  
  
                                   [1X [5XAutomata[105X [101X
  
  
                            [1X A package on automata [101X
  
  
                                  Version 1.14
  
  
                                 Manuel Delgado
  
                                  Steve Linton
  
                                José João Morais
  
  
  
  Manuel Delgado
      Email:    [7Xmailto:mdelgado@fc.up.pt[107X
      Homepage: [7Xhttps://cmup.fc.up.pt/cmup/mdelgado/[107X
      Address:  [33X[0;14YDepartamento de Matemática - Faculdade de Ciências[133X
                [33X[0;14YPorto[133X
                [33X[0;14YPortugal[133X
  
  
  Steve Linton
      Email:    [7Xmailto:steve.linton@st-andrews.ac.uk[107X
      Homepage: [7Xhttp://www-groups.dcs.st-and.ac.uk/~sal/[107X
      Address:  [33X[0;14YSchool of Computer Science[133X
                [33X[0;14YUniversity of St Andrews[133X
                [33X[0;14YJack Cole Building, North Haugh[133X
                [33X[0;14YSt Andrews, Fife, KY16 9SX[133X
                [33X[0;14YUnited Kingdom[133X
  
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2004 by Manuel Delgado, Steve Linton and José Morais[133X
  
  [33X[0;0Y[5XAutomata[105X  package is free software; you can redistribute it and/or modify it
  under     the     terms     of    the    GNU    General    Public    License
  ([7Xhttps://www.fsf.org/licenses/gpl.html[107X)  as  published  by the Free Software
  Foundation;  either  version 2 of the License, or (at your option) any later
  version.  For details, see the file 'GPL' included in the package or see the
  FSF's own site.[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThe  first  author wishes to acknowledge Cyril Nicaud and Paulo Varandas for
  their  help  in programming some functions of the very first version of this
  package.  He  wishes  also to acknowledge useful discussions and comments by
  Cyril Nicaud, Vítor H. Fernandes, Jean-Eric Pin and Jorge Almeida.[133X
  
  [33X[0;0YThe  first  author also acknowledges support of FCT through CMUP and the FCT
  and  POCTI  Project POCTI/32817/MAT/2000 which is funded in cooperation with
  the European Community Fund FEDER.[133X
  
  [33X[0;0YThe third author acknowledges financial support of FCT and the POCTI program
  through  a  scholarship  given  by  Centro  de Matemática da Universidade do
  Porto.[133X
  
  [33X[0;0YThe  authors  would  like  to  thank  Mark  Kambites for his contribution in
  finding bugs and making suggestions for the improvement of this package.[133X
  
  [33X[0;0YConcerning the maintenance:[133X
  
  [33X[0;0YThe first author was/is (partially) supported by:[133X
  
  [33X[0;0Ythe    Fundação    para   a   Ciência   e   a   Tecnologia   (FCT)   project
  PTDC/MAT/65481/2006;[133X
  
  [33X[0;0Ythe  [13XCentro de Matemática da Universidade do Porto[113X (CMUP), in particular for
  those periods in which CMUP was funded by[133X
  
  [33X[0;0Ythe  European  Regional  Development Fund through the program COMPETE and by
  the  Portuguese  Government  through  the  FCT - Fundação para a Ciência e a
  Tecnologia under the project PEst-C/MAT/UI0144/2011;[133X
  
  [33X[0;0YThe   first   author   wants   to   acknowledge   partial  support  by  CMUP
  (UID/MAT/00144/2013   and   UID/MAT/00144/2019),  which  is  funded  by  FCT
  (Portugal)  with  national  (MEC)  and European structural funds through the
  programs FEDER, under the partnership agreement PT2020.[133X
  
  
  -------------------------------------------------------
  [1XColophon[101X
  [33X[0;0YThis  work  started  in  1998, when the first author was in the LIAFA at the
  University  of Paris 7, in a post-doc. Encouraged by J. E. Pin, he began the
  implementation in [5XGAP[105X3 of an algorithm obtained some time before to answer a
  question from the realm of Finite Semigroups proposed by J. Almeida.[133X
  
  [33X[0;0YThe  first  version  of  this  package on automata was prepared by the first
  author  who  gave  it  the form of a [5XGAP[105X share package. In a second version,
  prepared  by the first and third authors, many functions have been added and
  the  performance  of  many  of  the existing ones has been improved. Further
  important improvements, specially concerning performance, have been achieved
  when  the  second  author joined the group. The first author is particularly
  grateful to Max Horn whose help made the release of Versions 1.14 and 1.15 a
  lot easier.[133X
  
  [33X[0;0YSince  Version 1.12, the package is maintained by the first two authors. Bug
  reports,  suggestions  and  comments are, of course, welcome. Please use our
  email addresses to this effect.[133X
  
