Yoshihiro Mizoguchi

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.

Relational graph rewritings

This note presents a new formalization of graph rewritings which generalizes traditional graph rewritings. Relational notions of graphs and their rewritings are introduced and several properties about graph rewritings are discussed using relational calculus (theory of binary relations). Single pushout approaches to graph rewritings proposed by Raoult and Kennaway are compared with our rewritings of relational (labeled) graph. Moreover, a more general sufficient condition for two rewritings to commute and a theorem concerning critical pairs useful to demonstrate the confluency of graph rewriting systems are also given.

Period lengths of cellular automata on square lattices with rule 90

This paper studies two‐dimensional cellular automata ca−90(m,n) having states 0 and 1 and working on a square lattice of size (m−1)×(n−1). All their dynamics, driven by the local transition rule 90, can be simply formulated by representing their configurations with Laurent polynomials over a finite field F2={0,1}. The initial configuration takes the next configuration to a particular configuration whose cells all have the state 1. This paper answers the question of whether the initial configuration lies on a limit cycle or not, and, if that is the case, some properties on period lengths of such limit cycles are studied.

Mathematical analysis on affine maps for 2D shape interpolation

This paper gives a simple mathematical framework for 2D shape interpolation methods that preserve rigidity. An interpolation technique in this framework works for given the source and target 2D shapes, which are compatibly triangulated. Focusing on the local affine maps between the corresponding triangles, we describe a global transformation as a piecewise affine map. Several existing rigid shape interpolation techniques are discussed and mathematically analyzed through this framework. This gives us not only a useful comprehensive understanding of existing approaches, but also new algorithms and a few improvements of previous approaches.

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.

Categorical Assertion Semantics in Toposes

Summary A categorical interpretation of assertion (axiomatic) semantics of programming languages is proposed. All of the preconditions, postconditions, and programs are interpreted as (binary) relations in topoi with the use of relational calculus, and several fundamental properties of Dijkstras weakest preconditions are proved. Assertions in the semantics depend on intuitionistic logic, so this is an extension of the assertion semantics due to E. G. Manes and M. A. Arbib

Relational Structures and Their Partial Morphisms in View of Single Pushout Rewriting

In this paper we present a basic notion of relational structures which includes simple graphs, labelled graphs and hypergraphs, and introduce a notion of partial morphisms between them. An existence theorem of pushouts in the category of relational structures and their partial morphisms is proved under a certian functorial condition, and it enables us to discuss single pushout rewritings of relational structures.

Wang Tile Modeling of Wall Patterns

Wall patterns are essential in the creation of textures for visually rich buildings. Particularly, irregular wall patterns give an organic and lively feeling to the building. In this chapter, we introduce a modeling method for wall patterns using Wang tiles which are known for creating aperiodic tiling of the plane under certain conditions. We introduce a class of Wang tiles and prove that any rectangle with border constraints and bigger than a \(2\times 2\) rectangle can be tiled. We use this proof to derive a tiling algorithm that is in linear time. Finally, we give some results of our algorithm and compare the computation time with previous Wang tiling algorithms introduced in computer graphics.

Uniqueness of Butson Hadamard matrices of small degrees

Let BH n × n ( m ) be the set of n × n Butson Hadamard matrices where all the entries are m-th roots of unity. For H 1 , H 2 ? BH n × n ( m ) , we say that H 1 is equivalent to H 2 if H 1 = P H 2 Q for some monomial matrices P and Q whose nonzero entries are m-th roots of unity. In the present paper we show by computer search that all the matrices in BH 17 × 17 ( 17 ) are equivalent to the Fourier matrix of degree 17. Furthermore we shall prove that, for a prime number p, a matrix in BH p × p ( p ) which is not equivalent to the Fourier matrix of degree p gives rise to a non-Desarguesian projective plane of order p.

Properties of graphs preserved by relational graph rewritings

Abstract We formulate graphs and graph rewritings using binary relations and call them relational graphs and relational graph rewritings. In this framework, rewriting is defined using a pushout in a category of relational graphs. It is known that an important theorem of rewriting systems called critical pairs lemma can be proved using simple and clear categorical properties. In this paper, we construct treelike graphs and Raoult Graphs by some relational conditions. We give a sufficient condition for rewriting rules and matchings which guarantees the closedness of those graph rewritings. These results show that the critical pairs lemma also holds under some conditions for a graph rewriting system in which graphs are restricted to treelike graphs or Raoult Graphs.