On-line Tessellation Automata Research Articles

Two-dimensional pattern matching against local and regular-like picture languages

Given a two-dimensional array of symbols and a picture language over a finite alphabet, we investigate how to find rectangular subarrays that belong to the picture language. Two-dimensional pattern matching problems can be formulated by interpreting subarrays as matches and picture languages as patterns. We formulate four particular problems – finding maximum, minimum, any or all match(es) – and describe algorithms solving the problems for basic classes of picture languages, which include local picture languages and picture languages accepted by various types of deterministic two-dimensional finite automata such as four-way, three-way and two-way automata, on-line tessellation automata, and returning finite automata. We also prove that the pattern matching problems cannot be solved for the class of local picture languages in time linear in the input area unless the well known problem of triangle finding in a graph is solvable in quadratic time. This shows a fundamental difference between the complexity of one-dimensional and two-dimensional pattern matching.

Complexity of Matching Sets of Two-Dimensional Patterns by Two-Dimensional On-Line Tessellation Automaton

We study the two-dimensional pattern matching implemented using the deterministic two-dimensional on-line tessellation automaton. This restricted two-dimensional cellular automaton is able to simulate the Baker–Bird algorithm, which was proposed as the first algorithm for the two-dimensional pattern matching. We explore capabilities of this automaton to carry out the matching task against an arbitrary set of equal-sized patterns. To measure amount of resources needed to accomplish it, we introduce the pattern complexity of a picture language. We show that this complexity ranges from a constant to exponential one. All of these are illustrated by giving examples of two-dimensional on-line tessellation automata matching sets of patterns, describing general techniques of how to construct them and proving lower bounds on the pattern complexity of some picture languages as well as operations over them.

Non-recursive trade-offs between two-dimensional automata and grammars

We study succinctness of descriptional systems for picture languages. Basic models of two-dimensional finite automata and generalizations of context-free grammars are considered. They include the four-way automaton of Blum and Hewitt, the two-dimensional online tessellation automaton of Inoue and Nakamura and the context-free Kolam grammar of Siromoney et al. We show that non-recursive trade-offs between the systems are very common. Basically, each separation result proving that one system describes a picture language which cannot be described by another system can usually be turned into a non-recursive trade-off result between the systems. These findings are strongly based on the ability of the systems to simulate Turing machines.

Recognizability of iso‐picture languages by Wang systems

AbstractIn the context of a syntactic approach to pattern recognition, there have been several studies in the last few decades ontheoretical 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 iso‐triangular Wang systems (IWS) to recognize iso‐picture languages. We prove that the family of iso‐picture languages recognized by iso‐triangular 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) = 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. © 2009 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 19, 140–145, 2009.

Two-dimensional cellular automata and deterministic on-line tessalation automata

In this work we consider the relationships between the classes of two-dimensional languages defined by deterministic on-line tessellation automata and by real time two-dimensional cellular automata...