Quasi-recognizable vs MSO Definable Languages of One-Dimensional Overlapping Tiles

Abstract

It has been shown [6] that, within the McAlister inverse monoid [10], whose elements can be seen as overlapping one-dimensional tiles, the class of languages recognizable by finite monoids collapses compared with the class of languages definable in Monadic Second Order Logic (MSO).

This paper aims at capturing the expressive power of the MSO definability of languages of tiles by means of a weakening of the notion of algebraic recognizability which we shall refer to as quasi-recognizability. For that purpose, since the collapse of algebraic recognizability is intrinsically linked with the notion of monoid morphism itself, we propose instead to use premorphisms, monotonic mappings on ordered monoids that are only required to be sub-multiplicative with respect to the monoid product, i.e. mapping φ so that for all x and y, φ(xy) ≤ φ(x) φ(y).

In doing so, we indeed obtain, with additional but relatively natural closure conditions, the expected quasi-algebraic characterization of MSO definable languages of positive tiles. This result is achieved via the axiomatic definition of an original class of well-behaved ordered monoid so that quasi-recognizability implies MSO definability. An original embedding of any (finite) monoid S into a (finite) well-behaved ordered monoid ${\mathcal Q}(S)$ is then used to prove the converse.