Permutation pattern classes can be determined using their corresponding basis.
The basis of a pattern class is the anti-chain of the class,
under the order of containment. A permutation <M>\pi</M> contains another
permutation <M>\sigma</M> (of shorter length) if there is a subsequence
in <M>\pi</M>, which is isomorphic to <M>\sigma</M>. <P/>
With the rational language of the rank encoded class, it is also possible
to find the rational language of the basis and vice versa.
Several specific kinds of transducers are used in this process.
<Cite Key="RegCloSetPerms"/>
<Section><Heading>Transducers</Heading>
<ManSection>
<Func Name="Transducer" Arg="states,init,transitions,accepting"/>
<Returns>A record that represents a transducer.</Returns>
<Description>
A transducer is essentially an automaton, where running through the process does not
determine whether the input is accepted, but the input is translated to another language,
over a different alphabet. <P/>
Formally a transducer is a six-tuple with:
<M>Q</M> being the set of states, <M>S</M> the input alphabet, <M>G</M> the output alphabet,
<M>I \in Q</M> the start state, <M>A \subseteq Q</M> the set of accept states, and with the transition function
<M>f: Q \times (S \cup \{e\}) \rightarrow Q \times (G \cup \{e\})</M>, where <M>e</M> is the empty word. <P/>
In this function the transducer is stored by defining how many states there are, which one (by index) is the start
or initial state, the transitions are of the form <C>[inputletter,outputletter,fromstate,tostate]</C> and
a list of accept states. The input and output alphabet are determined by the input and output letters on
the transitions.
<Example><![CDATA[
gap> trans:=Transducer(3,1,[[1,2,1,2],[1,2,2,2],[2,2,1,3],[2,2,2,3],
> [1,1,3,3],[2,2,3,3]],[2]);
rec( accepting := [ 2 ], initial := 1, states := 3,
transitions := [ [ 1, 2, 1, 2 ], [ 1, 2, 2, 2 ], [ 2, 2, 1, 3 ],
[ 2, 2, 2, 3 ], [ 1, 1, 3, 3 ], [ 2, 2, 3, 3 ] ] )
gap> ]]></Example>
<Alt Only="LaTeX">
This transducer can be visualised as the following graph:
\begin{figure}[H] \begin{center} \leavevmode \includegraphics[scale=0.75]{img/trans.jpg} \end{center} \end{figure}
</Alt>
<Alt Only="HTML">
This transducer can be visualised as the following graph:
<br><center><img src="img/trans.jpg" WIDTH=241 HEIGHT=296 ></center><br>
</Alt>
</Description>
</ManSection>
<ManSection>
<Func Name="DeletionTransducer" Arg="k"/>
<Returns>A transducer over <C>k</C>+1 states.</Returns>
<Description>
A deletion transducer is a transducer that deletes one letter of a rank encoded permutation
and returns the correct rank encoding of the new permutation. The deletion transducer over <C>k</C>
deletes letters over the set of all rank encoded permutations with highest rank <C>k</C>.
Specifically, running through a deletion transducer with a rank encoded permutation, of highest rank <C>k</C>,
will lead to the set of all rank encoded permutations that have one letter of the initial permutation removed.
It is important to note that computing these shorter permutations with the transducer, is done by reading the
input permutation from right to left.
For example the deletion transducer with <C>k</C>=3, looks as follows:
<Example><![CDATA[
gap> DeletionTransducer(3);
rec( accepting := [ 1 .. 3 ], initial := 4, states := 4,
transitions := [ [ 1, 1, 4, 4 ], [ 1, 0, 4, 1 ], [ 2, 1, 1, 1 ],
[ 1, 1, 1, 2 ], [ 3, 2, 1, 1 ], [ 1, 1, 2, 3 ], [ 1, 1, 3, 3 ],
[ 2, 2, 4, 4 ], [ 2, 0, 4, 2 ], [ 3, 2, 2, 2 ], [ 2, 2, 2, 3 ],
[ 2, 2, 3, 3 ], [ 3, 3, 4, 4 ], [ 3, 0, 4, 3 ], [ 3, 3, 3, 3 ] ] )
gap> ]]></Example>
<Alt Only="LaTeX">
\begin{figure}[H] \begin{center} \leavevmode \includegraphics[scale=0.75]{img/dt3.jpg} \end{center} \end{figure}
</Alt>
<Alt Only="HTML">
<br><center><img src="img/dt3.jpg" WIDTH=288 HEIGHT=317 ></center><br>
</Alt>
</Description>
</ManSection>
<ManSection>
<Func Name="TransposedTransducer" Arg="t"/>
<Returns>A new transducer with interchanged input and output letters on each transition.</Returns>
<Description>
A transducer is transposed when all origins and destinations of transitions are left the same as before
but the input and output letters on each transition are interchanged.
