR \fIsym\fR
Returns the set of states which can be reached by a single transition
originating in a state in the set \fIstateset\fR and labeled with the
symbol \fIsym\fR\&.
.sp
In other words, this is the union of
[\fIfaName\fR next \fBs\fR \fIsymbol\fR]
for all states \fBs\fR in \fIstateset\fR\&.
.TP
\fIfaName\fR \fBis\fR \fBdeterministic\fR
A predicate\&. It tests whether the FA in \fIfaName\fR is a
deterministic FA or not\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
FA is deterministic, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBis\fR \fBcomplete\fR
A predicate\&. It tests whether the FA in \fIfaName\fR is a complete FA
or not\&. A FA is complete if it has at least one transition per state
and symbol\&. This also means that a FA without symbols, or states is
also complete\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
FA is deterministic, and \fBfalse\fR otherwise\&.
.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\&.
.TP
\fIfaName\fR \fBis\fR \fBuseful\fR
A predicate\&. It tests whether the FA in \fIfaName\fR is an useful FA
or not\&. A FA is useful if all states are \fIreachable\fR
and \fIuseful\fR\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
FA is deterministic, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBis\fR \fBepsilon-free\fR
A predicate\&. It tests whether the FA in \fIfaName\fR is an
epsilon-free FA or not\&. A FA is epsilon-free if it has no epsilon
transitions\&. This definition means that all deterministic FAs are
epsilon-free as well, and epsilon-freeness is a necessary
pre-condition for deterministic'ness\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
FA is deterministic, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBreachable_states\fR
Returns the set of states which are reachable from a start state by
one or more transitions\&.
.TP
\fIfaName\fR \fBunreachable_states\fR
Returns the set of states which are not reachable from any start state
by any number of transitions\&. This is
.sp
.CS


	 [faName states] - [faName reachable_states]

.CE
.TP
\fIfaName\fR \fBreachable\fR \fIs\fR
A predicate\&. It tests whether the state \fIs\fR in the FA \fIfaName\fR
can be reached from a start state by one or more transitions\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
state can be reached, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBuseful_states\fR
Returns the set of states which are able to reach a final state by
one or more transitions\&.
.TP
\fIfaName\fR \fBunuseful_states\fR
Returns the set of states which are not able to reach a final state by
any number of transitions\&. This is
.sp
.CS


	 [faName states] - [faName useful_states]

