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<h1>kbmag</h1>


<h2>Knuth-Bendix on Monoids and Automatic Groups</h2>

<p>
    1.5.11</p>

<p>
    3 January 2023
  </p>

</div>
<p><b>
    Derek Holt




  </b>
<br />Email: <span class="URL"><a href="mailto:D.F.Holt@warwick.ac.uk">D.F.Holt@warwick.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://homepages.warwick.ac.uk/staff/D.F.Holt/">https://homepages.warwick.ac.uk/staff/D.F.Holt/</a></span>
<br />Address: <br />Mathematics Institute<br /> University of Warwick<br /> Coventry CV4 7AL<br /> UK<br />
</p>

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<h3>Abstract</h3>
<p>The <strong class="pkg">KBMag</strong> package is a <strong class="pkg">GAP</strong> interface to some `C' programs for running the Knuth-Bendix completion program on finite semigroup, monoid or group presentations, and for attempting to compute automatic structures of finitely presented groups.



<p>Bug reports, comments, suggestions for additional features, and offers to implement some of these, will all be very welcome.</p>

<p>Please submit any issues at <span class="URL"><a href="https://github.com/gap-packages/kbmag/issues/">https://github.com/gap-packages/kbmag/issues/</a></span>.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 1997 by Derek Holt</p>

<p>This package may be distributed under the terms and conditions of the GNU Public License Version 2.</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>This documentation was prepared with the <strong class="pkg">GAPDoc</strong> <a href="chapBib.html#biBGAPDoc">[LN17]</a> and <strong class="pkg">AutoDoc</strong> <a href="chapBib.html#biBAutoDoc">[GH17]</a> packages.</p>

<p>The procedure used to produce new releases uses the package <strong class="pkg">GitHubPagesForGAP</strong> <a href="chapBib.html#biBGitHubPagesForGAP">[Hor17]</a> and the package <strong class="pkg">ReleaseTools</strong>.</p>

