Two-dimensional On-line Tessellation Research Articles

Deterministic fuzzy two-dimensional on-line tessellation automata and their languages

The main work of this paper is to extend deterministic two-dimensional on-line tessellation automata (D2OTAs) to the fuzzy setting. We call these new automata models deterministic fuzzy two-dimensional on-line tessellation automata (DF2OTAs) and focus on some of their properties. Concretely, we firstly give the definitions of DF2OTAs and the fuzzy picture languages recognized by them. Next, the closure properties of the collection of fuzzy picture languages recognized by DF2OTAs are concentrated on under some familiar operations. The decomposition theorem, the representation theorem and the Pumping lemma, which are studied in the theory of fuzzy string automata and the related languages, are also considered scrupulously in the framework of DF2OTAs and their languages. Then, we put forward the concepts of transition-accessible DF2OTAs and state-accessible DF2OTAs, and conclude that transition-accessible DF2OTAs are special state-accessible DF2OTAs and DF2OTAs are equivalent to state-accessible DF2OTAs. Finally, state reduction relations on state-accessible DF2OTAs are defined to study the problem of the state reduction of DF2OTAs. Given a state-accessible DF2OTA A, in order to construct the factor DF2OTA of A with respect to a state reduction relation on A such that this factor one is equivalent to A and possesses a fewer states, we design a polynomial-time algorithm to compute the largest state reduction relation and then construct the corresponding factor automaton.

Relative approximate bisimulations for fuzzy picture automata

In this paper, we firstly give the notion of relative ϵ-approximate bisimulations between two fuzzy two-dimensional on-line tessellation automata (F2OTAs) and elaborate that the relative error between two F2OTAs, accompanying with a relative ϵ-approximate bisimulation between them, is less than or equal to ϵ where ϵ∈[0,1]. Moreover, if relative ϵ-approximate bisimulations between two F2OTAs are restricted to be surjective functional, then some properties are drawn and by which relative ϵ-approximate bisimulations on a F2OTA are defined. We construct the factor F2OTA of a F2OTA with respect to a relative ϵ-approximate bisimulation on it and describe the relationship between this factor automaton and the original F2OTA. Whereafter, we novelly design two algorithms to compute all maximal relative ϵ-approximate bisimulations on a given F2OTA. Finally, we discuss some interesting properties of bisimulations on F2OTAs.

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.

A Nivat theorem for weighted picture automata and weighted MSO logics

Picture languages have been investigated by several research groups. Here, we define weighted two-dimensional on-line tessellation automata (w2ota) taking weights from a new weight structure called picture valuation monoid. The behavior of this automaton model is a picture series mapping pictures to values. As our first main result, we prove a Nivat theorem for w2ota. It shows a connection of mappings and recognizability of certain simple series. Then, we introduce a weighted MSO logic which models average density of pictures. In the second main result, we show that w2ota and a fragment of our weighted MSO logics are expressively equivalent.

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.

LEARNING OF RECOGNIZABLE PICTURE LANGUAGES

Learning of certain classes of two-dimensional picture languages is considered in this paper. Linear time algorithms that learn in the limit, from positive data the classes of local picture languages and locally testable picture languages are presented. A crucial step for obtaining the learning algorithm for local picture languages is an explicit construction of a two-dimensional on-line tessellation acceptor for a given local picture language. A polynomial time algorithm that learns the class of recognizable picture languages from positive data and restricted subset queries, is presented in contrast to the fact that this class is not learnable in the limit from positive data alone.

A CHARACTERIZATION OF RECOGNIZABLE PICTURE LANGUAGES

This paper first shows that REC, the family of recognizable picture languages in Giammarresi and Restivo,3 is equal to the family of picture languages accepted by two-dimensional on-line tessellation acceptors in Inoue and Nakamura.5 By using this result, we then solve open problems in Giammarresi and Restivo,3 and show that (i) REC is not closed under complementation, and (ii) REC properly contains the family of picture languages accepted by two-dimensional nondeterministic finite automata even over a one letter alphabet.

Two-dimensional on-line tessellation acceptors are not closed under complement

A two-dimensional on-line tessellation acceptor is a two-dimensional array of identical finite automata, each connected to its four nearest neighbors. Each symbol of the input is placed on one automaton. The computation starts at the top-left corner of the input and a transition “wave” passes once diagonally across the array. In each cell the new state is computed from the symbol of the input placed in the cell and from the new states of northern and western neighbors. After m + n − 1 steps the input is accepted or rejected depending on the new state entered at the bottom-right corner of the input. In this paper it is shown that the class of languages accepted by nondeterministic two-dimensional on-line tessellation acceptors is not closed under complement. Also the relations between languages accepted by two-dimensional on-line tessellation acceptors and finite automata are considered.

