On the Power of Accepting Networks of Evolutionary Processors with Special Topologies and Random Context Filters

Abstract

In this paper, we approach the problem of accepting all recursively enumerable languages by accepting networks of evolutionary processors (ANEPs, for short) with a fixed architecture. More precisely, we show that every recursively enumerable language can be accepted by an ANEP with an underlying graph in the form of a star with 13 nodes or by an ANEP with an underlying grid with 13 × 4 = 52 nodes as well as by ANEPs having underlying graphs in the form of a chain, a ring, or a wheel with 29 nodes each. In all these cases, the size and form as well as the general working strategy of the constructed networks do not depend on the accepted language; only the rewriting rules and the filters associated to each node of the networks depend on this language. Noteworthy is also the fact that the filtering process is implemented using random context conditions only. Our results answer problems which were left open in a paper published by J. Dassow and F. Manea at the conference on Descriptional Complexity of Formal Systems (DCFS) 2010 and improve a result published by B. Truthe at the conference on Non-Classical Models of Automata and Applications (NCMA) 2013. This paper is an extended version of the contributions to the conferences DCFS 2010 ([5]) and NCMA 2013 ([17]). ‡ The work of Florin Manea was supported by the DFG grant 596676. § The results were obtained while Bianca Truthe worked at the Otto-von-Guericke-Universität Magdeburg, Germany. J. Dassow et al. / On the Power of ANEPs with Special Topologies and RC Filters 3 DefinitionsWe assume that the reader is familiar with the basic concepts of formal language theory (see e. g. [16]). We only recall here some notations used in the paper.An alphabet is a finite and nonempty set of symbols. The set of all words over V is denoted by V * and the empty word is denoted by λ. The length of a word x is denoted by |x| while alph(x) denotes the minimal alphabet W such that x ∈ W * . For a word x ∈ W * , we denote by x r the reversal of the word.We denote by ⊆ the inclusion relation of sets and by ⊂ the relation of strict inclusion. A phrase structure grammar is specified as a quadruple G = (N, T, P, S)where N and T are alphabets (the elements of N are called non-terminal symbols; the elements of T are called terminal symbols), P is a finite set of production rules which are written in the form

Variants of Networks of Evolutionary Processors with Polarizations and a Small Number of Processors

“…The communication graph H consists of the set of nodes i, 1 ≤ i ≤ 12, and of the following set of directed edges: (8,2), (2,9), (9, 1), (1,3), (3,1), (3,11), (11,12), (12,5), (5,1), (1,10), (10,4), (4,1), (5,6), (6,7)…”

“…Computations in such a network consist of alternatingly performing two types of steps -an evolution step where in each cell all possible operations on all strings currently present in the cell are performed, and a communication step in which strings are sent from one cell to another cell provided specific conditions are fulfilled. Examples of such conditions are (output and input) filters which have to be passed, and these (output and input) filters can be specific types of regular languages or permitting and forbidden context conditions [9,10]. The set of strings obtained as results of computations by the NEP is defined as the set of objects which appear in some distinguished node in the course of a computation.…”