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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
</div>
<div class="ContChap"><a href="chap2.html#X86F080117DE22242">2 <span class="Heading">The Knuth-Bendix program on semigroups, monoids and groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7F474DEE787325CD">2.1 <span class="Heading">Creating a rewriting system</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8722C57284F51940">2.1-1 KBMAGRewritingSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X85BFCE4B79A782D0">2.2 <span class="Heading">Elementary functions on rewriting systems</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X879AEB2885DB4988">2.2-1 IsKBMAGRewritingSystemRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8006790B86328CE8">2.2-2 IsConfluent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X832A66088276451E">2.2-3 SemigroupOfRewritingSytem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8643F8C480036EC6">2.2-4 ExternalWordToInternalWordOfRewritingSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7CF67C5E7F31F19A">2.2-5 Alphabet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X833EAA8C86356F42">2.2-6 Rules</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7AB017987DAE15E7">2.2-7 ResetRewritingSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7BDD01C183CD3234">2.3 <span class="Heading">Setting the ordering</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7E89CFE87973DD14">2.3-1 SetOrderingOfKBMAGRewritingSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7BB411528630D4E9">2.4 <span class="Heading">Control parameters</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X80C0185D8035B7F8">2.4-1 InfoRWS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7C1BF15280B3CE5B">2.4-2 OptionsRecordOfKBMAGRewritingSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X830D97B5805251E0">2.5 <span class="Heading">The Knuth-Bendix program</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8412C40B7B2DC8E0">2.5-1 KnuthBendix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7ED0190D7BDA74D7">2.5-2 ReductionAutomaton</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8786DA3679BA75C8">2.6 <span class="Heading">The automatic groups program</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X828FA0177E4C5733">2.6-1 AutomaticStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X82CAA53A7926DA74">2.6-2 WordAcceptor</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X807029A8841FCAF3">2.7 <span class="Heading">Word reduction</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84A3462B826AD47C">2.7-1 IsReducedWord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A5773A77C6BAC3F">2.7-2 ReducedWord</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7B8F6EBC87AF42C6">2.8 <span class="Heading">Counting and enumerating irreducible words</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X858ADA3B7A684421">2.8-1 Size</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X84F59A2687C62763">2.8-2 Order</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7CD646CD7AA2B718">2.8-3 EnumerateReducedWords</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7981EA28835C7A46">2.8-4 GrowthFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7E98CC0078047A36">2.9 <span class="Heading">Rewriting System Examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8388E29680F31ABD">2.9-1 <span class="Heading">Example 1</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7A18778D836BC971">2.9-2 <span class="Heading">Example 2</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7D680484821C7835">2.9-3 <span class="Heading">Example 3</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X848E4DDE845A6EE9">2.9-4 <span class="Heading">Example 4</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83FE3ED7852DDFAD">2.9-5 <span class="Heading">Example 5</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X7828AE2881251C6A">3 <span class="Heading">The Knuth-Bendix program on cosets</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8128360785343F6A">3.1 <span class="Heading">Subgroups, cosets and subgroup presentations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X834DC6477EE1482B">3.1-1 SubgroupOfKBMAGRewritingSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CDC9FDA855596C4">3.1-2 ResetRewritingSystemOnCosets</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7828AE2881251C6A">3.2 <span class="Heading">The Knuth-Bendix program on cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X822268027EF50066">3.2-1 KnuthBendixOnCosets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X857B2B40860C7F33">3.2-2 RewritingSystemOfSubgroupOfKBMAGRewritingSystem</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7832C01283EA446E">3.2-3 IsConfluentOnCosets</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X83CE52E17BB34E5F">3.3 <span class="Heading">The automatic cosets program</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A2595DB84579665">3.3-1 AutomaticStructureOnCosets</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7823571D7CD7F81D">3.3-2 PresentationOfSubgroupOfKBMAGRewritingSystem</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X83A702237F6BA69A">3.4 <span class="Heading">Word reduction on cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C27E5EE8462A720">3.4-1 IsReducedCosetRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X794181377A59DAC1">3.4-2 ReducedCosetRepresentative</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X827BF1CF85337C8B">3.5 <span class="Heading">Counting and enumerating irreducible words for cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83A0356F839C696F">3.5-1 Index</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84596FE47AFBCC91">3.5-2 EnumerateReducedCosetRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D0F6F3781DF343D">3.5-3 GrowthFunctionOfCosetRepresentatives</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X79BA526D818D1DE7">3.6 <span class="Heading">Examples of the use of Rewriting System on Cosets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8388E29680F31ABD">3.6-1 <span class="Heading">Example 1</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A18778D836BC971">3.6-2 <span class="Heading">Example 2</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X7E2A9D617D06547C">4 <span class="Heading">The stand-alone package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7C0DD0867AF616E6">4.1 <span class="Heading">Functions for manipulating finite state automata</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87AE05A77CDCE20F">4.1-1 IsInitializedFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8566C07585AB9BFE">4.1-2 InitializeFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E54AFBF87E6178D">4.1-3 FSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83AB55DB784F6919">4.1-4 WriteFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8095E72D7F4BED3B">4.1-5 IsDeterministicFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7BB0BDD785791A32">4.1-6 AlphabetFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85E8D92D7EA2AA72">4.1-7 NumberOfStatesFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X843A7B7681142B68">4.1-8 NumberOfLettersFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A1A2E47792F0C27">4.1-9 AcceptingStatesFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79C1FD457AD765AD">4.1-10 InitialStatesFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E583EFD877E5C76">4.1-11 DenseDTableFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C105C9E8549F356">4.1-12 SparseTableFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E64B44A7F30EC6D">4.1-13 TargetDFA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85ED262883274F58">4.1-14 TargetsFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C036453817D1D81">4.1-15 SourcesFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X847110228341E4A0">4.1-16 WordTargetDFA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D605322790678FA">4.1-17 IsAcceptedWordDFA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87E76EC885C533E8">4.1-18 AddStateFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X879CB95C86BC78AC">4.1-19 DeleteStateFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X849090A687418D32">4.1-20 PermuteStatesFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79244F2C7C3BF1A0">4.1-21 AddLetterFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E49F0E27D62BA29">4.1-22 DeleteLetterFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F660FCC7C845197">4.1-23 PermuteLettersFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X861E772582C50021">4.1-24 AddEdgeFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B15A2B5879610C8">4.1-25 DeleteEdgeFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86B8A7F58083F872">4.1-26 SetAcceptingFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79719ADF82A605B8">4.1-27 SetInitialFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X873C8BBF814C1561">4.1-28 IsAccessibleFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8682C2477B038D80">4.1-29 AccessibleFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86941B6A7C95B91F">4.1-30 IsTrimFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7DABF5A87D9CF3F0">4.1-31 TrimFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F06649787E078F6">4.1-32 IsBFSFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X868F0428795F2594">4.1-33 BFSFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83612F7678F790C6">4.1-34 LSizeDFA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8753DD8E7FAD4030">4.1-35 LEnumerateDFA</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X79A6F86484B6DAFE">4.2 <span class="Heading">Functions calling external programs</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B8AF0EF79125730">4.2-1 DeterminizeFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X858E2B79803E4320">4.2-2 MinimizeFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X78FF5D83848C9B8A">4.2-3 NotFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F14B8B1817981B5">4.2-4 StarFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C474BA67D9B0273">4.2-5 ReverseFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X837FC04D82EF7638">4.2-6 ExistsFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X788520037939B0F1">4.2-7 SwapCoordsFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X813DA14C7DD4B370">4.2-8 AndFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D4FB82082DD4474">4.2-9 OrFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C13E593822442B7">4.2-10 ConcatFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8526528584A70717">4.2-11 LanguagesEqualFSA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F7939727D8E8F39">4.2-12 GrowthFSA</a></span>
</div></div>
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<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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