Taking the deletion transducer from above, its transpose looks as follows:
<Example><![CDATA[
gap> TransposedTransducer(DeletionTransducer(3));
rec( accepting := [ 1 .. 3 ], initial := 4, states := 4,
transitions := [ [ 1, 1, 4, 4 ], [ 0, 1, 4, 1 ], [ 1, 2, 1, 1 ],
[ 1, 1, 1, 2 ], [ 2, 3, 1, 1 ], [ 1, 1, 2, 3 ], [ 1, 1, 3, 3 ],
[ 2, 2, 4, 4 ], [ 0, 2, 4, 2 ], [ 2, 3, 2, 2 ], [ 2, 2, 2, 3 ],
[ 2, 2, 3, 3 ], [ 3, 3, 4, 4 ], [ 0, 3, 4, 3 ], [ 3, 3, 3, 3 ] ] )
gap> ]]></Example>
<Alt Only="LaTeX">
\begin{figure}[H] \begin{center} \leavevmode \includegraphics[scale=0.75]{img/dtt3.jpg} \end{center} \end{figure}
</Alt>
<Alt Only="HTML">
<br><center><img src="img/dtt3.jpg" WIDTH=293 HEIGHT=319 ></center><br>
</Alt>
</Description>
</ManSection>
<ManSection>
<Func Name="CombineAutTransducer" Arg="aut,trans"/>
<Returns>An automaton consisting of a combination of <C>aut</C> and <C>trans</C>.</Returns>
<Description>
Combining automata and transducers is done over the natural "translation" of the by the automaton accepted language
through the transducer and then building a new automaton that accepts the new language. The function
<C>CombineAutTransducer</C> does this process and returns the new non-deterministic automaton.
<Example><![CDATA[
gap> a:=Automaton("det",1,1,[[1]],[1],[1]);
< deterministic automaton on 1 letters with 1 states >
gap> AutToRatExp(a);
a*
gap> t:=Transducer(2,1,[[1,2,1,2],[2,1,1,2],
> [1,1,2,1],[2,2,2,1]],[1]);
rec( accepting := [ 1 ], initial := 1, states := 2,
transitions := [ [ 1, 2, 1, 2 ], [ 2, 1, 1, 2 ], [ 1, 1, 2, 1 ],
[ 2, 2, 2, 1 ] ] )
gap> res:=CombineAutTransducer(a,t);
< non deterministic automaton on 2 letters with 2 states >
gap> AutToRatExp(res);
(ba)*
gap> Display(res);
| 1 2
-------------------
a | [ 1 ]
b | [ 2 ]
Initial state: [ 1 ]
Accepting state: [ 1 ]
gap> ]]></Example>
</Description>
</ManSection>
</Section>
<Section><Heading>From Class to Basis and vice versa</Heading>
<ManSection>
<Func Name="BasisAutomaton" Arg="a"/>
<Returns>An automaton that accepts the rank encoded permutations of the basis
of the input automaton <C>a</C>.</Returns>
<Description>
Every pattern class has a basis that consists of the smallest set of permutations which do not
belong to the class. Using
<Display Mode="M">B(L)=(L^{r} D^{t})^{r} \cap L^{c}</Display>
it is possible using the deletion transducer <M>D</M> and the language of rank encoded permutations <M>L</M>
to find the language of the rank encoded permutations of the basis <M>B(L)</M>, and thus the basis.
