<!DOCTYPEReturns=keyw>true codejava.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 177 "http://www.w3.org/TRdiv class="example"><pre>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[java.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 0
<p id="mathjaxlink" class="pcenter"><a href="chap2_mj.<5>2.16String</h5>
<p><a id="X811E5FC2849C5644" name="X811E5FC2849C5644
<div ="< java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 112
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#p>The way <strong class="pkg">GAP</strong> displays an automaton is quite natural, but when one wants to do small</> code class="><java.lang.StringIndexOutOfBoundsException: Range [273, 272) out of bounds for length 293
</span
<div class="ContSSBlock">
<span class""gap&t</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 135
<span =ContSS> > classnocss>nbsp;&;/java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 133
<span class="ontSS"br /><spanclass"ocss>nbsp; </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"nocss"> nbsp;</>< href="hap2htmlX81C5331790D6022"2.1-5 IsEpsilonAutomaton<<a>span>
<span class="ContSS"><brAutomaton"\,33,""[ , 2, ,[,, ] ][ ],[ 2 ];;
<span class="ContSS<pan class="GAPprompt">gap></span> <span class="GAPinput">Print(String(x));</span>
</div></div>
< class=ContSect"<spanclass"tocline><class">nbsp</pan><ahref="chap2.html#X80AB906D86BBC153">2.2 <span class="Heading">Automata internals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="java.lang.StringIndexOutOfBoundsException: Range [40, 45) out of bounds for length 0
<span class<pan class="GAPprompt">gap></span> <span class="GAPinput">z:=Automaton("epsilon",2,2,[[[1,2],],[[2],[1]]],[1,2],[1,2]);;</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X872BB42A81E4D0E7">2.2-3 TransitionMatrixOfAutomaton</a></span>
<ass"ContSS"<r />span =nocss>nbsp;nbsp;/><ahref"chap2.html#X7B5C3CFA83FF80EA">2.2-4 InitialStatesOfAutomaton</a></span>
<span class="ContSS"><br /><span java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 30
<""<table= "%><r>td =tdleft><code="unc">8227; java.lang.StringIndexOutOfBoundsException: Range [117, 116) out of bounds for length 287
< classclass=ContSS /=>n;nbsp<span>ahref"html#">2.2- SetFinalStatesOfAutomatona>/span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D22AD207A3D5FF4">2.2-8 NumberStatesOfAutomaton</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8454E24E7D9FC1C2">2.3 <span class="Heading">Comparison of automata</span></a>
<java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X867887A683961C63">2.4 <span class="Heading">Tests involving automata</span></a>
</span>
<div java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 "";java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 138
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8676D8388053F1E7 2 34 5
<span class="ContSS">----------
<span a | 2
<span class="ContSS"><br /><span java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 10
<java.lang.StringIndexOutOfBoundsException: Range [12, 5) out of bounds for length 143
java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 12
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#java.lang.StringIndexOutOfBoundsException: Index 114 out of bounds for length 26
</
<div class="ContSSBlock">
<span class="ContSS"><br /><lt non deterministicautomaton 3letterswith 3states&;
<span class="ContSS"><br /><span class="nocss"> <spanclass">gap&;span>< classGAPinput">andomAutomatonepsilon"2"";/>
<span class"ContSS"<br /><span class="nocss"> </><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 /><spanlt with states &;
<span class="ContSS"><span=GAPpromptgap;</><span">last);/span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A72DDF0782E8D5E">2| 1
< ">br /spanclass=n>n /java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 141
<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#java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 23
<span class="ContSS"><br /><span class="nocss"> </span><a href<panjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 97
</div></div>
<div class="ContSect"><span class="tocline"><span lt; deterministic automaton on 2 letters with 4 states >
</span
<div class="ContSSBlock">
<java.lang.StringIndexOutOfBoundsException: Range [45, 5) out of bounds for length 141
<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="h4>2.<spanclass="Heading">utomata internals<span>/h4
</div></div>
</div>
<pjava.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 146