  
  -------------------------------------------------------
  
  
  [1XContents (Automata)[101X
  
  1 [33X[0;0YIntroduction[133X
  2 [33X[0;0YFinite Automata[133X
    2.1 [33X[0;0YAutomata generation[133X
      2.1-1 Automaton
      2.1-2 IsAutomaton
      2.1-3 IsDeterministicAutomaton
      2.1-4 IsNonDeterministicAutomaton
      2.1-5 IsEpsilonAutomaton
      2.1-6 String
      2.1-7 RandomAutomaton
    2.2 [33X[0;0YAutomata internals[133X
      2.2-1 AlphabetOfAutomaton
      2.2-2 AlphabetOfAutomatonAsList
      2.2-3 TransitionMatrixOfAutomaton
      2.2-4 InitialStatesOfAutomaton
      2.2-5 SetInitialStatesOfAutomaton
      2.2-6 FinalStatesOfAutomaton
      2.2-7 SetFinalStatesOfAutomaton
      2.2-8 NumberStatesOfAutomaton
    2.3 [33X[0;0YComparison of automata[133X
    2.4 [33X[0;0YTests involving automata[133X
      2.4-1 IsDenseAutomaton
      2.4-2 IsRecognizedByAutomaton
      2.4-3 IsPermutationAutomaton
      2.4-4 IsInverseAutomaton
      2.4-5 AddInverseEdgesToInverseAutomaton
      2.4-6 IsReversibleAutomaton
    2.5 [33X[0;0YBasic operations[133X
      2.5-1 CopyAutomaton
      2.5-2 NullCompletionAutomaton
      2.5-3 ListSinkStatesAut
      2.5-4 RemovedSinkStates
      2.5-5 ReversedAutomaton
      2.5-6 PermutedAutomaton
      2.5-7 ListPermutedAutomata
      2.5-8 NormalizedAutomaton
      2.5-9 UnionAutomata
      2.5-10 ProductAutomaton
      2.5-11 ProductOfLanguages
    2.6 [33X[0;0YLinks with Semigroups[133X
      2.6-1 TransitionSemigroup
      2.6-2 SyntacticSemigroupAut
      2.6-3 SyntacticSemigroupLang
  3 [33X[0;0YRational languages[133X
    3.1 [33X[0;0YRational Expressions[133X
      3.1-1 RationalExpression
      3.1-2 RatExpOnnLetters
      3.1-3 RandomRatExp
      3.1-4 SizeRatExp
      3.1-5 IsRationalExpression
      3.1-6 AlphabetOfRatExp
      3.1-7 AlphabetOfRatExpAsList
      3.1-8 CopyRatExp
    3.2 [33X[0;0YComparison of rational expressions[133X
    3.3 [33X[0;0YOperations with rational languages[133X
      3.3-1 UnionRatExp
      3.3-2 ProductRatExp
      3.3-3  StarRatExp
  4 [33X[0;0YAutomata [13Xversus[113X rational expressions[133X
    4.1 [33X[0;0YFrom automata to rational expressions[133X
      4.1-1 AutomatonToRatExp 
    4.2 [33X[0;0YFrom rational expression to automata[133X
      4.2-1 RatExpToNDAut
      4.2-2 RatExpToAutomaton
    4.3 [33X[0;0YSome tests on automata[133X
      4.3-1 IsEmptyLang
      4.3-2 IsFullLang
      4.3-3 AreEqualLang
      4.3-4 IsContainedLang
      4.3-5 AreDisjointLang
  5 [33X[0;0YSome functions involving automata[133X
    5.1 [33X[0;0YFrom one type to another[133X
      5.1-1 EpsilonToNFA
      5.1-2 EpsilonToNFASet
      5.1-3 EpsilonCompactedAut
      5.1-4 ReducedNFA
      5.1-5 NFAtoDFA
      5.1-6 FuseSymbolsAut
    5.2 [33X[0;0YMinimalization of an automaton[133X
      5.2-1 UsefulAutomaton
      5.2-2 MinimalizedAut
      5.2-3  MinimalAutomaton
      5.2-4 AccessibleStates
      5.2-5 AccessibleAutomaton
      5.2-6 IntersectionLanguage
      5.2-7 AutomatonAllPairsPaths
  6 [33X[0;0YFinite regular languages[133X
    6.1 [33X[0;0YDealing with finite regular languages[133X
      6.1-1 IsFiniteRegularLanguage
      6.1-2 FiniteRegularLanguageToListOfWords
      6.1-3 ListOfWordsToAutomaton
  A [33X[0;0YDirected graphs[133X
    A.1 [33X[0;0YDirected graphs[133X
      A.1-1 RandomDiGraph
      A.1-2 VertexInDegree
      A.1-3 VertexOutDegree
      A.1-4 AutoVertexDegree
      A.1-5 ReversedGraph
      A.1-6 AutoConnectedComponents
      A.1-7 GraphStronglyConnectedComponents
      A.1-8 UnderlyingMultiGraphOfAutomaton
      A.1-9 UnderlyingGraphOfAutomaton
      A.1-10 DiGraphToRelation
      A.1-11 MSccAutomaton
      A.1-12 AutoIsAcyclicGraph
  B [33X[0;0YDrawing automata[133X
    B.1 [33X[0;0YInstalling some external programs[133X
    B.2 [33X[0;0YFunctions to draw automata[133X
      B.2-1 DrawAutomaton
      B.2-2 DotForDrawingAutomaton
      B.2-3 DrawSubAutomaton
      B.2-4 DotStringForDrawingSubAutomaton
      B.2-5 DotStringForDrawingGraph
      B.2-6 DrawSCCAutomaton
  C [33X[0;0YInverse automata and subgroups of the free group[133X
    C.1 [33X[0;0YFrom subgroups to inverse automata[133X
      C.1-1 GeneratorsToListRepresentation
      C.1-2 ListToGeneratorsRepresentation
      C.1-3 FlowerAutomaton
      C.1-4 FoldFlowerAutomaton
      C.1-5 SubgroupGenToInvAut
    C.2 [33X[0;0YFrom inverse automata to subgroups[133X
      C.2-1 GeodesicTreeOfInverseAutomaton
      C.2-2 InverseAutomatonToGenerators
  
  
  [32X
