Leader election on two-dimensional periodic cellular automata

Abstract

This article explores the computational power of bi-dimensional cellular automata acting on periodical configurations. It extends in some sense the results of a similar paper dedicated to the one-dimensional case. More precisely, we present an algorithm that computes a "minimal pattern network", i.e. a minimal pattern and the two translation vectors it can use to tile the entire configuration. This problem is equivalent to the computation of a leader, which is one equivalence class of the cells of the periodical configuration.

Inductive definitions in logic versus programs of real-time cellular automata

“…Besides, it includes an additional restriction -the "monotonicity condition" -that reflects the geometrical consideration above-mentioned. [3] proves that this logic exactly characterizes the linear time complexity class of cellular automata: more precisely, for each integer d ≥ 1, a set L of d-dimensional pictures can be decided in linear time on a d-dimensional CA -written L ∈ DLIN d…”

“…The present paper is a considerably extended version of the conference paper [18] which is in some sense the sequel of the paper [3] (see also [19]). First, [3] observes that the inductive processes defining the considered problems (product of integers, product of matrices, sorting, etc.) are "local" and are naturally formalized by Horn formulas, that is by conjunctions of first-order Horn clauses.…”

“…Moreover, on every concrete problem defined by a Horn formula with d + 1 first-order variables, this inductive computation by Horn rules can be geometrically modeled as the displacement of a d-dimensional hyperplane along some fixed line in a space of dimension d + 1. To capture these inductive behaviors, [3] defines a logic denoted monot-ESO-HORN d (∀ d+1 , arity d+1 ) obtained from the logic ESO-HORN tailored by Grädel [14] to characterize P, by restricting both the number of first-order variables and the arity of second-order predicate symbols. Besides, it includes an additional restriction -the "monotonicity condition" -that reflects the geometrical consideration above-mentioned.…”

“…Real-time complexity appears as a sensitive/fragile notion and one generally thinks it is so for CA of dimension 2 or more [35,15]. However, maybe surprisingly, one knows that real-time complexity is a robust notion for one-dimensional CA in the following sense: according to the many natural variants of the definition of a one-dimensional CA, which essentially rest on the choice of the neighborhood of the CA and the parallel or sequential presentation of its input word, exactly three real-time classes of one-dimensional CA 2 have been proved to be distinct [5,4,20,31,34,39,40] The final and decisive step to establish this classification is a nice dichotomy of [31] on admissible neighborhoods 3 of CA, which can be rephrased as follows: for each neighborhood N admissible with respect to the first cell as output cell, the real-time complexity class of one-dimensional CA with parallel input mode and neighborhood N ,…”

“…We now show that school multiplication is naturally described in logic pred-ESO-HORN. Since our logics only define decision problems, the problem should be formalized as Product := {(a, b, p) ∈ (N >0 ) 3 | a × b = p}, where each integer is represented in binary notation. More precisely, let p = a × b be a product of positive integers and let p = p n .…”