<>have statesofan are named <span class="SimpleMath">1,2,3,...;</span> the alphabet may be given by the user. By default it is <span class="SimpleMath">{a,b,c,...}</span> (or <span class="SimpleMath">{a_1,a_2,a_3,...}</span> in the case of alphabets with more than <spanp>Returns the number of symbols in the alphabet of automaton <code class="code">aut</code>.</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><>< id"java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<" "821java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
<h4>2.1 <java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
<p>The way to create an automaton in <strong class="pkg">GAP</strong> is the following</p>
<div class= =java.lang.StringIndexOutOfBoundsException: Range [32, 22) out of bounds for length 103
<p><code class="code">Type</code> may be <code;java.lang.StringIndexOutOfBoundsException: Range [31, 28) out of bounds for length 60
<div class="example"><pre>
<span class="GAPprompt">gap></span> span"APprompt>java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 98
< deterministic automaton on 2 letters with 4 class="GAPprompt">gap></span> <span class="GAPinput,20)<span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(span class="GAPprompt">gap></span> <span class="GAPinput">AlphabetOfAutomatonjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 92
| 12 <class"GAPprompt">gap></span> <span class="GAPinput">AlphabetOfAutomatonAsList(a);</span>
----------------- a | 334 344
Initial state: [ 1 ]
Accepting states classGAPprompt>java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 92
</pre></div>
<p>The alphabetclass=>&t;span"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 98
ample"java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 127
&java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 60
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(aut);</span>
| 1234
----java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17 0 | pre</ivjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 1 | 344
nitialstate:[ 1]
Accepting state: [ 4 ]
</pre></div>
<p>Instead of leaving a hole in the transition matrix, we may write a <java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class">&;/span<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 138
l; automaton lwith 4 states gjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(ndaut);</span>
| 1234
----------------------------------------- a | [ 3 ] [ 1, 2 ] [ 3 ] b | [ 3 ] [ 4 ] class"example"<>
Initial span class=GAPprompt">gap></span> <span class="GAPinput">aut:=Automaton("det",4,2,[[3,,3,4],[3,4,0,]],[1],[4]);;</span>
tingjava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 22
</pre></div>
<p>Also in the same way can be given pre<div
<div class="examplep< id=X7B5C3CFA83FF80EA "7B5C3CFA83FF80EA<p
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Automaton("java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">></span> <span class="GAPinput">[2]]],[2],[2,3]);</span>
< epsilonautomaton on3 with g
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(x);</span>
| 123
------------------------------ 0 | [ 2 ] [ 3 ] 1 | [ 1, 3 ] [ 1 ]
@ | [ 1 ] [ 2 ] [ 2 ]
Initialstate 2 java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
Accepting states: [ 2, 3 ]
java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 12
<p>Bigger automata are java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 5
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut
<span class="GAPprompt">></span> <span class="GAPinput">[3,4,0,0,java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 42
< deterministic on2 letters 16 statesgt;
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(aut);</span> 1a4 1b3 2b4
... some more lines 15a6 15b8 16b7
Initial state: [ 1 ]
Accepting state: 4 ]
</pre></div>
<div class="func" java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<p>In the presence of an object <var class="Arg">O</var>, one may want to test whether <code class="code">O</code> is an automaton. This may be done using the function <code class="codespan =G">></<spanclass=GAPinput"(aut;<span
<div class="example"><pre>
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAutomaton(x);</spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
</pre></div>
<div class="func"><table<>Returnsthe final < =""aut/ode></p
<p>Returns <code
<div class="< "">pre>
<span class="GAPprompt">gap></span> <<span class="GAPprompt">gap></span> <spanGAPinput"aut:=Automaton(det",4,2,[,3434,0,][,4;/java.lang.StringIndexOutOfBoundsException: Range [124, 123) out of bounds for length 124
