Remarks on Four-Dimensional Probabilistic Finite Automata (original) (raw)

Four Dimensional Multi-Inkdot Finite Automata

During the past about thirty-five years, many automata on a two- or three-dimensional input tape have been proposed and a lot of properties of such automata have been obtained. On the other hand, we think that recently, due to the advances in computer animation, motion image processing, and so forth, it is very useful for analyzing computational complexity of multi-dimensional information processing to explicate the properties of four-dimensional automata, i.e., three-dimensional automata with the time axis. In this paper, we propose a four- dimensional multi-inkdot finite automaton and mainly investigate its recognizability of four-dimensional connected pictures. Moreover, we briefly investigate some basic accepting powers of four-dimensional multi-inkdot finite automata.

Some properties of four-dimensional parallel Turing machines

2010

Some properties of four-dimensional parallel Turing machines lel computer can multiply themselves in the course of computation. Wiedermann 1 showed, for example, that every PTM can be simulated by an STM in polynomial time, and that a PTM cannot be simulated by any sequential Turing machine in linear space. Previously, 2-5 two-and threedimensional versions of PTM have been investigated. On the other hand, owing to the advances in many application areas such as moving image processing, computer animation, and so on, it has become increasingly apparent that the study of four-dimensional pattern processing is of crucial importance. Therefore, we think that the study of fourdimensional automata as a computational model of fourdimensional pattern processing is also meaningful. From this viewpoint, we fi rst introduced four-dimensional automata. 6,7 We have previously proposed a four-dimensional parallel Turing machine (4-PTM), and investigated some of its properties. 8 In particular, we dealt with a hardware-bounded 4-PTM, a variant of the 4-PTM, in which each side-length of each input tape is equivalent. The hardware-bounded 4-PTM is a 4-PTM the number of whose processors is bounded by a constant or a variable depending on the size of the inputs. The investigation of hardware-bounded 4-PTM's is more useful than that of 4-PTM's from a practical point of view. Here, we continue the study of the 4-PTM, 8 and investigate some accepting powers of its parallel computational model in which each side-length of each input tape is equivalent. 2 Preliminaries Defi nition 1: Let Σ be a fi nite set of symbols, a fourdimensional tape over Σ is a four-dimensional rectangular array of elements of Σ. The set of all four-dimensional tapes over Σ is denoted by Σ (4). Given a tape x ∈ Σ (4) , for each integer j (1 ≤ j ≤ 4), we let l j (x) be the length of x along the j-th axis. The set of all x ∈ Σ (4) with l 1 (x) = n 1 , l 2 (x) = n 2 , l 3 (x) = n 3 and l 4 (x) = n 4 is denoted by Σ (n1,n2,n3,n4). When 1 ≤ i j ≤ l j (x) for each j (1 ≤ j ≤ 4), let x(i 1 , i 2 , i 3 , i 4) denote the symbol in x with coordinates (i 1 , i 2 , i 3 , i 4). Furthermore, we defi ne Abstract Informally, the parallel Turing machine (PTM) proposed by Wiedermann is a set of identical usual sequential Turing machines (STMs) cooperating on two common tapes: a storage tape and an input tape. Moreover, STMs which represent the individual processors of a parallel computer can multiply themselves in the course of computation. On the other hand, during the past 7 years or so, automata on a four-dimensional tape have been proposed as computational models of four-dimensional pattern processing, and several properties of such automata have been obtained. We proposed a four-dimensional parallel Turing machine (4-PTM), and dealt with a hardware-bounded 4-PTM in which each side-length of each input tape is equivalent. We believe that this machine is useful in measuring the parallel computational complexity of three-dimensional images. In this work, we continued the study of the 3-PTM, in which each side-length of each input tape is equivalent, and investigated some of its accepting powers.

