Two-dimensional Finite 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.

On the power of nondeterminism and Las Vegas randomization for two-dimensional finite automata

The goal of this work is to investigate the computational power of nondeterminism and Las Vegas randomization for two-dimensional finite automata. The following three results are the main contribution of this paper: (i) Las Vegas (three-way) two-dimensional finite automata are more powerful than (three-way) two-dimensional deterministic ones. (ii) Three-way two-dimensional nondeterministic finite automata are more powerful than three-way two-dimensional Las Vegas finite automata. (iii) There is a strong hierarchy based on the number of computations (as measure of the degree of nondeterminism) for three-way two-dimensional finite automata. These results contrast with the situation for one-way and two-way finite automata, where all these computation modes have the same acceptance power, and the differences may occur only in the sizes of automata. Results (i) and (ii) provide the first such simultaneous acceptance separation between nondeterminism, Las Vegas, and determinism for a computing model.

A NOTE ON THREE-WAY TWO-DIMENSIONAL PROBABILISTIC TURING MACHINES

This paper introduces a three-way two-dimensional probabilistic Turing machine (tr2-ptm), and investigates several properties of the machine. The tr2-ptm is a two-dimensional probabilistic Turing machine (2-ptm) whose input head can only move left, right, or down, but not up. Let 2-ptms (resp. tr2-ptms) denote a 2-ptm (resp. tr2-ptm) whose input tape is restricted to square ones, and let 2-PTMs(S(n)) (resp. TR2-PTMs(S(n))) denote the class of sets recognized by S(n) space-bounded 2-ptms's (resp. tr2-ptms's) with error probability less than ½, where S(n): N→N is a function of one variable n (= the side-length of input tapes). Let TR2-PTM(L(m,n)) denote the class of sets recognized by L(m,n) space-bounded tr2-ptm's with error probability less than ½, where L(m,n): N2→N is a function of two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). The main results of this paper are: (1) 2-NFAs - TR2-PTMs(S(n))≠ϕ for any S(n)=o(log n), where 2-NFAs denotes the class of sets of square tapes accepted by two-dimensional nondeterministic finite automata, (2) TR2-PTMsS(n)[Formula: see text]2-PTMs(S(n)) for any S(n)=o(log n), and (3) for any function g(n)=o(log n) (resp. g(n)=o(log n/log log n)) and any monotonic nondecreasing function f(m) which can be constructed by some one-dimensional deterministic Turing machine, TR2-PTM(f(m)+g(n)) (resp. TR2-PTM(f(m)×g(n))) is not closed under column catenation, column closure, and projection. Additionally, we show that two-dimensional nondeterministic finite automata are equivalent to two-dimensional probabilistic finite automata with one-sided error in accepting power.

A note on two-dimensional probabilistic finite automata

This note introduces two-dimensional probabilistic finite automata (2-pfa's), and investigates several properties of them. We first show that the class of sets recognized by 2-pfa's with bounded error probability, 2-PFA, is incomparable with the class of sets accepted by two-dimensional alternating finite automata. We then show that 2-PFA is not closed under row catenation, column catenation, row +, and column + operations in Siromoney et al. (G. Siromoney, R. Siromoney, K. Krithivasan, Inform. and Control 22 (1973) 447).

COOPERATING SYSTEMS OF THREE-WAY, TWO-DIMENSIONAL FINITE AUTOMATA

This paper introduces a cooperating system of three-way, two-dimensional finite automata (CS-TR2-FA) which is a restricted version of a cooperating system of four- way, two-dimensional finite automata (CS-2-FA), and mainly investigates several fundamental properties of this system as a two-dimensional language acceptor whose input tapes are restricted to square ones. We show that (1) CS-TR2-FA’s are equivalent in accepting power to three-way, two-dimensional simple multihead finite automata, (2) CS-2-FA’s are more powerful than CS-TR2-FA’s, (3) £[CS-TR2-DFA(k)S] ⊊ £[CS-TR2-NFA(k)S], (4) ⋃1≤k<∞ £[CS-TR2-DFA(k)S] ⊊ ⋃1≤k<∞ £[CS-TR2-NFA(k)S], and (5) £[CS-TR2-DFA(k)S] (£[CS-TR2-NFA(k)S]) ⊊ £[CS-TR2-DFA(k+1)S] (£[CS-TR2-NFA(k+1)S]), where £[CS-TR2-DFA(k)S] (£[CS-TR2-NFA(k)S]) denotes the class of sets of square input tapes accepted by CS-TR2-FA’s which consist of k deterministic (non-deterministic) finite automata.

