<!-- $Id: rational.xml,v 1.09  Exp $ -->

<Chapter><Heading>Rational languages</Heading>
Rational languages are conveniently represented through rational expressions. These are finite expressions involving letters of the alphabet; <C>concatenation</C>, corresponding to the <E>product</E>; the symbol <C>U</C>, corresponding to the <E>union</E>; and  the symbol <C>*</C>, corresponding to the Kleene's star.
<Index>rational expressions</Index>

<Section><Heading>Rational Expressions</Heading>
The expressions <C>@</C> and <C>"empty\_set"</C> are used to represent the empty word and the empty set respectively.


 <ManSection> 
<Func Name="RationalExpression" Arg=" expr[, alph] "/>
<Description>
A rational expression can be created using the function <C>RationalExpression</C>. <A>expr</A> is a string representing the desired expression literally and <A>alph</A> (may or may not be present) is the alphabet of the expression. Of course <A>alph</A> must not contain the symbols '@', '(', ')', '*' nor 'U'. When <A>alph</A> is not present, the alphabet of the rational expression is the set of symbols (other than '"', etc...) occurring in the expression. (The alphabet is then ordered with the alphabetical order.)
<Example><![CDATA[
gap> RationalExpression("abUc");
abUc
gap> RationalExpression("ab*Uc");
ab*Uc
gap> RationalExpression("001U1*");
001U1*
gap> RationalExpression("001U1*","012");
001U1*
]]></Example>
</Description>
</ManSection>


 <ManSection> 
<Func Name="RatExpOnnLetters" Arg=" n, operation, list "/>
<Description>
This is another way to construct a rational expression over an alphabet. The user may specify the alphabet or just give the number <M>n</M> of letters (in this case the alphabet <M>\{a,b,c,\ldots\}</M> is considered).  <A>operation</A> is the name of an operation, the possibilities
being: <C>product</C>, <C>union</C> or <C>star</C>. <A>list</A> is a list
 of rational expressions, a rational expression in the case of
``star'', or a 
list consisting of an integer when the rational expression is a single
letter. The empty list <C>[  ]</C> and <C>empty\_set</C> are other possibilities for
<C>list</C>. 

An example follows
<Example><![CDATA[
gap> RatExpOnnLetters(2,"star",RatExpOnnLetters(2,"product",
> [RatExpOnnLetters(2,[],[1]),RatExpOnnLetters(2,"union",
> [RatExpOnnLetters(2,[],[1]),RatExpOnnLetters(2,[],[2])])]));      
(a(aUb))*
]]></Example>
 
The empty word and the empty set are the rational expressions:
<Example><![CDATA[
gap> RatExpOnnLetters(2,[],[]);         
@
gap> RatExpOnnLetters(2,[],"empty_set");
empty_set
]]></Example>
 </Description></ManSection> 

<ManSection> 
<Func Name="RandomRatExp" Arg="arg"/>
<Description>
Given the number of symbols of the alphabet and (possibly) a factor <M>m</M> which is intended to increase the randomality of the expression,
returns a pseudo random rational expression over that alphabet.
<Example><![CDATA[
gap> RandomRatExp(2);
b*(@Ua)
gap> RandomRatExp("01");
empty_set
gap> RandomRatExp("01");
(0U1)*
gap> RandomRatExp("01",7);
0*1(@U0U1)
]]></Example>
</Description> 
</ManSection> 

<ManSection> 
<Func Name="SizeRatExp" Arg="r"/>
<Description>
Returns the size, i.e. the number of symbols of the alphabet, of the rational expression <A>r</A>.
<Example><![CDATA[
gap> a:=RationalExpression("0*1(@U0U1)");
0*1(@U0U1)
gap> SizeRatExp(a);
4
]]></Example>
</Description> 
</ManSection> 

<ManSection> 
<Func Name="IsRationalExpression" Arg="r"/>
<Description>
Tests whether an object is a rational expression.
<Example><![CDATA[
gap> IsRatExpOnnLettersObj(r3) and IsRationalExpressionRep(r3);
true
gap> IsRationalExpression(r1);
true
]]></Example>
</Description> 
</ManSection> 

<ManSection> 
<Func Name="AlphabetOfRatExp" Arg="R"/>
<Description>

Returns the alphabet of the rational expression <C>R</C>.
<Example><![CDATA[
gap> r:=RandomRatExp(2);
a*(ba*U@)
gap> AlphabetOfRatExp(r);
2
gap> r:=RandomRatExp("01");
1*(01*U@)
gap> AlphabetOfRatExp(r);
"01"
gap> a:=RandomTransformation(3);;
gap> b:=RandomTransformation(3);;
gap> r:=RandomRatExp([a,b]);
(Transformation( [ 1, 1, 3 ] )UTransformation( [ 1, 1, 2 ] ))*
gap> AlphabetOfRatExp(r);
[ Transformation( [ 1, 1, 3 ] ), Transformation( [ 1, 1, 2 ] ) ]
]]></Example>
</Description> </ManSection>


