Walking automata in free inverse monoids
Résumé
Walking automata, be they running over words, trees or even graphs, possibly extended with pebbles that can be dropped and lifted on vertices, have long been defined and studied in Computer Science. However, questions concerning walking automata are surprisingly complex to solve, as illustrated, for language of trees, by the pebble hierarchy problem. In this paper, we study a generic notion of walking automata over graphs which semantics, thanks to the notion of birooted graphs that are accepted by these automata, naturally lays within inverse semigroup theory. Then, from the simplest notion of walking automata on birooted trees, that is, elements of free inverse monoids, to the more general cases of walking automata on birooted finite subgraphs of Cayley’s graphs of groups, that is, elements of free E-unitary inverse monoids, we provide a robust algebraic framework into which various classes of recognizable or regular languages of birooted graphs can uniformly be defined and related one with the other.
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