GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<Chapter><Heading>From Automata to Networks</Heading>
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It is not only important to see how a TPN can be interpreted as a finite state automaton. 
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But also if an arbitrary automaton could represent the language of rank encoded permutations
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of a TPN. Currently within <C>PatternClass</C> it is only possible to check whether 
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deterministic automata are possible representatives of a TPN.
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<Section><Heading>Functions</Heading>
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<ManSection>
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  <Func Name="IsStarClosed" Arg="aut"/>
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  <Returns> <K>true</K> if the start and accept states in <C>aut</C> are one and the same state. </Returns>
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  <Description>
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    This has the consequence that the whole rational expression accepted by <C>aut</C> is always enclosed by the
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    Kleene star.
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<Example><![CDATA[
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gap> x:=Automaton("det",4,2,[[3,4,2,4],[2,2,2,4]],[1],[2]);
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< deterministic automaton on 2 letters with 4 states >
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gap> IsStarClosed(x);                                      
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false
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gap> AutToRatExp(x);        
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(a(aUb)Ub)b*
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gap> y:=Automaton("det",3,2,[[3,2,1],[2,3,1]],[1],[1]);
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< deterministic automaton on 2 letters with 3 states >
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gap> IsStarClosed(y);
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true
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gap> AutToRatExp(y);
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((ba*bUa)(aUb))*
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gap> ]]></Example>
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  </Description>
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</ManSection>
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<ManSection>
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  <Func Name="Is2StarReplaceable" Arg="aut"/>
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  <Returns> <K>true</K> if none of the states in the automaton <C>aut</C>, which 
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  are not the initial state, have a transition to the initial state labelled with 
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  the first letter of the alphabet. It also returns <C>true</C> if there is at least one 
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  state <M>i \in Q</M> with the first two transitions from <M>i</M> being 
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  <M>f(i,1)=1</M> and <M>f(i,2)=x</M>, where <M>x \in Q\setminus\{1\}</M> and 
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  <M>f(x,1)=1</M>.</Returns>
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  <Description>    
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<Example><![CDATA[
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gap> x:=Automaton("det",3,2,[[1,2,3],[2,2,3]],[1],[2]);
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< deterministic automaton on 2 letters with 3 states >
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gap> Is2StarReplaceable(x);
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true
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gap> y:=Automaton("det",4,2,[[4,1,1,2],[1,3,3,2]],[1],[1]);
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< deterministic automaton on 2 letters with 4 states >
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gap> Is2StarReplaceable(y);
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true
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gap> z:=Automaton("det",4,2,[[4,1,1,2],[1,4,2,2]],[1],[4]);
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< deterministic automaton on 2 letters with 4 states >
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gap> Is2StarReplaceable(z);
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false
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gap> ]]></Example>
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  </Description>
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</ManSection>
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<ManSection>
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  <Func Name="IsStratified" Arg="aut"/>
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  <Returns><K>true</K> if <C>aut</C> has a specific "layered" form.</Returns>
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  <Description>
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    A formal description of the most basic stratified automaton is: <P/>
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    <M>(S,Q,f,q_{0},A)</M> with 
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    <M>S:=\{1,...,n\},\,\, Q:=\{1,...,m\},\,\, n&lt;m,\,\, q_{0}:=1,\,\, 
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    A:=Q\setminus \{n+1\},\,\, f: Q \times S \rightarrow Q </M> and
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    <M>n+1</M> is a sink state. <P/>
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    If <M>i,j \in Q \setminus \{ n+1 \}</M>,with <M>i &lt; j</M>, and <M>i',j' \in S</M>, 
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    <M>i=i',\,\, j=j'</M> then 
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    <Display Mode="M">f(i,i')=i,\,\, f(i,j')=j,\,\, f(j,j')=j,\,\, f(j,i')=j-1 \,\, or \,\, n+1.</Display>
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<Example><![CDATA[
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gap> x:=Automaton("det",4,2,[[1,3,1,4],[2,2,4,4]],[1],[2]);
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< deterministic automaton on 2 letters with 4 states >
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gap> IsStratified(x);                                      
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true
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gap> y:=Automaton("det",4,2,[[1,3,2,4],[2,4,1,4]],[1],[2]);
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< deterministic automaton on 2 letters with 4 states >
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gap> IsStratified(y);                                      
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false
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gap> ]]></Example>
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  </Description>
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</ManSection>
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<ManSection>
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  <Func Name="IsPossibleGraphAut" Arg="aut"/>
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  <Returns> <K>true</K> if <C>aut</C> returns <C>true</C> in <C>IsStratified</C>, <C>Is2StarReplaceable</C> and 
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  <C>IsStarClosed</C>.</Returns>
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  <Description>
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    An automaton that fulfils the three functions above has the right form to be an automaton representing
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    rank encoded permutations, which are output from a TPN.
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<Example><![CDATA[
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gap> x:=Automaton("det",2,2,[[1,2],[2,2]],[1],[1]);
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< deterministic automaton on 2 letters with 2 states >
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gap> IsPossibleGraphAut(x);
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true
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gap> y:=Automaton("det",2,2,[[1,2],[1,2]],[1],[1]);
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< deterministic automaton on 2 letters with 2 states >
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gap> IsPossibleGraphAut(y);                        
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false
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gap> IsStarClosed(y);
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true
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gap> Is2StarReplaceable(y);
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true
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gap> IsStratified(y);
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false
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gap> ]]></Example>
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  </Description>
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</ManSection>
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</Section>
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</Chapter>
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