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<p><a id="X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5"></a></p>
<div class="pcenter">
<h1>Automata</h1>
<p>( Version
1.12
)</p>
</div>
<p><b> Manuel Delgado
</b>
<br />Email: <span class="URL"><a href="mailto:mdelgado@fc.up.pt">mdelgado@fc.up.pt</a></span>
<br />Homepage: <span class="URL"><a href="http://www.fc.up.pt/cmup/mdelgado">http://www.fc.up.pt/cmup/mdelgado</a></span>
</p><p><b> Steve Linton
</b>
<br />Email: <span class="URL"><a href="mailto:sal@dcs.st-and.ac.uk">sal@dcs.st-and.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="http://www.dcs.st-and.ac.uk/~sal/">http://www.dcs.st-and.ac.uk/~sal/</a></span>
</p><p><b> José João Morais
</b>
<br />Email: <span class="URL"><a href="mailto:josejoao@fc.up.pt">josejoao@fc.up.pt</a></span>
</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2004 by Manuel Delgado, Steve Linton and José Morais</p>
<p>We adopt the copyright regulations of <strong class="pkg">GAP</strong> as detailed in the copyright notice in the <strong class="pkg">GAP</strong> manual.</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>The 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.</p>
<p>The 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.</p>
<p>The third author acknowledges financial support of FCT and the POCTI program through a scholarship given by Centro de Matemática da Universidade do Porto.</p>
<p>The authors would like to thank Mark Kambites (http://www.theory.informatik.uni-kassel.de/~kambites/) for his contribution in finding bugs and making suggestions for the improvement of this package.</p>
<p><a id="X7982162280BC7A61" name="X7982162280BC7A61"></a></p>
<h3>Colophon</h3>
<p>This 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 <strong class="pkg">GAP</strong>3 of an algorithm obtained some time before to answer a question from the realm of Finite Semigroups proposed by J. Almeida. It is now part of a separate package: <code class="code">finsemi</code>.</p>
<p>The first version of this package on automata was prepared by the first author who gave it the form of a <strong class="pkg">GAP</strong> 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. Bug reports, suggestions and comments are, of course, welcome. Please contact any of the authors to this effect. Our e-mail addresses are <span class="URL"><a href="mailto:mdelgado@fc.up.pt">mdelgado@fc.up.pt</a></span> or <span class="URL"><a href="mailto:sal@dcs.st-and.ac.uk">sal@dcs.st-and.ac.uk</a></span> or <span class="URL"><a href="mailto:josejoao@fc.up.pt">josejoao@fc.up.pt</a></span>.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents</h3>
<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">
Introduction
</span></a>
</div>
<div class="ContChap"><a href="chap2.html#X811E5FC2849C5644">2 <span class="Heading">Finite Automata</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X821C3B3687B1F2FF">2.1 <span class="Heading">Automata generation</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X87D8222D7B50731E">2.1-1 Automaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83CCDEF9814F1E6D">2.1-2 IsAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D39CECC7E12DD8A">2.1-3 IsDeterministicAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83C1148481BAA3DD">2.1-4 IsNonDeterministicAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81EC5331790D6022">2.1-5 IsEpsilonAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81FB5BE27903EC32">2.1-6 String</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X801019097C93BCCC">2.1-7 RandomAutomaton</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X80AB906D86BBC153">2.2 <span class="Heading">Automata internals</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A34B47778B50FFE">2.2-1 AlphabetOfAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8044B24B82C59BBF">2.2-2 AlphabetOfAutomatonAsList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X872BB42A81E4D0E7">2.2-3 TransitionMatrixOfAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B5C3CFA83FF80EA">2.2-4 InitialStatesOfAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8408FE8487028B7F">2.2-5 SetInitialStatesOfAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X78CDDCC27D085F00">2.2-6 FinalStatesOfAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80689F1480F9D959">2.2-7 SetFinalStatesOfAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D22AD207A3D5FF4">2.2-8 NumberStatesOfAutomaton</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X8454E24E7D9FC1C2">2.3 <span class="Heading">Comparison of automata</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X867887A683961C63">2.4 <span class="Heading">Tests involving automata</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8356E41086482483">2.4-1 IsDenseAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8676D8388053F1E7">2.4-2 IsRecognizedByAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80CCDD438258CD25">2.4-3 IsPermutationAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B7CA23680888C9C">2.4-4 IsInverseAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A4CDEFB84B97849">2.4-5 AddInverseEdgesToInverseAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8321BCE57E55FB30">2.4-6 IsReversibleAutomaton</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X82EB5BE77F9F686A">2.5 <span class="Heading">Basic operations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8225A1B886131707">2.5-1 CopyAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80D423A584246E2E">2.5-2 NullCompletionAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79F052EC81135807">2.5-3 ListSinkStatesAut</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8240136E7A26B1A6">2.5-4 RemovedSinkStates</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C0526217BFE7A65">2.5-5 ReversedAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A4A066583C71ABE">2.5-6 PermutedAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A72DDF0782E8D5E">2.5-7 ListPermutedAutomata</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7FA7DF6D87D63D67">2.5-8 NormalizedAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A94A77A7C65BA90">2.5-9 UnionAutomata</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83E772F2878546A4">2.5-10 ProductAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X85F6AD697DCA5765">2.5-11 ProductOfLanguages</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X79F21CB37B34A354">2.6 <span class="Heading">Links with Semigroups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B9994827CF94CC7">2.6-1 TransitionSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7E3F29DF86A26347">2.6-2 SyntacticSemigroupAut</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D058F0D83D7B49B">2.6-3 SyntacticSemigroupLang</a></span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X833D315483172905">3 <span class="Heading">Rational languages</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7C144D368043DE01">3.1 <span class="Heading">Rational Expressions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X801EC6F38568426D">3.1-1 RationalExpression</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7EE5A70F7F237C41">3.1-2 RatExpOnnLetters</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DA59CBE8571796C">3.1-3 RandomRatExp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E3AA84784019E6C">3.1-4 SizeRatExp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7DDB46817D6E79BE">3.1-5 IsRationalExpression</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8773359880149A98">3.1-6 AlphabetOfRatExp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X84B9922B7C006158">3.1-7 AlphabetOfRatExpAsList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X786A096681CAC3CD">3.1-8 CopyRatExp</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7FB9270D7E8FABF3">3.2 <span class="Heading">Comparison of rational expressions</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X83A280D47DB3A033">3.3 <span class="Heading">Operations with rational languages</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8206BD4E82A81D8F">3.3-1 UnionRatExp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E29107587611CE2">3.3-2 ProductRatExp</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X83D8DAE6862C8A96">3.3-3 StarRatExp</a></span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X7B5CD9B7796BD926">4 <span class="Heading">Automata <em>versus</em> rational expressions</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7E631B92873300C1">4.1 <span class="Heading">From automata to rational expressions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8751E3927CA4DEA1">4.1-1 AutomatonToRatExp </a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X8138227D7E65FC8C">4.2 <span class="Heading">From rational expression to automata</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X840EEB7B7DD8B03D">4.2-1 RatExpToNDAut</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X866BCCB2788E8561">4.2-2 RatExpToAutomaton</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X85DCEFB88712056E">4.3 <span class="Heading">
Some tests on automata
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84E0143A860889A6">4.3-1 IsEmptyLang</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X86AA1A5F7E1EEAFE">4.3-2 IsFullLang</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8346D1B17DBF96E7">4.3-3 AreEqualLang</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7FCB176285FA5BBB">4.3-4 IsContainedLang</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X83F1DE067C2D31A5">4.3-5 AreDisjointLang</a></span>
</div>
</div>