<span class="GAPprompt">gap></<pa id"=java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
true
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsNonDeterministicAutomaton</code>( <var class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Returns <code class="keyw">true</code> when <code class="code">aut</code> is a non deterministic automaton and <code class="keyw">false</code> otherwise.</p>
<div class="example"><pre>
<class=GAPprompt">>/><spanclass=GAPinputy:Automaton("nondet",3,2,[[,[1,3],],[,[2,3],[1,3]]],[1,2],[1,3]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNonDeterministicAutomaton(y);</span"GAPprompt">gap&t<span>< "APinput>(aut;<>
true
</prespanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 97
< =java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<h5>2.1-5 IsEpsilonAutomaton</h5>
<div class="func"><table class="func" width="100%"><java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 0
< an <span class="SimpleMath">ϵ</span>-automaton and <code class="keyw">false</code> otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></<>Returns numberof of java.lang.StringIndexOutOfBoundsException: Range [46, 44) out of bounds for length 79
<span class="GAPprompt">gap
</pre></divspan=>gap<java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 124
<div class="func"><table class="func" width="100%"><tr><td class=" class="GAPprompt">gap></span> <span class="GAPinput">NumberStatesOfAutomaton
<p>The way <strong class="pkg">GAP</java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <spanh4>. spanjava.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 64
< GAPpromptgap&;</spanspan class"APinput>String()span> "Automaton(\"det\",3,\"ab\",[ [ 0, 2, 0 ], [ 0, 1, 0 ] ],[ 3 ],[ 2 ]);;"
<span class="GAPprompt">gap&
Automaton("det",3,"abdiv "<re
</pre></div>
<div class="example"><pre>/ <span class=GAPinputy=utomaton"et[[ 0 0][1,3,0 ]][ ,[,3 ];;span
<span class="GAPprompt">gap></span> <span class="GAPinput">z:=Automaton("epsilon",2,2,[[[1,2],],[[2],[1]]],[1,2],[1,2]);;</span>
<span class="GAPprompt">gap></span> <span span class="GAPprompt">gap></span> <span class(et([,,]);span
Automaton("epsilon",2,"a@",[ [ [ 1, 2 ], 1, ;java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
</pre></div>
<div class="func"><table class="func" widthp>This tests automata<p
<p>Given the <var class="Arg">Type</var>, the <var class="<p><a id="X8356E41086482483" name="X8356E41086482483"></a></p
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomAutomaton("det",5,"ac");</span>
< deterministic automaton on 2 letters with 5 states >
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
| 235
-------------------- a | 23
c | 23
Initial state: [ 4 ]
Accepting states: [ 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RandomAutomaton("nondet",3,["a","b","c"]);</span>
< non deterministic automaton on 3 letters with 3 states >
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 104
< epsilon automaton on 4 letters with 2 states >
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 4
< epsilon automaton on
<span class="GAPprompt">gap<><a id="X80CCDD438258CD25" name="X80CCDD438258CD25"></a></p>
| 12
---------------------- a | [ 1, 2 ] b | [ 2 ] [ 1 ]
@ | [ 1, 2 ]
Initial: ]
Accepting states: [ 1, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=RandomTransformation(3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=RandomTransformation(3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut:=RandomAutomaton("det",4,[a,b]);</
< deterministic automaton on 2 letters with 4 states >
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AlphabetOfAutomaton</code>( <var class="Arg">aut</var> )</td><tdclass="tdright">( function )</td></tr></table></div>
<p>Returns the number of symbols in the alphabet of automaton <code class="code">aut</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="class="GAPprompt">gap></span> <span class="GAPinput">IsInverseAutomaton(x);</span>
<p>Returns the alphabet of automaton <code p>Frequently an inverse automaton is thought asjava.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 294
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=RandomAutomaton("det",5,"cat");</span>
< deterministic automaton on 3 letters with 5 states >
<span class="GAPprompt">gap></span> <span class="GAPinput">AlphabetOfAutomaton(a);</span> 3
<span class="GAPprompt">gap></span> <span class="GAPinput">AlphabetOfAutomatonAsList(a);</span> "cat"
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=RandomAutomaton("det",5,20);</span>
< deterministic automaton on 20 letters with 5 states >
