Naoki Kamikawa

A FAMILY OF SMALLEST SYMMETRICAL FOUR-STATE FIRING SQUAD SYNCHRONIZATION PROTOCOLS FOR RING ARRAYS

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

A synchronization problem on 1-bit communication cellular automata

We study a classical firing squad synchronization problem for a large sale of one- and two-dimensional cellular automata having 1-bit inter-cell communications (CA1-bit). First, it is shown that there exists a one-dimensional CA1-bit that can synchronize n cells with the general on the kth cell in n + max (k, n - k + 1) steps, where the performance is two steps larger than the optimum one that was developed for O(1)-bit communication model. Next, we give a two-dimensional CA1-bit which can synchronize any n × n square and m × n rectangular arrays in 2n - 1 and m + n + max (m, n) steps, respectively. Lastly, we propose a generalized synchronization algorithm that operates in m + n + max (r + s, m + n - r - s) + O(1) steps on two-dimensional m × n rectangular arrays with the general located at an arbitrary position (r, s) of the array, where 1 ≤ r ≤ m and 1 ≤ s ≤ n. The time complexities for the first three algorithms developed are one to four steps larger than optimum ones proposed for O(1)-bit communication models. We show that there still exist several new interesting synchronization algoritms on CA1-bit although more than 40 years have passed since the development of the problem.

Some algorithms for real-time generation of non-regular sequences on one-bit inter-cell-communication cellular automata

A model of cellular automata (CA) is considered to be a non-linear model of complex systems in which an infinite one-dimensional array of finite state machines (cells) updates itself in a synchronous manner according to a uniform local rule. We study a sequence generation problem on a special restricted class of cellular automata having 1-bit inter-cell communications (CA1-bit) and propose several state-efficient real-time sequence generation algorithms for non-regular sequences. The 1-bit CA can be thought to be one of the most powerless and simplest models in a variety of CAs.

An infinite prime sequence can be generated in real-time by a 1-bit inter-cell communication cellular automaton

It is shown that an infinite prime sequence can be generated in real-time by a cellular automaton having 1-bit inter-cell communications.

A Transition Rule Set for the First 2-D Optimum-Time Synchronization Algorithm

The firing squad synchronization problem on cellular automata has been studied extensively for more than forty years, and a rich variety of synchronization algorithms have been proposed for two-dimensional cellular arrays. In the present paper, we reconstruct a real-coded transition rule set for an optimum-time synchronization algorithm proposed by Shinahr [11], known as the first optimum-time synchronization algorithm for two-dimensional rectangle arrays. Based on our computer simulation, it is shown that the proposed rule set consists of 28-state, 12849 transition rules and has a validity for the synchronization for any rectangle arrays of size m ×n such that 2 ≤ m, n ≤ 500.

A note on sequence generation power o two-states cellular automata

Cellular automaton (CA) are considered to be a non-linear model of complex systems in which an infinite one-dimensional array of finite state machines (cells) updates itself in a synchronous manner according to a uniform local rule. It is studied in many fields such as complex systems. We study a sequence generation problem on the CA. Arisawa, Fischer and Korec studied generation of a class of natural numbers on CA. In this paper, we study the sequence generation power of CA with 2 internal states.

About 4-States Solutions to the Firing Squad Synchronization Problem

We present some elements of a new family of time-optimal solutions to a less restrictive firing squad synchronization problem. These solutions are all built on top of some elementary algebraic cellular automata. Thus, this gives a very new insight on the problem and a more general way of computing on cellular automata.

Implementations of FSSP Algorithms on Fault-Tolerant Cellular Arrays.

The firing squad synchronization problem (FSSP, for short) on cellular automata has been studied extensively for more than fifty years, and a rich variety of FSSP algorithms has been proposed. Here we study the classical FSSP on a model of fault-tolerant cellular automata that might have possibly some defective cells and present the first state-efficient implementations of fault-tolerant FSSP algorithms for one-dimensional (1D) and two-dimensional (2D) arrays. It is shown that, under some constraints on the distribution and length of defective cells, any 1D cellular array of length n with p defective cell segments can be synchronized in \(2n-2+p\) steps and the algorithm is realized on a 1D cellular automaton with 164 states and 4792 transition rules. In addition, we give a smaller implementation for the 2D FSSP that can synchronize any 2D rectangular array of size \( m \times n\), including O(mn) rectangle-shaped isolated defective zones, exactly in \(2(m+n)-4\) steps on a cellular automaton with only 6 states and 939 transition rules.

The Smallest FSSP Partial Solutions for One-Dimensional Ring Cellular Automata: Symmetric and Asymmetric Synchronizers

A synchronization problem in cellular automata has been known as the Firing Squad Synchronization Problem (FSSP) since its development, where the FSSP gives a finite-state protocol for synchronizing a large scale of cellular automata. A quest for smaller state FSSP solutions has been an interesting problem for a long time. Umeo, Kamikawa and Yunes (2009) answered partially by introducing a concept of partial FSSP solutions and proposed a full list of the smallest four-state symmetric powers-of-2 FSSP protocols that can synchronize any one-dimensional (1D) ring cellular automata of length \(n=2^{k}\) for any positive integer \(k \ge 1\). Afterwards, Ng (2011) also added a list of asymmetric FSSP partial solutions, thus completing the four-state powers-of-2 FSSP partial solutions. The number four is the lower bound in the class of FSSP protocols. A question: are there any four-state partial solutions other than powers-of-2? has remained open. In this paper, we answer the question by proposing a new class of the smallest symmetric and asymmetric four-state FSSP protocols that can synchronize any 1D ring of length \(n=2^{k}-1\) for any positive integer \(k \ge 2\). We show that the class includes a rich variety of FSSP protocols that consists of 39 symmetric and 132 asymmetric solutions, ranging from minimum-time to linear-time in synchronization steps. In addition, we make an investigation into several interesting properties of these partial solutions, such as swapping general states, reversal protocols, and a duality property between them.

A New Class of the Smallest Four-State Partial FSSP Solutions for One-Dimensional Ring Cellular Automata

The synchronization in cellular automata has been known as the firing squad synchronization problem (FSSP) since its development, where the FSSP gives a finite-state protocol for synchronizing a large scale of cellular automata. A quest for smaller state FSSP solutions has been an interesting problem for a long time. Umeo, Kamikawa and Yunes [9] answered partially by introducing a concept of partial FSSP solutions and proposing a full list of the smallest four-state symmetric powers-of-2 FSSP protocols that can synchronize any one-dimensional (1D) ring cellular automata of length \(n=2^{k}\) for any positive integer \(k \ge 1\). Afterwards, Ng [7] also added a list of asymmetric FSSP partial solutions, thus completing the four-state powers-of-2 FSSP partial solutions. The number four is the smallest one in the class of FSSP protocols proposed so far. A question remained is that “are there any other four-state partial solutions?”. In this paper, we answer to the question by proposing a new class of the smallest four-state FSSP protocols that can synchronize any 1D ring of length \(n=2^{k}-1\) for any positive integer \(k \ge 2\). We show that the class includes a rich variety of FSSP protocols that consists of 39 symmetric solutions and 132 asymmetric ones, ranging from minimum-time to linear-time in synchronization steps. In addition, we make an investigation into several interesting properties of these partial solutions such as swapping general states, a duality between them, inclusion of powers-of-2 solutions, reflected solutions and so on.