Hiroshi Umeo

Deterministic one-way simulation of two-way real-time cellular automata and its related problems

With the advent of VLSI technology there has been increasing interest in the study or” cellular I ,usomata as a model of parallel co_mputation. Dyer [2] considered the capability of qhe cellul.ar automata in which the direction of the information flow was restricted. In [2] it was shown that oneway and two-way cellular automata were computationally equivalent in the nondeterministic case. Such a relation in the deterministic case, however, has been left open. In this paper we show that for any deterministic two-way real-time cellular automaton, M, there exists a deterministic one-way cellular automaton which can simulate M in twice real-time. Moreover we present a new type of deterministic one-way cellular automata, called circular cellular automata, which are computationally equivalent to deterministic two-way celhtlar automata.

An Efficient Mapping Scheme for Embedding Any One-Dimensional Firing Squad Synchronization Algorithm onto Two-Dimensional Arrays

An efficient mapping scheme is proposed for embedding any one-dimensional firing squad synchronization algorithm onto 2-D arrays, and some new 2-D synchronization algorithms based on the mapping scheme are presented. The proposed mapping scheme can be readily applied to the design of synchronization algorithms with fault tolerance, algorithms operating on multi-dimensional cellular arrays, and for the generalized case where the general is located at an arbitrary position on the array. A six-state algorithm is developed that can synchronize any m × n rectangular array in 2(m + n) - 4 steps. In addition, we develop a nine-state optimum-time synchronization algorithm on square arrays. We progressively reduce the number of internal states of each cellular automaton on square and rectangular arrays, achieving nine states for a square array and six states for a rectangular array. These are the smallest number of states reported to date for synchronizing rectangular and square arrays.

A twelve-state optimum-time synchronization algorithm for two-dimensional rectangular cellular arrays

The firing squad synchronization problem has been studied extensively for more than 40 years [1-18]. The present authors are involved in research on firing squad synchronization algorithms on two-dimensional (2-D) rectangular cellular arrays. Several synchronization algorithms on 2-D arrays have been proposed, including Beyer [2], Grasselli [3], Kobayashi [4], Shinahr [10], Szwerinski [12] and Umeo et al. [13, 15]. To date, the smallest number of cell states for which an optimum-time synchronization algorithm has been developed is 14 for rectangular array, achieved by Umeo et al. [15]. In the present paper, we propose a new optimum-time synchronization algorithm that can synchronize any 2-D m × n rectangular arrays in m + n + max(m, n) –3 steps. We progressively reduce the number of internal states of each cellular automaton on rectangular arrays, achieving twelve states. This is the smallest number of states reported to date for synchronizing rectangular arrays in optimum-step.

Several new generalized linear- and optimum-time synchronization algorithms for two-dimensional rectangular arrays

We propose several new generalized synchronization algorithms for 2-D cellular arrays. Firstly, a generalized linear-time synchronization algorithm and its 14-state implementation are given. It is shown that there exists a 14-state 2-D CA that can synchronize any m × n rectangular array in m + n + max(r + s , m + n – r – s + 2) – 4 steps with the general at an arbitrary initial position (r, s),where 1 ≤ r ≤ m, 1 ≤ s ≤ n. The generalized linear-time synchronization algorithm is interesting in that it includes an optimum-step synchronization algorithm as a special case where the general is located at one corner. In addition, we propose a noveloptimum-time generalized synchronization scheme that can synchronize any m × n array in m+n+max (m, n)− min (r, m−r+1)− min (s, n−s+1)−1 optimum steps.

Linear-time recognition of connectivity of binary images on 1-bit inter-cell communication cellular automaton

Abstract We introduce a new class of two-dimensional cellular automata (CAs) whose inter-cell communication is restricted to 1-bit and propose a linear-time connectivity recognition algorithm for two-dimensional binary images. Precisely, it is shown that a set of two-dimensional connected binary images of size m×n can be recognized in 2(m+n)+O(1) steps by a two-dimensional CA with 1-bit inter-cell communication.

A design of symmetrical six-state 3 n -step firing squad synchronization algorithms and their implementations

In 1994, Yunes [19] began to explore 3n-step firing squad synchronization algorithms and developed two seven-state synchronization algorithms for one-dimensional cellular arrays His algorithms were so interesting in that he progressively decreased the number of internal states of each cellular automaton.In this paper, we propose a new symmetrical six-state 3n-step firing squad synchronization algorithm Our result improves the seven-state 3n-step synchronization algorithms developed by Yunes [19] The number six is the smallest one known at present in the class of 3n–step synchronization algorithms A non-trivial and new symmetrical six-state 3n-step generalized firing squad synchronization algorithm is also given In addition, we study a state-change complexity in 3n-step firing squad synchronization algorithms We show that our algorithms have O(n2) state-change complexity, on the other hand, the thread-like 3n-step algorithms developed so far have O(n logn) state-change complexity.

A seven-state time-optimum square synchronizer

The firing squad synchronization problem on cellular automata has been studied extensively for more than fifty years, and a rich variety of synchronization algorithms have been proposed for not only one-dimensional arrays but two-dimensional arrays. In the present paper, we propose a seven-state optimum-time synchronization algorithm that can synchronize any square arrays of size n × n with a general at one corner in 2n - 2 steps, which is a smallest realization of time-optimum square synchronizer known at present. The implementation is based on a new, simple zebra-like mapping scheme which embeds synchronization operations on one-dimensional arrays onto square arrays.

Some New Generalized Synchronization Algorithms and Their Implementations for Large Scale Cellular Automata

In this paper, we study a generalized synchronization problem for large scale cellular automata (CA) on one- and two- dimensional arrays. Some new generalized synchronization algorithms will be designed both on O(1)-bit and 1-bit inter-cell communication models of cellular automata. We give a 9-state and 13-state CA that can solve the generalized synchronization problem in optimum- and linear-time on O(1)-bit 1-D and 2-D CA, respectively. The number of internal states of the CA implemented is the smallest one known at present. In addition, it is shown that there exists a 1-bit inter-cell communication CA that can synchronize 1-D n cells with the general on the kth cell in n+max(k, n - k + 1) steps, which is two steps larger than the optimum time. We show that there still exist several new generalized synchronization algorithms, although more than 40 years have passed since the development of the problem.

Correction, Optimization and Verification of Transition Rule Set for Waksman's Firing Squad Synchronization Algorithm

In 1966, A. Waksman proposed a 16-state firing squad synchronization algorithm, which is known, together with an unpublished Goto’s algorithm, as the first-in-the-world optimum-time firing algorithm. Waksman described his algorithm in terms of a finite state transition table, however, it has been reported in the talks of cellular automata researchers that some fatal errors were included in the Waksman’s transition table. In this paper we correct all errors included in his original transition table and give a complete list of transition rules which yield successful firings for any array less than 10000 cells. In our correction, ninety-three percent reduction has been made in the number of Waksman’s original transition rules. It has been shown that two-hundred and two rules are necessary and sufficientones for the Waksman’s optimum-time firing squad synchronization.

A smallest five-state solution to the firing squad synchronization problem

An existence or non-existence of five-state firing squad synchronization protocol has been a long-standing, famous open problem for a long time. In this paper, we answer partially to this problem by proposing a smallest five-state firing squad synchronization algorithm that can synchronize any one-dimensional cellular array of length n = 2k in 3n-3 steps for any positive integer k. The number five is the smallest one known at present in the class of synchronization protocols proposed so far.