<span class="GAPprompt">gap></span> <span class="GAPinput">AlphabetOfAutomaton(a);</span> 20
<span class="GAPprompt">gap></span> <span class="GAPinput">AlphabetOfAutomatonAsList(a);</span> "abcdefghijklmnopqrst"
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=RandomAutomaton("det",5,30);</span>
< deterministic automaton on 30 letters with 5 states >
<span class="GAPprompt">gap></span> <span class="GAPinput">AlphabetOfAutomaton(a);</span> 30
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ "a1", "a2", "a3", "a4", "a5", "a6", "a7", "a8", "a9", "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 "a12", "a13", "a14"java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
|java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
</pre></div>
<>a =X8321" name"java.lang.StringIndexOutOfBoundsException: Range [58, 52) out of bounds for length 62
<div class="func"><table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>Returns transitionmatrix of automatoncodeclass="ode><code.<>
<div class="example"><pre>
<class"GAPprompt>></> span class=G"a:Automaton"det"4,[3,4],3,,0][][4);/spanjava.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ 0, 0 ]]
</pre></div>
="java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 233
<p>Returns the initial states of automaton <code class="code">aut</code>.</p>
<div class="example"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
=>&;<span> <span class="APinput">ut:=Automaton("et,,2,[[3,,,,0,],[1,4);</panjava.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInitialStatesOfAutomaton(aut,4);</span>
<span class="GAPprompt">gap></span> <spanp< ="">aut/ isadeterministic it iscomplete returns< ="><var,otherwise the (with a state of< ="Arg"aut</var>. Notice that thewords recognizedby <var class="rg">aut</var and its completion are the same.</p>
[ 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput
<span class"">gap&;</pan> spanclass"APinput
[ 2, 3 ]
</pre></div>
<div class="func"><table class="func" width="100%"><trjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<p>Returns the final states of automaton <code class="code">aut</code>.</p>
<div class="example"><pre>
<span class="GAPprompt | 3 java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
s ="">gap;<span < =""Faut);/>
[ 4 ]
</pre></div>
<div class="func"><table<> emsinkstate</>ie <class">/ notinitial accepting ""q/> (Notice java.lang.StringIndexOutOfBoundsException: Range [379, 376) out of bounds for length 426
<p>Sets the final states of automaton <code class="code">aut</code>. <code class="code">F</code> may be a positive integer or a list of positive integers.</p>
<
<span class="GAPprompt<p><aid"79 =X79F052EC81135807">/a>pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<span class="GAPprompt">gap><5>2.-3ListSinkStatesAut/h5>
[ 4 ]
<span class="GAPprompt<ivclass"func"><ableclass=func ="100%><tr>td class=tdleft">class=""‣ ListSinkStatesAut</code>( <var class="Arg">aut</var> )</td><td class="tdright">( function )</td></tr></table></div>
< class="GAPinput"FinalStatesOfAutomatonaut)</>
[ 2 ]
</pre></div>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'><span class="GAPprompt">gap></span> <span class="GAPinput">ListSinkStatesAut(x);</span>
the number of statesof code =code>ut/code>.p>
<h4>2.3 <span class="Heading">Comparison of automata
<p>Although there is no standard java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
<div class="example"><pre>
<span classp> allsinkstates ofautomaton var ="Arg">aut/>./p>
<span java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
<span classdivclass""<>
false
< class=GAPinput>Set[,,]);span 2
</pre></div>
divclass=func>table class=""width="100"><>tdclass="dleft"< class=func"&8227;IsDenseAutomaton/(<class"">ut/var>)/td< class="tdright">( function )</td></tr></table></div>
<p>Tests whether a deterministic automaton <code class="code">aut</code> is complete b | 12
<="example"><re
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 22
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
false
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 70
<p>The arguments are: java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<div class="example"><pre>
spanclass=""gapgt;span span ">:A(det"3,2[1,,][2,13],[][];/span
<span class="GAPpromptspanclass"APprompt"><span><class=">:=(y;Djava.lang.StringIndexOutOfBoundsException: Range [95, 94) out of bounds for length 105
true
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 0