<Example><![CDATA[
gap> x:=Parstacks(2,2);
[ [ 2, 4 ], [ 3, 6 ], [ 2 ], [ 5, 6 ], [ 4 ], [ ] ]
gap> xa:=GraphToAut(x,1,6);
< epsilon automaton on 5 letters with 66 states >
gap> ma:=MinimalAutomaton(xa);
< deterministic automaton on 4 letters with 9 states >
gap> Display(ma);
| 1 2 3 4 5 6 7 8 9
--------------------------------
a | 1 2 1 3 2 2 6 6 3
b | 3 2 3 3 4 3 6 9 3
c | 9 2 9 4 6 6 4 9 9
d | 8 2 8 7 5 5 7 8 8
Initial state: [ 1 ]
Accepting state: [ 1 ]
gap> ba:=BasisAutomaton(ma);
< deterministic automaton on 4 letters with 9 states >
gap> Display(ba);
| 1 2 3 4 5 6 7 8 9
--------------------------------
a | 2 2 1 3 4 2 2 2 2
b | 2 2 2 2 2 4 1 2 8
c | 2 2 2 2 2 2 1 2 2
d | 2 2 2 9 2 2 5 6 2
Initial state: [ 7 ]
Accepting states: [ 1, 5 ]
gap> AutToRatExp(ba);
d(a(dbdb)*aaU@)UbUc
gap> ]]></Example>
Ignoring the trailing <C>UbUc</C> which essentially are noise, the basis of the pattern class indicates which
permutations are avoided in this particular class. The shortest permutation in the basis, looking at the rational expression,
is <M>daaa</M>, which can be translated to 4111 and decoded to the permutation 4123.
</Description>
</ManSection>
<ManSection>
<Func Name="ClassAutomaton" Arg="a"/>
<Returns>The automaton that accepts permutations of the class in their rank encoding.</Returns>
<Description>
The function <C>ClassAutomaton</C> does the reverse process of <C>BasisAutomaton</C>. Namely it
takes the automaton that represents the language of the rank encoded basis of a permutation class,
and using the formula
<Display Mode="M">L=B_{k} \cap ((B(L)^{r} I^{t})^{c} )^{r}</Display>
returns the automaton that accepts the rank encoded permutations of the class.
In the formula, <M>B_{k}</M> is the automaton that accepts all permutations of any length with
highest rank <M>k</M>, <M>B(L)</M> is the automaton that represents the basis and <M>I</M> is
the involement transducer.
<Example><![CDATA[
gap> xa:=Automaton("det",9,4,[[2,2,1,3,4,2,2,2,2],[2,2,2,2,2,4,1,2,8],
> [2,2,2,2,2,2,1,2,2],[2,2,2,9,2,2,5,6,2]],[7],[1,5]);
< deterministic automaton on 4 letters with 9 states >
gap> ca:=ClassAutomaton(xa);
< deterministic automaton on 4 letters with 9 states >
gap> Display(ca);
| 1 2 3 4 5 6 7 8 9
--------------------------------
a | 1 2 1 3 2 2 6 6 3
b | 3 2 3 3 4 3 6 9 3
c | 9 2 9 4 6 6 4 9 9
d | 8 2 8 7 5 5 7 8 8
Initial state: [ 1 ]
Accepting state: [ 1 ]
gap> IsPossibleGraphAut(ca);
true
gap> ]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="BoundedClassAutomaton" Arg="k"/>
<Returns>An automaton that accepts all rank encoded permutations, with highest rank being <C>k</C>.</Returns>
<Description>
The bounded class automaton, is an automaton that accepts all rank encoded permutations, of any length, with
highest rank <C>k</C>.
<Example><![CDATA[
gap> BoundedClassAutomaton(3);
< deterministic automaton on 3 letters with 3 states >
gap> Display(last);
| 1 2 3
--------------
a | 1 1 2
b | 2 2 2
c | 3 3 3
Initial state: [ 1 ]
Accepting state: [ 1 ]
gap> ]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="ClassAutFromBaseEncoding" Arg="base,k"/>
<Returns>The class automaton from a list of rank encoded basis permutations.</Returns>
<Description>
Given the permutations in the basis, in their rank encoded form, and the bound of the rank
of the permutations of the class, <C>ClassAutFromBaseEncoding</C> builds the automaton that
accepts rank encoded permutations of the class.
<Example><![CDATA[
gap> ClassAutFromBaseEncoding([[4,1,1,1]],4);
< deterministic automaton on 4 letters with 7 states >
gap> Display(last);
| 1 2 3 4 5 6 7
--------------------------
a | 1 2 1 3 2 2 6
b | 3 2 3 3 4 3 4
c | 4 2 4 4 6 6 4
d | 7 2 7 7 5 5 7
Initial state: [ 1 ]
Accepting state: [ 1 ]
gap> ]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="ClassAutFromBase" Arg="perms,k"/>
<Returns>The class automaton from a list of permutations of the basis.</Returns>
<Description>
Taking <C>perms</C> which is the list of permutations in the basis and <C>k</C> an
integer which indicates the highest rank in the encoded permutations of the class,
<C>ClassAutFromBase</C> constructs the automaton that accepts the language
of rank encoded permutations of the class.
