Ubuntu Manpage: grammar::fa - Create and manipulate finite automatons

NAME

       grammar::fa - Create and manipulate finite automatons

SYNOPSIS

       package require Tcl  8.4

       package require snit  1.3

       package require struct::list

       package require struct::set

       package require grammar::fa::op  ?0.2?

       package require grammar::fa  ?0.4?

       ::grammar::fa faName ?=|:=|<--|as|deserialize src|fromRegex re ?over??

       faName option ?arg arg ...?

       faName destroy

       faName clear

       faName = srcFA

       faName --> dstFA

       faName serialize

       faName deserialize serialization

       faName states

       faName state add s1 ?s2 ...?

       faName state delete s1 ?s2 ...?

       faName state exists s

       faName state rename s snew

       faName startstates

       faName start add s1 ?s2 ...?

       faName start remove s1 ?s2 ...?

       faName start? s

       faName start?set stateset

       faName finalstates

       faName final add s1 ?s2 ...?

       faName final remove s1 ?s2 ...?

       faName final? s

       faName final?set stateset

       faName symbols

       faName symbols@ s ?d?

       faName symbols@set stateset

       faName symbol add sym1 ?sym2 ...?

       faName symbol delete sym1 ?sym2 ...?

       faName symbol rename sym newsym

       faName symbol exists sym

       faName next s sym ?--> next?

       faName !next s sym ?--> next?

       faName nextset stateset sym

       faName is deterministic

       faName is complete

       faName is useful

       faName is epsilon-free

       faName reachable_states

       faName unreachable_states

       faName reachable s

       faName useful_states

       faName unuseful_states

       faName useful s

       faName epsilon_closure s

       faName reverse

       faName complete

       faName remove_eps

       faName trim ?what?

       faName determinize ?mapvar?

       faName minimize ?mapvar?

       faName complement

       faName kleene

       faName optional

       faName union fa ?mapvar?

       faName intersect fa ?mapvar?

       faName difference fa ?mapvar?

       faName concatenate fa ?mapvar?

       faName fromRegex regex ?over?

_________________________________________________________________

DESCRIPTION

       This  package provides a container class for finite automatons (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
       grammar::fa::op), it does not have the ability to execute a definition  for  a  stream  of
       symbols.   Use  the  packages  grammar::fa::dacceptor  and  grammar::fa::dexec  for  that.
       Another package related to this is grammar::fa::compiler. 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.

       For more information about what a finite automaton is see section FINITE AUTOMATONS.

API

       The package exports the API described here.

       ::grammar::fa faName ?=|:=|<--|as|deserialize src|fromRegex re ?over??
              Creates a new finite automaton with an associated global Tcl command whose name  is
              faName.  This command may be used to invoke various operations on the automaton. It
              has the following general form:

              faName option ?arg arg ...?
                     Option and the args determine the exact behavior of the command. See section
                     FA  METHODS for more explanations. The new automaton will be empty if no src
                     is specified. Otherwise it will contain a copy of the  definition  contained
                     in  the  src.   The  src  has  to be a FA object reference for all operators
                     except deserialize and fromRegex. The deserialize operator requires  src  to
                     be  the  serialization  of  a  FA  instead,  and  fromRegex  takes a regular
                     expression in the form a of a syntax tree. See  ::grammar::fa::op::fromRegex
                     for more detail on that.

FA METHODS

All automatons provide the following methods for their manipulation:

faName destroy
Destroys the automaton, including its storage space and associated command.

faName clear
Clears out the definition of the automaton contained in faName, but does not
destroy the object.

faName = srcFA
Assigns the contents of the automaton contained in srcFA to faName, overwriting any
existing definition. This is the assignment operator for automatons. It copies the
automaton contained in the FA object srcFA over the automaton definition in faName.
The old contents of faName are deleted by this operation.

This operation is in effect equivalent to

faName deserialize [srcFA serialize]

faName --> dstFA
This is the reverse assignment operator for automatons. It copies the automation
contained in the object faName over the automaton definition in the object dstFA.
The old contents of dstFA are deleted by this operation.

This operation is in effect equivalent to

dstFA deserialize [faName serialize]

faName serialize
This method serializes the automaton stored in faName. In other words it returns a
tcl value 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.

