arXiv subjects

Katsunobu Imai

Publications and source records attributed to Katsunobu Imai.

6 recordsLinked to original sources

arXiv2019

A hierarchical structure in the motion representation of 2-state number-conserving cellular automata

A one-dimensional two-state number-conserving cellular automaton (NCCA) is a cellular automaton whose states are 0 or 1 and where cells take states 0 and 1 and updated their states by the rule which keeps overall sum of states constant. It can be regarded as a kind of particle based modeling of physical systems and has another intuitive representation, motion representation, based on the movement of each particle. We introduced a kind of hierarchical interpretation of motion representations to understand the necessary pattern size to each motion. We show any NCCA of its neighborhood size n can be hierarchically represented by NCCAs of their neighborhood size from n-1 to 1.

nlin.CG↗

arXiv2018

Corona limits of tilings : Periodic case

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic tiling is a centrally symmetric convex polyhedron (see [Zhuravlev 2001], [Maleev-Shutov 2011]).

math.MG↗

arXiv2018

On weak universality of three-dimensional Larger than Life cellular automaton

Larger than Life cellular automaton (LtL) is a class of cellular automata and is a generalization of the game of Life by extending its neighborhood radius. We have studied the three-dimensional extension of LtL. In this paper, we show a radius-4 three-dimensional LtL rule is a candidate for weakly universal one.

nlin.CG↗

arXiv2016

5-State Rotation-Symmetric Number-Conserving Cellular Automata are not Strongly Universal

We study two-dimensional rotation-symmetric number-conserving cellular automata working on the von Neumann neighborhood (RNCA). It is known that such automata with 4 states or less are trivial, so we investigate the possible rules with 5 states. We give a full characterization of these automata and show that they cannot be strongly Turing universal. However, we give example of constructions that allow to embed some boolean circuit elements in a 5-states RNCA.

cs.FL↗

arXiv2012

A Universal Semi-totalistic Cellular Automaton on Kite and Dart Penrose Tilings

In this paper we investigate certain properties of semi-totalistic cellular automata (CA) on the well known quasi-periodic kite and dart two dimensional tiling of the plane presented by Roger Penrose. We show that, despite the irregularity of the underlying grid, it is possible to devise a semi-totalistic CA capable of simulating any boolean circuit on this aperiodic tiling.

cs.FL↗

arXiv2008

Simulations between triangular and hexagonal number-conserving cellular automata

A number-conserving cellular automaton is a cellular automaton whose states are integers and whose transition function keeps the sum of all cells constant throughout its evolution. It can be seen as a kind of modelization of the physical conservation laws of mass or energy. In this paper, we first propose a necessary condition for triangular and hexagonal cellular automata to be number-conserving. The local transition function is expressed by the sum of arity two functions which can be regarded as 'flows' of numbers. The sufficiency is obtained through general results on number-conserving cellular automata. Then, using the previous flow functions, we can construct effective number-conserving simulations between hexagonal cellular automata and triangular cellular automata.

cs.DM↗