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<p><a id="X7B5CD9B7796BD926" name="X7B5CD9B7796BD926"></a></p>
<div class="ChapSects"><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>
<h3>4 <span class="Heading">Automata <em>versus</em> rational expressions</span></h3>
<p>A remarkable theorem due to Kleene shows that a language is recognized by a finite automaton precisely when it may be given by means of a rational expression, i.e. is a rational language.</p>
<p>An automaton and a rational expression are said to be <em>equivalent</em> when both represent the same language. In this chapter we describe functions to go from automata to equivalent rational expressions and <em>vice-versa</em>.</p>
<p><a id="X7E631B92873300C1" name="X7E631B92873300C1"></a></p>
<h4>4.1 <span class="Heading">From automata to rational expressions</span></h4>
<p><a id="X8751E3927CA4DEA1" name="X8751E3927CA4DEA1"></a></p>
<h5>4.1-1 AutomatonToRatExp </h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AutomatonToRatExp </code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AutToRatExp</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> FAtoRatExp</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>From a finite automaton, <code class="code">FAtoRatExp</code>, <code class="code">AutToRatExp</code> and <code class="code">AutomatonToRatExp</code>, which are synonyms, compute one equivalent rational expression, using the state elimination algorithm.</p>
<table class="example">
<tr><td><pre>
gap> x:=Automaton("det",4,2,[[ 0, 1, 2, 3 ],[ 0, 1, 2, 3 ]],[ 3 ],[ 1, 3, 4 ]);;
gap> FAtoRatExp(x);
(aUb)(aUb)U@
gap> aut:=Automaton("det",4,"xy",[[ 0, 1, 2, 3 ],[ 0, 1, 2, 3 ]],[ 3 ],[ 1, 3, 4 ]);;
gap> FAtoRatExp(aut);
(xUy)(xUy)U@
</pre></td></tr></table>
<p><a id="X8138227D7E65FC8C" name="X8138227D7E65FC8C"></a></p>
<h4>4.2 <span class="Heading">From rational expression to automata</span></h4>
<p><a id="X840EEB7B7DD8B03D" name="X840EEB7B7DD8B03D"></a></p>
<h5>4.2-1 RatExpToNDAut</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> RatExpToNDAut</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given a rational expression <var class="Arg">R</var>, computes the equivalent NFA</p>
<table class="example">
<tr><td><pre>
gap> r:=RationalExpression("aUab");
aUab
gap> Display(RatExpToNDAut(r));
| 1 2 3 4
--------------------------------
a | [ 1, 2 ]
b | [ 3 ]
Initial state: [ 4 ]
Accepting states: [ 1, 3 ]
gap> r:=RationalExpression("xUxy");
xUxy
gap> Display(RatExpToNDAut(r));
| 1 2 3 4
--------------------------------
x | [ 1, 2 ]
y | [ 3 ]
Initial state: [ 4 ]
Accepting states: [ 1, 3 ]
</pre></td></tr></table>
<p><a id="X866BCCB2788E8561" name="X866BCCB2788E8561"></a></p>
<h5>4.2-2 RatExpToAutomaton</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> RatExpToAutomaton</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> RatExpToAut</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given a rational expression <var class="Arg">R</var>, these functions, which are synonyms, use <code class="func">RatExpToNDAut</code> (<a href="chap4.html#X840EEB7B7DD8B03D"><b>4.2-1</b></a>)) to compute the equivalent NFA and then returns the equivalent minimal DFA</p>
<table class="example">
<tr><td><pre>
gap> r:=RationalExpression("@U(aUb)(aUb)");
@U(aUb)(aUb)
gap> Display(RatExpToAut(r));
| 1 2 3 4
-----------------
a | 3 1 3 2
b | 3 1 3 2
Initial state: [ 4 ]
Accepting states: [ 1, 4 ]
gap> r:=RationalExpression("@U(0U1)(0U1)");
@U(0U1)(0U1)
gap> Display(RatExpToAut(r));
| 1 2 3 4
-----------------
0 | 3 1 3 2
1 | 3 1 3 2
Initial state: [ 4 ]
Accepting states: [ 1, 4 ]
</pre></td></tr></table>
<p><a id="X85DCEFB88712056E" name="X85DCEFB88712056E"></a></p>
<h4>4.3 <span class="Heading">
Some tests on automata