The result of this method has to be semantically identical over all implementations
of the grammar::fa interface. This is what will enable us to copy automatons
between different implementations of the same interface.

The result is a list of three elements with the following structure:

[1] The constant string grammar::fa.

[2] A list containing the names of all known input symbols. The order of
elements in this list is not relevant.

[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.

The value of each dictionary entry is a list of three elements describing
the state in more detail.

[1] A boolean flag. If its value is true then the state is a start state,
otherwise it is not.

[2] A boolean flag. If its value is true then the state is a final state,
otherwise it is not.

[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.

Assuming the following FA (which describes the life of a truck driver in a very simple way
:)

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

a possible serialization is

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}}}

A possible one, because I did not care about creation order here

faName deserialize serialization
This is the complement to serialize. It replaces the automaton definition in faName
with the automaton described by the serialization value. The old contents of faName
are deleted by this operation.

faName states
Returns the set of all states known to faName.

faName state add s1 ?s2 ...?
Adds the states s1, s2, et cetera to the FA definition in faName. The operation
will fail any of the new states is already declared.

faName state delete s1 ?s2 ...?
Deletes the state s1, s2, et cetera, and all associated information from the FA
definition in faName. 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
s is not known to the FA.

faName state exists s
A predicate. It tests whether the state s is known to the FA in faName. The result
is a boolean value. It will be set to true if the state s is known, and false
otherwise.

faName state rename s snew
Renames the state s to snew. Fails if s is not a known state. Also fails if snew is
already known as a state.

faName startstates
Returns the set of states which are marked as start states, also known as initial
states. See FINITE AUTOMATONS for explanations what this means.

faName start add s1 ?s2 ...?
Mark the states s1, s2, et cetera in the FA faName as start (aka initial).

faName start remove s1 ?s2 ...?
Mark the states s1, s2, et cetera in the FA faName as not start (aka not
accepting).

faName start? s
A predicate. It tests if the state s in the FA faName is start or not. The result
is a boolean value. It will be set to true if the state s is start, and false
otherwise.

faName start?set stateset
A predicate. It tests if the set of states stateset 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 true if a start state is
present in stateset, and false otherwise.

faName finalstates
Returns the set of states which are marked as final states, also known as accepting
states. See FINITE AUTOMATONS for explanations what this means.

faName final add s1 ?s2 ...?
Mark the states s1, s2, et cetera in the FA faName as final (aka accepting).

faName final remove s1 ?s2 ...?
Mark the states s1, s2, et cetera in the FA faName as not final (aka not
accepting).

faName final? s
A predicate. It tests if the state s in the FA faName is final or not. The result
is a boolean value. It will be set to true if the state s is final, and false
otherwise.

faName final?set stateset
A predicate. It tests if the set of states stateset 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 true if a final state is
present in stateset, and false otherwise.

faName symbols
Returns the set of all symbols known to the FA faName.

faName symbols@ s ?d?
Returns the set of all symbols for which the state s has transitions. If the empty
symbol is present then s has epsilon transitions. If two states are specified the
result is the set of symbols which have transitions from s to t. This set may be
empty if there are no transitions between the two specified states.

faName symbols@set stateset
Returns the set of all symbols for which at least one state in the set of states
stateset has transitions. In other words, the union of [faName symbols@ s] for all
states s in stateset. If the empty symbol is present then at least one state
contained in stateset has epsilon transitions.

faName symbol add sym1 ?sym2 ...?
Adds the symbols sym1, sym2, et cetera to the FA definition in faName. The
operation will fail any of the symbols is already declared. The empty string is not
allowed as a value for the symbols.

faName symbol delete sym1 ?sym2 ...?
Deletes the symbols sym1, sym2 et cetera, and all associated information from the
FA definition in faName. 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.

faName symbol rename sym newsym
Renames the symbol sym to newsym. Fails if sym is not a known symbol. Also fails if
newsym is already known as a symbol.

faName symbol exists sym
A predicate. It tests whether the symbol sym is known to the FA in faName. The
result is a boolean value. It will be set to true if the symbol sym is known, and
false otherwise.

faName next s sym ?--> next?
Define or query transition information.