<p>The argument is a deterministic automaton. Returns <code class="keyw"><div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code="func">‣ PermutedAutomaton</code>( <var class="Arg">aut</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Automaton("det",3,2<>Given an automaton <var class"">utvar and list<class=A">representing permutation of the states, outputs the equivalent permuted automaton.</p>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPermutationAutomaton(x);</span>
true
</pre></div>
<p><a id="X7B7CA23680888C9C" name=" | 4
<h5>2.4-4 IsInverseAutomaton</h5>
<div class="func"><table class="func" width="100%"><tr>< 1
<p>The argumentisa automaton.Returns<ode ="">true<code ofthe alphabetinducesan functionthe verticesand <ode=""false/> java.lang.StringIndexOutOfBoundsException: Range [219, 217) out of bounds for length 222
<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The argument is an inverse java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
--------------
a | 13
b | 12
Initial state 2
Accepting state: [ 1 ]
<panclass"GAPprompt">apgt<span> <span class="GAPinput">AddInverseEdgesToInverseAutomaton(x);Display(x);</span>
|1 java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
-------
a | 13 2
A | 23
B | 23
state: [ ]
Accepting state: [ 1 ]
</pre</divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
< class="">
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Automaton("detjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span ="GAPprompt>></span> span ="GAPinput">IsReversibleAutomaton(x);</span>
true
</pre></div>
42.5class=""> <span></java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
<p><a id="X8225A1B886131707" namejava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
<h5>2.5-1 CopyAutomaton</h5>
<div class="func"><table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 13
<p>Returns a new automaton, which is a copy of automaton a | 1 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
E""java.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 62
<h5>2.5-2 NullCompletionAutomaton</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; NullCompletionAutomaton</code>( <var class="Arg">aut</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="code">aut</code> is a deterministic automaton. If it is complete returns <var class="Arg">aut</var>, otherwise returns the completion (with a null state) of <var class="Arg">aut</var>. Notice that the words recognized by <var class="Arg">aut</var> and its completion are the same.</p>
<p>The state added is a <em>sink state</em> i.e. it is a state <span class="SimpleMath">q</span> which is not initial nor accepting and forall letter <span class="SimpleMath">a</span> in the alphabet of the automaton, <span class="SimpleMath">q</span> is the result of the action of <span class="SimpleMath">a</span> in <span class="SimpleMath">q</span>. (Notice that reading a word, one does not go out of a sink state.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; ListSinkStatesAut</code>( <var class="Arg">aut</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Computes the list of all sink states of the automaton <var class="Arg">aut</var>.</p>
<div class="func"java.lang.StringIndexOutOfBoundsException: Range [17, 12) out of bounds for length 12
<p>Removes all sink states of thep<id=X7A94A77A7C65BA90 name=java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; PermutedAutomaton</code>( <var class="Arg">aut</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given an automaton <var class="Arg">aut</var> and a list <var class="Arg">p</var> representing a permutation of the states, outputs the equivalent permuted automaton.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; ListPermutedAutomata</code>( <var class="Arg">aut</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given an automaton <var class="Arg">aut</var>, returns a list of automata with permuted states</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Automaton("det",3,2,[ [ 0, 2, 3 ], [ 1, 2, 3 ] ],[ 1 ],[ 2, 3 ]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ListPermutedAutomata(x);</span>
[ < deterministic automaton on 2 letters with 3 states >,
< deterministic automaton on 2 letters with 3 states >,
< deterministic automaton on 2 letters with 3 states >,
< deterministic automaton on 2 letters with 3 states >,
< deterministic automaton on 2 letters with 3 states >,
< deterministic automaton on 2 letters with 3 states > ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; NormalizedAutomaton</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Produces an equivalent automaton but in which the initial state is numbered 1 and the accepting states have the greatest numbers.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func"> b| [2]