</span></h4>
<p>This section describes functions that perform tests on automata, rational expressions and their accepted languages.</p>
<p><a id="X84E0143A860889A6" name="X84E0143A860889A6"></a></p>
<h5>4.3-1 IsEmptyLang</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsEmptyLang</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function tests if the language given through a rational expression or an automaton <var class="Arg"> R </var> is the empty language.</p>
<table class="example">
<tr><td><pre>
gap> r:=RandomRatExp(2);
empty_set
gap> IsEmptyLang(r);
true
gap> r:=RandomRatExp(2);
aUb
gap> IsEmptyLang(r);
false
</pre></td></tr></table>
<p><a id="X86AA1A5F7E1EEAFE" name="X86AA1A5F7E1EEAFE"></a></p>
<h5>4.3-2 IsFullLang</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsFullLang</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function tests if the language given through a rational expression or an automaton <var class="Arg"> R </var> represents the Kleene closure of the alphabet of <var class="Arg"> R </var> .</p>
<table class="example">
<tr><td><pre>
gap> r:=RationalExpression("aUb");
aUb
gap> IsFullLang(r);
false
gap> r:=RationalExpression("(aUb)*");
(aUb)*
gap> IsFullLang(r);
true
</pre></td></tr></table>
<p><a id="X8346D1B17DBF96E7" name="X8346D1B17DBF96E7"></a></p>
<h5>4.3-3 AreEqualLang</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AreEqualLang</code>( <var class="Arg">A1, A2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AreEquivAut</code>( <var class="Arg">A1, A2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>These functions, which are synonyms, test if the automata or rational expressions <var class="Arg">A1</var> and <var class="Arg">A2</var> are equivalent, i.e. represent the same language. This is performed by checking that the corresponding minimal automata are isomorphic. Note hat this cannot happen if the alphabets are different.</p>
<table class="example">
<tr><td><pre>
gap> y:=RandomAutomaton("det",4,2);;
gap> x:=RandomAutomaton("det",4,2);;
gap> AreEquivAut(x, y);
false
gap> a:=RationalExpression("(aUb)*ab*");
(aUb)*ab*
gap> b:=RationalExpression("(aUb)*");
(aUb)*
gap> AreEqualLang(a, b);
false
gap> a:=RationalExpression("(bUa)*");
(bUa)*
gap> AreEqualLang(a, b);
true
gap> x:=RationalExpression("(1U0)*");
(1U0)*
gap> AreEqualLang(a, x);
The given languages are not over the same alphabet
false
g</pre></td></tr></table>
<p><a id="X7FCB176285FA5BBB" name="X7FCB176285FA5BBB"></a></p>
<h5>4.3-4 IsContainedLang</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsContainedLang</code>( <var class="Arg">R1, R2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Tests if the language represented (through an automaton or a rational expression) by <var class="Arg"> R1 </var> is contained in the language represented (through an automaton or a rational expression) by <var class="Arg"> R2 </var> .</p>
<table class="example">
<tr><td><pre>
gap> r:=RationalExpression("aUab");
aUab
gap> s:=RationalExpression("a","ab");
a
gap> IsContainedLang(s,r);
true
gap> IsContainedLang(r,s);
false
</pre></td></tr></table>
<p><a id="X83F1DE067C2D31A5" name="X83F1DE067C2D31A5"></a></p>
<h5>4.3-5 AreDisjointLang</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AreDisjointLang</code>( <var class="Arg">R1, R2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Tests if the languages represented (through automata or a rational expressions) by <var class="Arg"> R1 </var> and <var class="Arg"> R2 </var> are disjoint.</p>
<table class="example">
<tr><td><pre>
gap> r:=RationalExpression("aUab");;
gap> s:=RationalExpression("a","ab");;
gap> AreDisjointLang(r,s);
false
</pre></td></tr></table>
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