Lattices of processes for graphs

Abstract

The structure of the lattice of processes of a finite digraph with entries is investigated. Necessary and sufficient conditions for the isomorphism of a given lattice to the lattice of processes of a certain digraph are obtained. Bibliography: 5 titles.In papers [!-4] the notion of a process in a finite digraph was introduced and discussed. This is a formalization of the computing process in an arbitrary pipeline. This paper contains some definitions and results from [1][2][3][4] and presents some new results related to the separation of the class of lattices of processes from the set of all distributive lattices.i ~ Definitions and necessary information Let G = (V, E) be a finite digraph with vertex set V and edge set E, and let the graph R(G) = (V x Z, U) be its unfolding, where u = = v., v'. t + 1, t e z, 9 E}.We assign to each vertex w = v 9 t E V x Z a set T w of its predecessors in the graph R(G) :Tw= {v" t-ll(v,v') E E}.We call a subset p E V x Z a process in the graph G when w E p if and only if the nonempty set T w c p for each w E V x Z. We denote by/~ the closure of a subset p C V x Z with respect to this property. Theorem 1 ([2]). The set P(G) of all processes in the graph G forms a bounded distributive lattice with respect to set-theoretic inclusion, and pap I = p M p I, p V p I = p u pqThe following definitions and results are borrowed from [2]. Sets p, p' C_ V x Z. are similar if i~here is r E Z such that p~ = {v 9 t + r I v 9 t E p}. In that case we shall write p~ =< p >". Obviously, similarity is an equivalence relation on P(G). A set p C_ V x Z is called periodic if the similarity class containing p is finite. The cardinality s(p) of this class is called the period of p. Lemma 1 ([2]). If p C V x Z is a periodic set, then p is Mso periodic and s(~) divides s(p).First we consider a strongly connected graph B = (VB, EB). Let k(B) be the splitting index of B, i.e., the number of connected components in the graph R(B). Lemma 2 ([2]). The vertex set VB of a strongly connected graph B has a unique partition into k(B) classes and a numeration of these classes VB = Vo( B ) U . .. U Vk(~)-I( B ) satisfying the following properties:(1) for each v E ld(B), from (v,v'), (v",v) E EB it follows that v' E V/+l(modk(B))(B) and v" E V~-l(moa k(m)(B); , ~Vj(B) e P(B);(3) the set {Wj(B)I 0 _< j < k(B)} is a similarity class in P(B);