intro.xml

<Chapter><Heading>Introduction</Heading> 

Token passing networks (TPNs) are directed graphs with nodes that can hold at most one token. Also 
each graph has a designated input node, which generates an ordered sequence of numbered tokens and
a designated output node that collects the tokens in the order they arrive at it. The input node
has no incoming edges, whereas the output node has no outgoing edges. A token <M>t</M> travels through the
graph, from node to node, if there is an edge connecting the nodes, if the node the token is moving
from is either the input node and the tokens <M>1, \ldots, t-1</M> have been released or the node is not
the output node, and lastly if the destination node contains no token or it is the output node. <Cite Key="PermGenTPGraph" /> <P/>
The set of permutations resulting from a TPN is closed under the property of containment. A permutation
<M>a</M> contains a permutation <M>b</M> of shorter length if in <M>a</M> there is a subsequence
that is isomorphic to <M>b</M>. This class of permutations can be represented by its anti-chain, which
in this context is called the basis. <Cite Key="RegCloSetPerms"/> <P/>
To enhance the computability of permutation pattern classes, each permutation can be encoded, using the
so called rank encoding. For a permutation <M>p_{1} \ldots p_{n}</M>, it is the sequence 
<M>e_{1}\ldots e_{n}</M> where <M>e_{i}</M> is the rank of <M>p_{i}</M> among <M>\{p_{i},p_{i+1},\ldots,p_{n}\}</M>.
It can be shown that the sets of encoded permutations of the class and the basis, both are a rational languages.
Rational languages can be represented by automata. <Cite Key="RegCloSetPerms"/> <P/>
There is another approach to get from TPNs to their corresponding automata. Namely building equivalence classes from
TPNs using the different dispositions of tokens within them. These equivalence classes of dispositions 
and the rank encoding of the permutations allow to build the same rational language as from the process above.
<Cite Key="PermGenTPGraph" />





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