Shuichi Inokuchi

On reversible cellular automata with finite cell array

Discrete quantum cellular automata are cellular automata with reversible transition. This paper deals with 1d cellular automata with finite cell array and triplet local transition rules. We present the necessary condition of local transition rules for cellular automata to be reversible, and prove the reversibility of some cellular automata.

Statistical properties of a quantum cellular automaton

We study a quantum cellular automaton (QCA) whose time evolution is defined using the global transition function of a classical cellular automaton (CA). In order to investigate natural transformations from CAs to QCAs, the present QCA includes the CA with Wolframs rules 150 and 105 as special cases. We first compute the time evolution of the QCA and examine its statistical properties. As a basic statistical value, the probability of finding an active cell averaged over spatial-temporal space is introduced, and the difference between the CA and QCA is considered. In addition, it is shown that statistical properties in QCAs are related to the classical trajectory in configuration space.

Cellular Automata and Formulae on Monoids

This paper studies cellular automata with binary states on monoids making use of formulae in propositional logic, instead of local functions. Also we prove that the multiplication of formulae, defined by monoid action, determines the composition of transition functions of CA. This result converts the reversibility of transition functions to the reversibility of formulae. Several examples of reversible formulae are illustrated. Finally, introducing the Stone topology on configuration spaces, we give a neat proof of Hedlund’s theorem for CA.

The Number of Orbits of Periodic Box-Ball Systems

A box-ball system is a kind of cellular automata obtained by the ultradiscrete Lotka-Volterra equation. Similarities and differences between behavious of discrete systems (cellular automata) and continuous systems (differential equations) are investigated using techniques of ultradiscretizations. Our motivations is to take advantage of behavious of box-ball systems for new kinds of computations. Especially, we tried to find out useful periodic box-ball systems(pBBS) for random number generations. Applicable pBBS systems should have long fundamental cycles. We focus on pBBS with at most two kinds of solitons and investigate their behaviours, especially, the length of cycles and the number of orbits. We showed some relational equations of soliton sizes, a box size and the number of orbits. Varying a box size, we also found out some simulation results of the periodicity of orbits of pBBS with same kinds of solitons.In this paper we suggest the use of light for performing useful computations. Namely, we propose a special device which uses light rays for solving the Hamiltonian path problem on a directed graph. The device has a graph-like representation and the light is traversing it following the routes given by the connections between nodes. In each node the rays are uniquely marked so that they can be easily identified. At the destination node we will search only for particular rays that have passed only once through each node. We show that the proposed device can solve small and medium instances of the problem in reasonable time.

Formal Proofs for Automata and Sticker Systems

We implemented operations appeared in the theory of automata using the Coq proof-assistant. A language which contains infinite elements is defined using ssreflect (a Small Scale Reflection Extension for the Coq system). We also implemented the modules for sticker systems. Paun and Rozenberg introduced a concrete method to transform an automaton to a sticker system in 1998. One of our aims is to present formal proofs of the correctness of their transformation. We modified some of their definitions to improve their insufficient results. We note that all of our formulation are written in Coq and we show some examples of machine-checkable proofs.

Continuity of Inverse Transition Relations of 2-Neighborhood Cellular Automata

In this paper, we discuss the continuousness of inverse transition relations of cellular automata. Generally, the inverse of a function is a relation and is not always a function. Our interest here lies in the inverse transition systems of cellular automata and cellular automata with transition relations, that is, non-deterministic cellular automata. Richardson investigated non-deterministic cellular automata and proved Richardsons theorem. The theorem provides necessary and sufficient conditions for a transition system to be a cellular automaton, and the continuousness of the transition relation is one of the conditions. We prove that the inverse relation of the transition functions of any 2-neighborhood deterministic cellular automata are continuous, and that for 2-neighborhood cellular automata of which the transition relation does not satisfy the totality, its inverse relation is not always continuous. We show an example that the transition system of the continuous inverse transition relation of a cellular automaton is not a cellular automaton.

Limit Cycle for Composited Cellar Automata

We know that a few uniform cellular automata have maximum cycle lengths. However, there are many uniform cellular automata, and checking the cycles of all uniform cellular automata is impractical. In this paper, we define a cellular automaton by composition and show how its cycles are related.

The number of orbits of periodic box-ball systems

A box-ball system is a kind of cellular automata obtained by the ultradiscrete Lotka-Volterra equation. Similarities and differences between behavious of discrete systems (cellular automata) and continuous systems (differential equations) are investigated using techniques of ultradiscretizations. Our motivations is to take advantage of behavious of box-ball systems for new kinds of computations. Especially, we tried to find out useful periodic box-ball systems(pBBS) for random number generations. Applicable pBBS systems should have long fundamental cycles. We focus on pBBS with at most two kinds of solitons and investigate their behaviours, especially, the length of cycles and the number of orbits. We showed some relational equations of soliton sizes, a box size and the number of orbits. Varying a box size, we also found out some simulation results of the periodicity of orbits of pBBS with same kinds of solitons.