A note on time-bounded bottom-up pyramid cellular acceptors

This paper investigates a relationship among the accepting powers of time-bounded bottom-up pyramid cellular acceptors, nondeterministic two-dimensional on-line tessellation acceptors, and two-dimensional finite automata. Let UPCA denote a bottom-up pyramid cellular acceptor. Then we show that (1) nondeterministic UPCAs which operate in O(diameter) time can simulate nondeterministic two-dimensional on-line tessellation acceptors, and thus such nondeterministic UPCAs are more powerful than nondeterministic two-dimensional finite automata, (2) O(diameter× log diameter) time is necessary for deterministic UPCAs to simulate deterministic two-dimensional finite automata, and (3) O((diameter) 2) time is necessary for deterministic UPCAs to simulate nondeterministic two-dimensional finite automata.

Deterministic two-dimensional on-line tessellation acceptors are equivalent to two-way two-dimensional alternating finite automata through 180°-rotation

A two-way two-dimensional alternating finite automaton (TW2-AFA) is a two- dimensional alternating finite automation whose input head can move right and down (or may not move), but cannot move left or up. Let 2-dota denote a two-dimensional deterministic on-line tesselation acceptor. In this paper, we show that 2-dota's are equivalent to TW2-AFA's through 180°-rotation.

Connected pictures are not recognizable by deterministic two-dimensional on-line tessellation acceptors

This paper shows that the set of all connected pictures is not accepted by any deterministic two-dimensional on-line tessellation acceptor.

Two-dimensional pattern matching by two-dimensional on-line tessellation acceptors

By reducing an array matching problem to a string matching problem in a natural way, it is shown that efficient string matching algorithms may be applied to arrays. In this paper, based on the ideas due to Baker, an application of the two-dimensional on-line tessellation acceptor (2-dota) is presented for very rapid on-line detection of occurrences of a fixed set of keyarrays as embedded subarrays in a text array. The main part of the algorithm described in this paper consists of constructing two finite state pattern (string) matching machines from the keyarrays. By combining these two finite state pattern matching machines, we construct the 2-dota which, given an m × n text array, solves the two-dimensional pattern matching problem in m + n−1 steps.

Connected pictures are not recognizable by deterministic two-dimensional on-line tessellation acceptors

Connected pictures are not recognizable by deterministic two-dimensional on-line tessellation acceptors

A remark on two-dimensional finite automata

Abstract : Let S2-APMOTA (m) be an area-preserving two-dimensional multipass on-line tessellation acceptor over square array input languages whose pass number is bounded by m. It is proved than an open problem 'Is L(2-NA) a subset but not coincident with L(S2-AMPOTA(1)) ?' proposed in a previous paper by Inoue and the present author has a positive solution. (Author)

Two-dimensional multipass on-line tessellation acceptors

The purpose of this paper is to propose a new type of acceptor on a two-dimensional tape, called a “two-dimensional multipass on-line tessellation acceptor” (denoted by “2-mpota”), and to examine some properties of the 2-mpota. It is shown that (1) nondeterministic 2-mpota's with the property that the sizes of the input and output tapes in each pass are the same, are equivalent to nondeterministic rectangular array bounded automata, and (2) nondeterministic 2-mpota's which are permitted to have an enlarged output tape in each pass are equivalent to nondeterministic Turing rectangular array automata.

Some properties of two-dimensional on-line tessellation acceptors

The purpose of this paper is to propose a new type acceptor called the “two-dimensional on-line tessellation acceptor” (denoted by “2-ota”) and to examine several properties of the 2-ota. The 2-ota might be considered as a real-time mode of rectangular array bounded cellular space, introduced by Smith and Beyer. First, several fundamental properties (e.g., closure properties) of the 2-ota are examined. Then, the accepting power of the 2-ota is compared with that of the two-dimensional finite automaton, introduced by Blum and Hewitt. The main results are 1. 1. The class of sets accepted by nondeterministic 2-ota's contains properly that of sets accepted by nondeterministic two-dimensional finite automata; 2. 2. The class of sets accepted by deterministic 2-ota's is incomparable with that of sets accepted by deterministic (or nondeterministic) two-dimensional finite automata.