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





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:







  

  
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:



  

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

  

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

  

  
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]






  

  
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]






  

  
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



  

  
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.

  

  
Returns the set of all states known to faName.

  

  
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.

  

  
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.

  

  
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.

  

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

  

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

  

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

  

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

  

  
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.

  

  
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.

  

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

  

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

  

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

  

  
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.

  

  
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.

  

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

  

  
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.

  

  
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.

  

  
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.

  

  
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.

  

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

  

  
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.

  

  
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.

  

  

  
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.

  

  

  
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.

  

  

  
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.

  

  
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.

  

  

  
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.

  

  
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.

  

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

  

  
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]






  

  
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.

  

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

  

  
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]






  

  
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.

  

  
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.

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  

  
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
    ... 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 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 Trackers
    [http://core.tcl.tk/tcllib/reportlist]. Please also report any ideas for
    enhancements you may have for either package and/or documentation.


When proposing code changes, please provide unified diffs,
    i.e the output of diff -u.


Note further that attachments are strongly preferred over
    inlined patches. Attachments can be made by going to the Edit form of
    the ticket immediately after its creation, and then using the left-most
    button in the secondary navigation bar.






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>








  

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