<ManSection> 
<Func Name="AlphabetOfRatExpAsList" Arg="R"/>
<Description>

Returns the alphabet of the rational expression <C>R</C> always
as a list.
If the alphabet of the rational expression is an integer
less than 27 it returns the list <C>"abcd...."</C>,
otherwise returns <C>[ "a1", "a2", "a3", "a4", ... ]</C>.
<Example><![CDATA[
gap> r:=RandomRatExp(2);
(aUb)((aUb)(bU@)U@)U@
gap> AlphabetOfRatExpAsList(r);
"ab"
gap> r:=RandomRatExp("01");
1*(0U@)
gap> AlphabetOfRatExpAsList(r);
"01"
gap> r:=RandomRatExp(30);;
gap> AlphabetOfRatExpAsList(r);
[ "a1", "a2", "a3", "a4", "a5", "a6", "a7", "a8", "a9", "a10", "a11", 
"a12", "a13", "a14", "a15", "a16", "a17", "a18", "a19", "a20", "a21", 
"a22", "a23", "a24", "a25", "a26", "a27", "a28", "a29", "a30" ]
]]></Example>
</Description> </ManSection>


<ManSection> 
<Func Name="CopyRatExp" Arg="R"/>
<Description>
Returns a new rational expression, which is a copy of <C>R</C>.
<Example><![CDATA[
gap> r:=RandomRatExp(2);
a*(bU@)
gap> CopyRatExp(r);
a*(bU@)
]]></Example>
</Description> </ManSection>

    </Section>

<Section><Heading>Comparison of rational expressions</Heading>
The way two rational expressions <C>r1</C> and <C>r2</C> are compared through the
<Alt Only="LaTeX">
$<![CDATA[<]]>$
</Alt>
<Alt Only="HTML">
&lt;
</Alt> operator is the following:
the empty set is lesser than everything else;
if r1 and r2 are letters, then the lesser is taken from the order in the alphabet;
if r1 is a letter an r2 a product, union or star, then r1 is lesser than r2;
a star expression is considered to be lesser than a product or union expression
and a product lesser than a union;
to compare two star expressions we compare the expressions under the stars;
to compare two product or union expressions we compare the subexpressions of
each expression from left to right;
</Section>


<Section><Heading>Operations with rational languages</Heading>

Only operations with rational languages over the same alphabet are allowed.
<P/>
We may compute expressions for the 
<C>product</C>, <C>union</C> and <C>star</C> (i.e.,
submonoid generated by) of rational sets. In some cases,
simpler expressions representing the same set are returned.    
Note that that two simplifications are always made, namely,
<Alt Only="LaTeX">
$r\cup"empty_set" = r$
</Alt>
<Alt Only="HTML">
rU"empty_set" = r
</Alt>
 and 
<Alt Only="LaTeX">
$r\epsilon = r$
</Alt>
<Alt Only="HTML">
r@ = r
</Alt>.


Of course, these operations may be used to construct more complex
expressions. 

For rational expressions we have the functions <C>UnionRatExp</C>, <C>ProductRatExp</C>, <C>StarRatExp</C>, that return respectively rational expressions for the <E>union</E> and <E>product</E> of 
the languages given by the rational expressions <C>r</C> and <C>s</C> and the <C>star</C> of
the language given by the rational expression <C>r</C>.
<ManSection> 
<Func Name="UnionRatExp" Arg="r,s"/>
 </ManSection> 

 <ManSection> 
<Func Name="ProductRatExp" Arg="r,s"/>
 </ManSection> 

 <ManSection> 
<Func Name=" StarRatExp" Arg="r"/>
<Description>


The expression <C>(a(aUb))*</C> may be produced in the following way
<Example><![CDATA[
gap> r1 := RatExpOnnLetters(2,[],[1]); 
a
gap> r2 := RatExpOnnLetters(2,[],[2]);
b
gap> r3 := UnionRatExp(r1,r2);
aUb
gap> r4 := ProductRatExp(r1,r3);
a(aUb)
gap> r5 := StarRatExp(r4);
(a(aUb))*
]]></Example>
</Description></ManSection> 
</Section>
  </Chapter>