.CE
.TP
\fIfaName\fR \fBuseful\fR \fIs\fR
A predicate\&. It tests whether the state \fIs\fR in the FA \fIfaName\fR
is able to reach a final state by one or more transitions\&.
The result is a boolean value\&. It will be set to \fBtrue\fR if the
state is useful, and \fBfalse\fR otherwise\&.
.TP
\fIfaName\fR \fBepsilon_closure\fR \fIs\fR
Returns the set of states which are reachable from the state \fIs\fR
in the FA \fIfaName\fR by one or more epsilon transitions, i\&.e
transitions over the empty symbol, transitions which do not consume
input\&. This is called the \fIepsilon closure\fR of \fIs\fR\&.
.TP
\fIfaName\fR \fBreverse\fR
.TP
\fIfaName\fR \fBcomplete\fR
.TP
\fIfaName\fR \fBremove_eps\fR
.TP
\fIfaName\fR \fBtrim\fR ?\fIwhat\fR?
.TP
\fIfaName\fR \fBdeterminize\fR ?\fImapvar\fR?
.TP
\fIfaName\fR \fBminimize\fR ?\fImapvar\fR?
.TP
\fIfaName\fR \fBcomplement\fR
.TP
\fIfaName\fR \fBkleene\fR
.TP
\fIfaName\fR \fBoptional\fR
.TP
\fIfaName\fR \fBunion\fR \fIfa\fR ?\fImapvar\fR?
.TP
\fIfaName\fR \fBintersect\fR \fIfa\fR ?\fImapvar\fR?
.TP
\fIfaName\fR \fBdifference\fR \fIfa\fR ?\fImapvar\fR?
.TP
\fIfaName\fR \fBconcatenate\fR \fIfa\fR ?\fImapvar\fR?
.TP
\fIfaName\fR \fBfromRegex\fR \fIregex\fR ?\fIover\fR?
These methods provide more complex operations on the FA\&.  Please see
the same-named commands in the package \fBgrammar::fa::op\fR for
descriptions of what they do\&.
.PP
.PP
.SH EXAMPLES
.PP
.SH "FINITE AUTOMATONS"
.PP
For the mathematically inclined, a FA is a 5-tuple (S,Sy,St,Fi,T) where
.IP \(bu
S is a set of \fIstates\fR,
.IP \(bu
Sy a set of \fIinput symbols\fR,
.IP \(bu
St is a subset of S, the set of \fIstart\fR states, also known as
\fIinitial\fR states\&.
.IP \(bu
Fi is a subset of S, the set of \fIfinal\fR states, also known as
\fIaccepting\fR\&.
.IP \(bu
T is a function from S x (Sy + epsilon) to {S}, the \fItransition function\fR\&.
Here \fBepsilon\fR denotes the empty input symbol and is distinct
from all symbols in Sy; and {S} is the set of subsets of S\&. In other
words, T maps a combination of State and Input (which can be empty) to
a set of \fIsuccessor states\fR\&.
.PP
.PP
In computer theory a FA is most often shown as a graph where the nodes
represent the states, and the edges between the nodes encode the
transition function: For all n in S' = T (s, sy) we have one edge
between the nodes representing s and n resp\&., labeled with sy\&. The
start and accepting states are encoded through distinct visual
markers, i\&.e\&. they are attributes of the nodes\&.
.PP
FA's are used to process streams of symbols over Sy\&.
.PP
A specific FA is said to \fIaccept\fR a finite stream sy_1 sy_2
\&.\&.\&. sy_n if there is a path in the graph of the FA beginning at a
state in St and ending at a state in Fi whose edges have the labels
sy_1, sy_2, etc\&. to sy_n\&.
The set of all strings accepted by the FA is the \fIlanguage\fR of
the FA\&. One important equivalence is that the set of languages which
can be accepted by an FA is the set of \fIregular languages\fR\&.
.PP
Another important concept is that of deterministic FAs\&. A FA is said
to be \fIdeterministic\fR if for each string of input symbols there
is exactly one path in the graph of the FA beginning at the start
state and whose edges are labeled with the symbols in the string\&.
While it might seem that non-deterministic FAs to have more power of
recognition, this is not so\&. For each non-deterministic FA we can
construct a deterministic FA which accepts the same language (-->
Thompson's subset construction)\&.
.PP
While one of the premier applications of FAs is in \fIparsing\fR,
especially in the \fIlexer\fR stage (where symbols == characters),
this is not the only possibility by far\&.
.PP
Quite a lot of processes can be modeled as a FA, albeit with a
possibly large set of states\&. For these the notion of accepting states
is often less or not relevant at all\&. What is needed instead is the
ability to act to state changes in the FA, i\&.e\&. to generate some
output in response to the input\&.
This transforms a FA into a \fIfinite transducer\fR, which has an
additional set OSy of \fIoutput symbols\fR and also an additional
\fIoutput function\fR O which maps from "S x (Sy + epsilon)" to
"(Osy + epsilon)", i\&.e a combination of state and input, possibly
empty to an output symbol, or nothing\&.
.PP
For the graph representation this means that edges are additional
labeled with the output symbol to write when this edge is traversed
while matching input\&. Note that for an application "writing an output
symbol" can also be "executing some code"\&.
.PP
Transducers are not handled by this package\&. They will get their own
package in the future\&.
.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) 2004-2009 Andreas Kupries <andreas_kupries@users\&.sourceforge\&.net>

.fi