<div class="ContChap"><a href="chap5.html#X7919AA9384EBC6A5">5 <span class="Heading">Some functions involving automata</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X8050E142796E0CBF">5.1 <span class="Heading">From one type to another</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X81E06D518428CA3C">5.1-1 EpsilonToNFA</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X81DC84E17A170270">5.1-2 EpsilonToNFASet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X871F807D79CE148C">5.1-3 EpsilonCompactedAut</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X83B0473278DC14F3">5.1-4 ReducedNFA</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X87D0F9F17F2BEB57">5.1-5 NFAtoDFA</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7B61945581FE4AC6">5.1-6 FuseSymbolsAut</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X862A34E9801BEB25">5.2 <span class="Heading">Minimalization of an automaton</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7B5B5B10868FB525">5.2-1 UsefulAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X83C26846866AEE46">5.2-2 MinimalizedAut</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F7AE088808A5D00">5.2-3 MinimalAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7F484D5A781BB643">5.2-4 AccessibleStates</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X804A6BC979DA6E61">5.2-5 AccessibleAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BAACCAF7E2D213B">5.2-6 IntersectionLanguage</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8460C44386EE6225">5.2-7 AutomatonAllPairsPaths</a></span>
</div>
</div>
<div class="ContChap"><a href="chap6.html#X7AF3E5D081126EBD">6 <span class="Heading">Finite regular languages</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X85643AEB7E7FB39A">6.1 <span class="Heading">Dealing with finite regular languages</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X82971FC2851B7B30">6.1-1 IsFiniteRegularLanguage</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7E48CD3E78277FF7">6.1-2 FiniteRegularLanguageToListOfWords</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7F9C5C6F815773E6">6.1-3 ListOfWordsToAutomaton</a></span>
</div>
</div>
<div class="ContChap"><a href="chapA.html#X82FB3D357E1BE288">A <span class="Heading">Directed graphs</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chapA.html#X82FB3D357E1BE288">A.1 <span class="Heading">Directed graphs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X86CF9F66788B2A24">A.1-1 RandomDiGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X868EE741872B932D">A.1-2 VertexInDegree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X84DF2E8E7A7B32C6">A.1-3 VertexOutDegree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X7FA6FAAE7AA8715D">A.1-4 AutoVertexDegree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X7BA5F1DF7DA89DC5">A.1-5 ReversedGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X7F23780E7A12A79E">A.1-6 AutoConnectedComponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X7D5288C982F92481">A.1-7 GraphStronglyConnectedComponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X859B7C517AFBD198">A.1-8 UnderlyingMultiGraphOfAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X78CF8E507E100C62">A.1-9 UnderlyingGraphOfAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X78869D478792B3AD">A.1-10 DiGraphToRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X7D63604A8413AAAF">A.1-11 MSccAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X7971EE367B6B7F36">A.1-12 AutoIsAcyclicGraph</a></span>
</div>
</div>
<div class="ContChap"><a href="chapB.html#X82D249F0793E6561">B <span class="Heading">
Drawing automata
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chapB.html#X7988DBAB78EA0C06">B.1 <span class="Heading">
Installing some external programs
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chapB.html#X84C97CA079719B11">B.2 <span class="Heading">
Functions to draw automata
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X7BC2FDA77FD0237B">B.2-1 DrawAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X7AEE146D8391CA3D">B.2-2 DrawAutomata</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X7D17B77F829F9CCB">B.2-3 DrawGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X7E478FDD807853CA">B.2-4 DrawSCCAutomaton</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chapB.html#X7F5419527FFCD1DF">B.3 <span class="Heading">Drawings output formats</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chapB.html#X795DD98D86A1A441">B.4 <span class="Heading">Drawings extra graph attributes</span></a>
</div>
</div>
<div class="ContChap"><a href="chapC.html#X7DBBB0537ADC9899">C <span class="Heading">Inverse automata and subgroups of the free group</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chapC.html#X7DDDA5127A3D170C">C.1 <span class="Heading">From subgroups to inverse automata</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapC.html#X85358D097C314EB5">C.1-1 GeneratorsToListRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapC.html#X80F3E10784590374">C.1-2 ListToGeneratorsRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapC.html#X7EAFF7E879D115C5">C.1-3 FlowerAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapC.html#X7F729A4E8784D92E">C.1-4 FoldFlowerAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapC.html#X826D581D794F1BFB">C.1-5 SubgroupGenToInvAut</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chapC.html#X85F2060A86DBE62B">C.2 <span class="Heading">From inverse automata to subgroups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapC.html#X81DA149779A167BD">C.2-1 GeodesicTreeOfInverseAutomaton</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapC.html#X7F117C43814F2CDE">C.2-2 InverseAutomatonToGenerators</a></span>
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