Overlapping Tile Automata

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, an extension of classical recognizability by morphisms, called quasi-recognizability, has already proved both its effectiveness and its power. In this paper, we complete the theory of such tile languages by providing a notion of (finite state) non deterministic tile automata that capture quasi-recognizability in the sense that quasi-recognizable languages correspond to finite boolean combinations of languages recognizable by finite state non deterministic tile automata. As a consequence, it is also shown that quasi-recognizable languages of tiles correspond to finite boolean combination of upward closed (in the natural order) languages of tiles definable in Monadic Second Order logic.