<p>Produces the disjoint union of the deterministic or non deterministic automata <code class="code">A</code> Accepting states: [ 1, 2, 4, 5, 6 ]
d class=example">pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Automaton("det",3,2,[ [ 1, 2, 0 ], [ 0, 1, 2 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class=""gapgt;</span spanclass"GAPinput>:=Automaton"det"32,[ 0,, 3 ] [ 0 0 ][ 1 ][ 1 2,3 ]);</>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnionAutomata(x,y);</span>
< non deterministic automaton on 2 letters with 6 states >
<span class="GAPprompt">gap></span> <span class="GAPinput"><p>The arguments must be deterministic automata. Returns the product <var class=Arg>A1</var and <var class="Arg">2</var./p>
| 123456
---------
a | [
b |
Initialdiv class">prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26 124 ,6
</pre></divspanclass"">gap;</ < class"GAPinput">x);/pan>
a=nameXjava.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
<h5>2.5-10 ProductAutomaton</h5>
<div class="func"><table class="func" width="100%"><tr><td classspan ">ap&t;/ span class""y: java.lang.StringIndexOutOfBoundsException: Range [78, 76) out of bounds for length 143
<p>The arguments must be deterministic automata. Returns the product of <var class="Arg">A1</var> and <var class="Arg">A2</var>.</p>
<p>Note: <span class="SimpleMath">(p-------
<div class="example"><pre>
<span class="GAPprompt">gap></span>
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 80
| 123
-----------<span class=GAPprompt">gap>/span> <span class="GAPinput">z:=ProductAutomaton(x, y);;Display(z);</span>
a | 12
b | 13
Initial state: [ 1 ]
cceptingstates: [1, 2]
<span class="GAPprompt">gap></span> <span class="GAPinput">y := Automaton( 245
<span class="GAPprompt">gap></span> java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 23
| 123
--------------
a | 12
b | 2
Initial state: [ 2 ]
Accepting states: [ 1, 2 ]
span> < ">:=ProductAutomaton(,y);(z)</spanjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
| 123456789
--------------------------------
a | 1245
b | 28
Initial state: [ 2 ]
Accepting states: [ 1, 2, 4, 5 ]
</pre></div>
<div class="example"><pre>
<span p><a id="X7B9994827CF94CC7name"X7B9994827CF94CC7">/>/pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<span class="GAPprompt">></span> <<h5>2.6-1TransitionSemigroup</h5>
<span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
spanclass"GAPprompt">gap></span> <span class="GAPinput">Size(s); </span> 30
</pre></java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
<iv =func"< =func width"100%>tr<td ="tdleft"<codeclass="func>&8227; SyntacticSemigroupAut</code>( <var class="Arg">aut</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the syntactic semigroup of the deterministic automaton <var sitionSemigroupaut); /pan>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Automaton("det",3,2,[ [ 1, 2, 0 ], [ 0, java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=<p>The transition semigroup of the minimal automaton recognizing a languagejava.lang.StringIndexOutOfBoundsException: Range [97, 96) out of bounds for length 133
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(S);</span> 3
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; SyntacticSemigroupLang</code>( <var class="Arg">rat</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the syntactic semigroup of the language given by the rational expression <var class="Arg">rat/var></
<div class="example"><pre>
<span classspanclass=GAPprompt>&<span><span ""x=(""32[ [ 12[012java.lang.StringIndexOutOfBoundsException: Index 138 out of bounds for length 138
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=SyntacticSemigroupLang(rat);;</span class=>&;/span span =GAPinput"()</>
<java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 7
</pre
<div class="chlinkprevnextbot"> <a href="chap0java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="chlinkbot>span class=chlink1"GotoChapter /pan< href"chap0">Top/><a href="chap1.html>1</> <a href="chap2.html">2/> < href".tml>3> <ahref="hap4.html"4/a> ahref"hap5."5/a < =chap6html>6/a> a =chapA.html"A/a> <a href="chapB.html">B</a> <a href="chapC.html">C</a> <a href="chapBib.html><a> <ahref"h"Ijava.lang.StringIndexOutOfBoundsException: Range [407, 406) out of bounds for length 418
<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></pjava.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 0
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