Characterizing Idempotency of Lazy Cellular Automata

Characterize the idempotency of a lazy cellular automaton τ: A^G → A^G in terms of its unique active transition p ∈ A^S and writing symbol a ∈ A \ {p(e)}. Provide necessary and sufficient conditions, expressed purely via the pattern p and the symbol a, that determine when τ is idempotent.

Background

The paper studies lazy cellular automata—maps τ: A^G → A^G that act as the identity except on a unique active transition p ∈ A^S, where they write a fixed symbol a ∈ A \ {p(e)}. Earlier work showed that certain forms of p (constant, symmetric, quasi-constant) ensure idempotency, and the present paper provides a general upper bound on order and sufficient idempotency conditions (Corollary 3).

In one dimension with interval neighborhoods, a conjecture suggested a self-overlapping criterion for non-idempotency, but the direct implication fails for some cases, motivating a complete characterization. The authors therefore explicitly pose the problem of characterizing idempotency of lazy cellular automata via p and a.