Overlapping tile automata

David Janin (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

Premorphisms are monotonic mappings between partially ordered monoids where the morphism condition φ(xy) = φ(x)φ(y) is relaxed into the condition φ(xy) ≤ φ(x)φ(y). Their use in place of morphisms has recently been advocated in situations where classical algebraic recognizability collapses. With languages of overlapping tiles, such an extension of classical recognizability by morphisms, called quasi-recognizability, has already proved both its effectiveness and its power; it is shown to essentially capture definability in monadic second order logic. In this paper, we complete the theory of languages of such tiles by providing a notion of (finite state) tile automaton that is proved to be both sound and complete with respect to quasi-recognizability, i.e. every quasi-recognizable languages of tiles is definable by a finite state tile automaton and, conversely, every language of tiles definable by a finite state tile automaton is quasi-recognizable.

Mots clés: Overlapping tile; automata; algebraic recognizability; premorphism; weakly adequate monoid

Domaines: Théorie et langage formel [cs.FL]

David Janin. Overlapping tile automata. The 8th International Computer Science Symposium in Russia (CSR), Jun 2013, Ekaterinburg, Russia. ⟨hal-00766261v1⟩