A relationship between Turing machines and finite automata on four-dimensional input tapes

Artificial Life and Robotics, 2008

A relationship between Turing machines and fi nite automata on four-dimensional input tapes of all four-dimensional tapes over Σ is denoted by Σ (4). Given a tape x ∈ Σ (4) , for each integer j(1 ≤ j ≤ 4), we let l j (x) be the length of x along the jth axis. The set of all x ∈ Σ (4) with l 1 (x) = n 1 , l 2 (x) = n 2 , l 3 (x) = n 3 , and l 4 (x) = n 4 , is denoted by Σ (n 1 ,n 2 ,n 3 ,n 4). When 1 ≤ i j ≤ l j (x) for each j(1 ≤ j ≤ 4), let x(i 1 , i 2 , i 3 , i 4) denote the symbol in x with coordinates (i 1 , i 2 , i 3 , i 4). Furthermore, we defi ne x[(i 1 , i 2 , i 3 , i 4), (i′ 1 , i′ 2 , i′ 3 , i′ 4)], when 1 ≤ i j ≤ i′ j ≤ l j (x) for each integer j(1 ≤ j ≤ 4), as the four-dimensional input tape y satisfying the following conditions: (i) for each j(1 ≤ j ≤ 4), l j (y) = i′ j − i j + 1; (ii) for each r

Path-bounded three-dimensional finite automata

Artificial Life and Robotics, 2008

Let Σ be a fi nite set of symbols. A three-dimensional tape over Σ is a three-dimensional rectangular array of elements of Σ. The set of all three-dimensional tapes over Σ is denoted by Σ (3). Given a tape x ∈ Σ (3) , for each integer j(1 ≤ j ≤ 3), we let l j (x) be the length of x along the jth axis. The set of all x ∈ Σ (3) with l 1 (x) = n 1 , l 2 (x) = n 2 , and l 3 (x) = n 3 is denoted by Σ (n 1 ,n 2 ,n 3). When 1 ≤ i j ≤ l j (x) for each j(1 ≤ j ≤ 3), let

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

Journal of Computer and System Sciences, 2004

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.

On languages defined by linear probabilistic automata

Information and Control, 1970

The class of mod-p linear probabilistic automata are defined. It is shown that rood-2 linear probabilistic automata driven by a single binomial process define regular languages for all cut-points, and that mod-3 linear probabilistic automata driven by a single trinomial process may define nonregular languages. It is also shown that mod-2 linear probabilistic automata driven by any number of binomial processes may define nonregular languages for only a finite number of cut-points.

Hierarchies based on the number of cooperating systems of three-dimensional finite automata

Artificial Life and Robotics, 2009

Defi nition 1: Let Σ be a fi nite set of symbols. A threedimensional tape over Σ is a three-dimensional rectangular array of elements of Σ. The set of all three-dimensional tapes over Σ is denoted by Σ (3). Given a tape x ∈ Σ (3) , for each integer j(1 ≤ j ≤ 3), we let l j (x) be the length of x along the j-th axis. The set of all x ∈ Σ (3) with l 1 (x) = n 1 , l 2 (x) = n 2 , and l 3 (x) = n 3 is denoted by Σ (n1,n2,n3). When 1 ≤ i j ≤ l j (x) for each j(1 ≤ j ≤ 3), let x(i 1 , i 2 , i 3) denote the symbol in x with coordinates (i 1 , i 2 , i 3). Furthermore, we defi ne Abstract The question of whether processing three-dimensional digital patterns is much more diffi cult than twodimensional ones is of great interest from both theoretical and practical standpoints. Recently, owing to advances in many application areas, such as computer vision, robotics, and so forth, it has become increasingly apparent that the study of three-dimensional pattern processing is of crucial importance. Thus, the study of three-dimensional automata as a computational model of three-dimensional pattern processing has become meaningful. This article introduces a cooperating system of three-dimensional fi nite automata as one model of three-dimensional automata. A cooperating system of three-dimensional fi nite automata consists of a fi nite number of three-dimensional fi nite automata and a three-dimensional input tape where these fi nite automata work independently (in parallel). Those fi nite automata whose input heads scan the same cell of the input tape can communicate with each other, i.e., every fi nite automaton is allowed to know the internal states of other fi nite automata on the cell it is scanning at the moment. In this article, we continue the study of cooperating systems of threedimensional fi nite automata, and mainly investigate hierarchies based on the number of their cooperating systems.

Parallel turing machines on four-dimensional input tapes

2010

The parallel Turing machine (PTM) proposed by Wiedermann is a set of identical usual sequential Turing machines (STMs) cooperating on two common tapes: storage tape and input tape. On the other hand, due to the advances in many application areas such as motion picture processing, computer animation, virtual reality systems, and so forth, it has become increasingly apparent that the study of four-dimensional patterns is of crucial importance. Therefore, we think that the study of four-dimensional automata as a computational model of four-dimensional pattern processing is also meaningful. In this article, we propose a four-dimensional parallel Turing machine (4-PTM), and investigate some of its properties based on hardware complexity.