<!-- Given the permutations of the basis and the highest rank of the encoded permutations of the class, <C>ClassAutFromBase</C> constructs the automaton that accepts the language
of rank encoded permutations of the class. -->
<Example><![CDATA[
gap> ClassAutFromBase([[4,1,2,3]],4);
< deterministic automaton on 4 letters with 7 states >
gap> Display(last);
| 1 2 3 4 5 6 7
--------------------------
a | 1 2 1 3 2 2 6
b | 3 2 3 3 4 3 4
c | 4 2 4 4 6 6 4
d | 7 2 7 7 5 5 7
Initial state: [ 1 ]
Accepting state: [ 1 ]
gap> ]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="ExpandAlphabet" Arg="a,newAlphabet"/>
<Returns>The automaton <C>a</C>, over the alphabet of size <C>newAlphabet</C>.</Returns>
<Description>
Given an automaton and the size of the new alphabet, which has to be bigger than the size of
the alphabet in <C>a</C> , <C>ExpandAlphabet</C> changes the automaton <C>a</C> to contain
an alphabet of size <C>newAlphabet</C>. The new letters have no transitions within the
automaton.
<Example><![CDATA[
gap> aut:=Automaton("det",3,2,[[2,2,3],[3,3,3]],[1],[3]);
< deterministic automaton on 2 letters with 3 states >
gap> Display(aut);
| 1 2 3
--------------
a | 2 2 3
b | 3 3 3
Initial state: [ 1 ]
Accepting state: [ 3 ]
gap> ExpandAlphabet(aut,4);
< deterministic automaton on 4 letters with 3 states >
gap> Display(last);
| 1 2 3
--------------
a | 2 2 3
b | 3 3 3
c |
d |
Initial state: [ 1 ]
Accepting state: [ 3 ]
gap> ]]></Example>
</Description>
</ManSection>
</Section>
<Section><Heading>Direct Sum of Regular Classes</Heading>
It is obvious that the direct sum of two rational pattern classes is also rational. But the skew sum of two rational
pattern classes does not imply that the resulting pattern class is rational, because if the second class in the
sum has infinitely many permutations, the alphabet of the skew sum class will be infinite and thusly the resulting
class will not be rational.
<ManSection>
<Func Name="ClassDirectSum" Arg="aut1,aut2"/>
<Returns>An automaton that corresponds to the direct sum of <C>aut1</C> with <C>aut2</C>.</Returns>
<Description>
<C>ClassDirectSum</C> builds the concatenation automaton of <C>aut1</C> with <C>aut2</C>, which corresponds
to the pattern class <C>aut1</C> <M>\oplus</M> <C>aut2</C>. <!-- EXAMPLE !!!!!!!! -->
<Example><![CDATA[
gap> a:=BasisAutomaton(GraphToAut(Parstacks(2,2),1,6));
< deterministic automaton on 4 letters with 9 states >
gap> AutToRatExp(a);
d(a(dbdb)*aaU@)UbUc
gap> b:=MinimalAutomaton(GraphToAut(Seqstacks(2,2),1,6));
< deterministic automaton on 3 letters with 3 states >
gap> AutToRatExp(b);
((cc*(aUb)Ub)(cc*(aUb)Ub)*aUa)*
gap> ab:=ClassDirectSum(a,b);
< deterministic automaton on 4 letters with 11 states >
gap> AutToRatExp(ab);
((d(acUc)c*(aUb)Ud(abUb))(cc*(aUb)Ub)*aUda(d(bdbd)*bdbaaUa)UbUc)((cc*(aUb)U\
b)(cc*(aUb)Ub)*aUa)*Ud(aU@)
gap> ]]></Example>
</Description>
</ManSection>
</Section>
<Section><Heading>Statistical Inspections</Heading>
It is of interest to see what permutations and how many of different length are accepted by the class,
respectively the basis. <P/>
In this section, the examples will be inspecting the basis automaton of the token passing network
containing 2 stacks of capacity 2, which are situated in parallel to each other.
