Transition Diagrams of Finite Cellular Automata

Abstract

This paper provides some simple recursive formulas generating tran sition diagrams of finite cellular automata with triplet local transition functions.

Tree Expressions and Their Product Formula

“…Kumamoto and Nohmi (in preparation) analysed behaviors of integral affine transition systems, and showed that the transition diagrams of integral affine transition systems are the cartesian product of a system whose nodes are all on a limit cycle and a system having a fixed point. Lee and Kawahara (1996) presented tree expressions and their product formula, and got the result that transition diagram of some cellular automata can be represented by the cartesian product of transition diagrams of cellular automata with smaller cells. But their tree expressions can represent only transition systems with a fixed point and their product formula is limited on transition systems such that they have a fixed point * Faculty of Mathematics , Kyushu University 33, Fukuoka 812-8581 Japan.…”

On Limit Cycles and Transient Lengths of Cellular Automata With Threshod Rules

“…Sato(1996aSato( ,1996bSato( ,1997Sato( ,1998 and Inokuchi(1996Inokuchi( ,1998Inokuchi( ,2000 investigated behaviors of 1-D cellular automata with triplet local transition rules and showed that the numbers of limit cycles and transient lengths obey simple rules like Fibonacci sequence. Furthermore, Lee and Kawahara(1996), and Inokuchi and Kawahara(1999) investigated transition diagrams of cellular automata and make an attempt to representing them by algebraic formulas called tree expressions.…”

On Behaviors of Cellular Automata With Rule 156

“…Sato (1996) investigated behaviors of nonlinear cellular automata with rule 27. Lee and Kawahara (1996) investigated transition diagrams of 1dimensional cellular automata and proved that transition diagrams of a cellular au tomaton can be represented by the cartesian product of transition diagrams of smaller cellular automata. Shingai (1978) showed that the period length of limit cycles of cel lular automata with a threshold transition rule is 4 or less.…”