The operation of a pipeline is considered as a special graph (the unfolding) associated to the pipeline graph. We investigate the decomposition of the unfolding, a particular case of which is its decomposition into minimal independent streams.i. Statement of the Problem. The aim of this paper is to elucidate some structural questions connected with the operation of pipelines in the framework of [i].We represent the operation of a fully loaded pipeline by its unfolding graph [2].A natural question in this context is, what subgraphs of the unfolding graph have independent computational meaning? In this paper, we consider as such subgraphs (computational processes) subgraphs that are closed in the following sense. Let a vertex have a nonempty set of immediate predecessors. Then if they all belong to the subgraph, the vertex itself also belongs to the subgraph, and conversely. Informally, this definition means the following. If at the moment t the arguments of some operation are ready for computation, then at the moment t + i the process will necessarily compute the result of this operation, and conversely. It is easy to see that the minimal independent streams (MIS) of [1]the connected components of the unfolding -are processes, but not only they.In this paper, we develop a decomposition of an arbitrary process into a union of finitely many processes of special types, which we call simple. This decomposition is more general than the decomposition into MIS. Thus, if the pipeline graph is strongly connected and the process is complete (the operation of a fully loaded pipeline), then this decomposition reduces to a MIS decomposition.It is interesting, however, that MIS may have proper subprocesses, some of which may be independent and some similar. Independence means that the processes have no shared vertices, while similarity means that one process is obtained by a time shift of the other.In addition to decomposition, we also highlight some other properties of these processes.2. Processes in Pipelines. Let G = (V, E) be a finite connected digraph. Its vertices will be denoted by ~ possibly with indices and primes, and the arc from ~ to ~' will be denoted by ~' . Below we only deal with digraphs, and we therefore simply call them graphs. To the graph G we associate another graph R=iV~Z~ { ~, ~%+I) : ~'~[~)-its unfolding. We use nonparenthesized notation for the pairs ~i~V~Z i arithmetic
For it a program is writ, ten which does not contain loops and has the following properties. The program's original data are a normal algorithm A (its notation) and a word ~ , while the results are a Boolean value ~ and a word ~ . If A is applicable to word ~ , then we can indicate a finite memory such that the program, having worked on this memory, yields the value true as % and A(~) as ~. However, if A is not applicable to ~ , then the program does not work on any (finite) memory and always yields the value false as ~ . If the program works on an infinite memory, then after its work ~ takes the value true if and only if A is applicable to ~ ; moreover, in the case of applicability ~ takes the value A~) The present paper contains a more detailed exposition of the result published in Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk USSR, 70 (1977).
Necessary and sufficient conditions for the homomorphicity of certain maps pertaining to the lattices of processes in graphs are presented. The results are stated in a graph-independent form. Bibliography: 2 titles.In the course of studying the lattices in graphs [1], the question concerning the homomorphicity of certain maps arises.The present paper provides necessary and sufficient conditions for the homomorphicity of these maps. The conditions are stated in a graph-independent form.Below we use the following notation: L is a distributive lattice; e, e I are arbitrary elements of L; DL is the lattice of filters of L; IL is the lattice of ideals of L; f~, F, are mappings determined by e' ---, e' A e, e' ---+ e' V e, which map L onto Proof. First we note that17^ 2 1-I1 u II~.Indeed, because of K^ < K1, K2 and the fact that 5 r [1] is antiisotone, we have H^ D I-I1, l-i^ _D II2.The relation a E [S) is equivalent to the existence of a finite .~I,,, M,, C S: AM,, _< a.
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