If next is specified, then the method will add a transition from the state s to the
successor state next labeled with the symbol sym to the FA contained in faName. The
operation will fail if s, or next are not known states, or if sym is not a known
symbol. An exception to the latter is that sym is allowed to be the empty string.
In that case the new transition is an epsilon transition which will not consume
input when traversed. The operation will also fail if the combination of (s, sym,
and next) is already present in the FA.

If next was not specified, then the method will return the set of states which can
be reached from s through a single transition labeled with symbol sym.

faName !next s sym ?--> next?
Remove one or more transitions from the Fa in faName.

If next was specified then the single transition from the state s to the state next
labeled with the symbol sym is removed from the FA. Otherwise all transitions
originating in state s and labeled with the symbol sym will be removed.

The operation will fail if s and/or next are not known as states. It will also fail
if a non-empty sym is not known as symbol. The empty string is acceptable, and
allows the removal of epsilon transitions.

faName nextset stateset sym
Returns the set of states which can be reached by a single transition originating
in a state in the set stateset and labeled with the symbol sym.

In other words, this is the union of [faName next s symbol] for all states s in
stateset.

faName is deterministic
A predicate. It tests whether the FA in faName is a deterministic FA or not. The
result is a boolean value. It will be set to true if the FA is deterministic, and
false otherwise.

faName is complete
A predicate. It tests whether the FA in faName 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 true if the FA is deterministic, and false otherwise.

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.

faName is useful
A predicate. It tests whether the FA in faName is an useful FA or not. A FA is
useful if all states are reachable and useful. The result is a boolean value. It
will be set to true if the FA is deterministic, and false otherwise.

faName is epsilon-free
A predicate. It tests whether the FA in faName 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 true if the FA is deterministic, and false otherwise.

faName reachable_states
Returns the set of states which are reachable from a start state by one or more
transitions.

faName unreachable_states
Returns the set of states which are not reachable from any start state by any
number of transitions. This is

[faName states] - [faName reachable_states]

faName reachable s
A predicate. It tests whether the state s in the FA faName can be reached from a
start state by one or more transitions. The result is a boolean value. It will be
set to true if the state can be reached, and false otherwise.

faName useful_states
Returns the set of states which are able to reach a final state by one or more
transitions.

faName unuseful_states
Returns the set of states which are not able to reach a final state by any number
of transitions. This is

[faName states] - [faName useful_states]

faName useful s
A predicate. It tests whether the state s in the FA faName is able to reach a final
state by one or more transitions. The result is a boolean value. It will be set to
true if the state is useful, and false otherwise.

faName epsilon_closure s
Returns the set of states which are reachable from the state s in the FA faName by
one or more epsilon transitions, i.e transitions over the empty symbol, transitions
which do not consume input. This is called the epsilon closure of s.

faName reverse

faName complete

faName remove_eps

faName trim ?what?

faName determinize ?mapvar?

faName minimize ?mapvar?

faName complement

faName kleene

faName optional

faName union fa ?mapvar?

faName intersect fa ?mapvar?

faName difference fa ?mapvar?

faName concatenate fa ?mapvar?

faName fromRegex regex ?over?
These methods provide more complex operations on the FA. Please see the same-named
commands in the package grammar::fa::op for descriptions of what they do.

EXAMPLES

FINITE AUTOMATONS

For the mathematically inclined, a FA is a 5-tuple (S,Sy,St,Fi,T) where

• S is a set of states,

• Sy a set of input symbols,

• St is a subset of S, the set of start states, also known as initial states.

• Fi is a subset of S, the set of final states, also known as accepting.

• T is a function from S x (Sy + epsilon) to {S}, the transition function. Here
epsilon 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 successor states.

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.

FA's are used to process streams of symbols over Sy.

A specific FA is said to accept a finite stream sy_1 sy_2 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 language of the FA. One important equivalence is that the set of
languages which can be accepted by an FA is the set of regular languages.

Another important concept is that of deterministic FAs. A FA is said to be deterministic
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).

While one of the premier applications of FAs is in parsing, especially in the lexer stage
(where symbols == characters), this is not the only possibility by far.

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 finite transducer,
which has an additional set OSy of output symbols and also an additional output function 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.

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".

Transducers are not handled by this package. They will get their own package in the
future.

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 SF Trackers
       [http://sourceforge.net/tracker/?group_id=12883].   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) 2004-2009 Andreas Kupries <andreas_kupries@users.sourceforge.net>