<Example><![CDATA[
gap> x:=Parstacks(2,2);
[ [ 2, 4 ], [ 3, 6 ], [ 2 ], [ 5, 6 ], [ 4 ], [ ] ]
gap> xa:=GraphToAut(x,1,6);
< epsilon automaton on 5 letters with 66 states >
gap> ma:=MinimalAutomaton(xa);
< deterministic automaton on 4 letters with 9 states >
gap> ba:=BasisAutomaton(ma);
< deterministic automaton on 4 letters with 9 states >
gap> AutToRatExp(ba);
d(a(dbdb)*aaU@)UbUc
gap> ]]></Example>
<ManSection>
<Func Name="Spectrum" Arg="aut [, int]"/>
<Returns>A list indicating how many words of each length from 1 to
<C>int</C> or 15 (default) are accepted by the automaton.</Returns>
<Description>
Each entry in the returned list indicates how many words of length the index of the entry
are accepted by the automaton <C>aut</C>. The length of the list is
by default 15, but if this is too much or too little the second optional argument
regulates the length of the list.
<Example><![CDATA[
gap> Spectrum(ba);
[ 3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0 ]
gap> Spectrum(ba,20);
[ 3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 ]
gap> ]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="NumberAcceptedWords" Arg="aut,len"/>
<Returns>The number of words of length <C>len</C> accepted by the automaton <C>aut</C>.</Returns>
<Description>
Given the automaton <C>aut</C> and the integer <C>len</C>, <C>NumberAcceptedWords</C> determines
how many words of length <C>len</C> are accepted by the automaton.
<Example><![CDATA[
gap> NumberAcceptedWords(ba,1);
3
gap> NumberAcceptedWords(ba,16);
1
gap> ]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="AutStateTransitionMatrix" Arg="aut"/>
<Returns>A matrix containing the number of transitions between states of the automaton <C>aut</C>.</Returns>
<Description>
In the matrix computed by <C>AutStateTransitionMatrix</C> the rows are indexed by the state the transitions
are originating from, the columnns are indexed by the states the transitions are ending at. Each entry
<M>a_{i,j}</M> of the matrix represents the number of transitions from the state <M>i</M> to the state <M>j</M>.
<Example><![CDATA[
gap> AutStateTransitionMatrix(ba);
[ [ 0, 4, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 4, 0, 0, 0, 0, 0, 0, 0 ],
[ 1, 3, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 2, 1, 0, 0, 0, 0, 0, 1 ],
[ 0, 3, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 3, 0, 1, 0, 0, 0, 0, 0 ],
[ 2, 1, 0, 0, 1, 0, 0, 0, 0 ], [ 0, 3, 0, 0, 0, 1, 0, 0, 0 ],
[ 0, 3, 0, 0, 0, 0, 0, 1, 0 ] ]
gap> ]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="AcceptedWords" Arg="aut,int"/>
<Returns>All words of length <C>int</C> that are accepted by the automaton <C>aut</C>.</Returns>
<Description>
<C>AcceptedWords</C> outputs all permutations accepted by the automaton <C>aut</C>, which have length
<C>int</C>, in a list. The permutations are output in their rank encoding.
<Example><![CDATA[
gap> AcceptedWords(ba,1);
[ [ 2 ], [ 3 ], [ 4 ] ]
gap> AcceptedWords(ba,16);
[ [ 4, 1, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 1, 1 ] ]
gap> ]]></Example>
</Description>
</ManSection>
<ManSection>
<Func Name="AcceptedWordsR" Arg="aut,int1 [,int2]"/>
<Func Name="AcceptedWordsReversed" Arg="aut,int1 [,int2]"/>
<Returns>The list of all by the automaton accepted words of length <C>int1</C>,
where each word is written in reverse.</Returns>
<Description>
The functions <C>AcceptedWordsR</C> and <C>AcceptedWordsReversed</C> are synonymous and take the following
arguments; an automaton <C>aut</C>, an integer <C>int1</C> which indicates the length of the words that are
accepted by the <C>aut</C> and another integer <C>int2</C> which is optional and represents the initial state
of <C>aut</C>. The return value of these functions is the list containing all permutations of length <C>int1</C>
that are accepted by <C>aut</C>. The permutations are rank encoded and written in reverse.
<Example><![CDATA[
gap> AcceptedWordsR(ba,1);
[ [ 2 ], [ 3 ], [ 4 ] ]
gap> AcceptedWordsReversed(ba,16);
[ [ 1, 1, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 1, 4 ] ]
gap> ]]></Example>
</Description>
</ManSection>
</Section>
</Chapter>
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