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.

Some remarks on two-dimensional finite automata

In this paper finite automata with two-dimensional inputs are considered. It is shown that 1. (1) the two-dimensional languages accepted by deterministic three-way finite automata are closed under complement and 2. (2) the two-dimensional languages accepted by nondeterministic three-way finite automata are closed under row reflection (rows in reverse order).

Three-way two-dimensional finite automata with rotated inputs

This paper introduces a new type of two-dimensional automaton called a three-way two-dimensional finite automaton with rotated inputs, and investigates a relationship among the accepting powers of these new automata, two-dimensional finite automata, and tape-bounded three-way two-dimensional Turing machines. We show, for example, that m log m space ( m 2 space) is necessary and sufficient for deterministic three-way two-dimensional Turing machines to simulate deterministic (nondeterministic) three-way two-dimensional finite automata with rotated inputs.

A relationship between two-dimensional finite automata and three-way tape-bounded two-dimensional Turing machines

This note supplements a result of Inoue and Takanami (1980) who showed that for any function L( m) such that (i) L( m)⩾log m, and (ii) lim m→∞ [ L(m) m 2 ]= 0 ( resp lim m→∞ [ L(m) m log m ] = 0) , the class of sets of square tapes accepted by deterministic three-way L( m) tape-bounded two-dimensional Turing machines is incomparable with the class of sets of square tapes accepted by nondeterministic (resp. deterministic) two-dimensional finite automata.

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)

A note on decision problems for three-way two-dimensional finite automata

A note on decision problems for three-way two-dimensional finite automata

Nonclosure property of nondeterministic two-dimensional finite automata under cyclic closure

It was unknown whether or not the class of sets of square tapes accepted by nondeterministic two-dimensional finite automata is closed under cyclic closure. This short paper solves this problem, and shows that the class is not closed under row or column cyclic closure.

Two-dimensional finite automata and unacceptable functions

In this note, we first introduce the concept of “functions acceptable by two-dimensional finite automata” and then give several functions unacceptable by two-dimensional finite automata.

A note on two-dimensional finite automata

During the past about thirty-five years, many types of twoor three-dimensional automata have been proposed and investigated the properties of them as the computational model of pattern processing. On the other hand, recently, due to the advances in many application areas such as computer animation, motion image processing, and so on, the study of three-dimensional pattern processing with the time axis has been of crucial importance. Thus, we think that it is very useful for analyzing computation of three-dimensional pattern processing with the time axis to explicate the properties of four-dimensional automata. In this paper, we propose a four-dimensional Turing machine and a four-dimensional finite automaton, and show the space complexities necessary and sufficient for seven-way four-dimensional Turing machines to simulate four-dimensional finite automata, where each sidelength of each input tape of these automata is equivalent. Key-Words: Turing machine, finite automaton, on-line tessellation acceptor, four-dimensional input tape, determinism, nondeterminism, space-bound, configuration, computation, space complexity

Cyclic closure properties of automata on a two-dimensional tape

In this paper, we investigate cyclic closure properties of several types of automata on a two-dimensional tape. It is shown that (1) the classes of sets accepted by two-dimensional finite automata, deterministic one-way parallel/sequential array automata, and two-dimensional deterministic on-line tessellation acceptors are not closed under row or column cyclic closure, and (2) the class of sets accepted by two-way parallel-sequential array automata is not closed under row cyclic closure.