grammar::fa(n)       Finite automaton operations and usage      grammar::fa(n)



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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?

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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

       o      S is a set of states,

       o      Sy a set of input symbols,

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

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

       o      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

CATEGORY
       Grammars and finite automata

COPYRIGHT
       Copyright (c) 2004-2009 Andreas Kupries <andreas_kupries@users.sourceforge.net>




grammar_fa                            0.4                       grammar::fa(n)
