Recognizability of iso‐picture languages by Wang systems

Abstract

In the context of a syntactic approach to pattern recognition, there have been several studies in the last few decades on theoretical models for generating or recognizing two-dimensional objects, pictures, and picture languages. Motivated by these studies, we have introduced a new notion of recognizability for a class of picture languages called iso-picture languages through iso-triangular tiling systems (ITS) and studied the properties of these languages [Kalyani et al., 2004]. In [Kalyani et al., 2005], we introduced iso-triangular domino systems (IDS) to recognize iso-picture languages, and proved the equivalence of ITS and IDS. In [Kalyani et al., 2006], we have constructed a two-dimensional online tessellation automaton (IPOTA) to recognize an iso-picture language and presented an algorithm to learn recognizable iso-picture languages from positive data and restricted subset queries. In this article, we introduce isotriangular Wang systems (IWS) to recognize iso-picture languages. We prove that the family of iso-picture languages recognized by isotriangular Wang systems L(IWS) coincides with the family of iso-picture languages recognized by iso-triangular tiling sytems L(ITS). We show that L(ITS) 5 L(IDS) as a corollary and obtain that the families of iso-picture languages L(ITS), L(IDS), L(IWS), and L(IPOTA) all coincide.