Extracting complexes that ensure sufficient structural conditions for system mutual informational agreement in multicomplex systems

Extraction of the structural environment of mutual system information agreement in multicomplex systems

“…This paper, as well as [1], presents the sufficient conditions for attaining the mutual information system agreement (MISA) in the multicomplex systems computing in parallel the set of interacting tasks with the given fault-tolerance characteristics both for solution of each of the tasks by its replication in the complex and for the environment of mutual exchanges between each pair of tasks.…”

“…Let from the system digraph G extracted be two disjoint complexes φ k (k = i, j) with T k , H k = (X k ∪ P k , D k ), X k , N k = P k ∪ Q k , where T k is the subdigraph of the complex φ k , H k = (X k ∪ P k , D k ) is the subdigraph from T k homeomorphic to the complete digraph where X k is the set of basic vertices, P k is the set of separating vertices, and D k is the set of arcs [1]. Additionally, let the subdigraph R φ i ↔φ j = G R φ i ↔φ j of the domain of intercomplex exchange T i and T j such that, first,R φ i ↔φ j ∩T i = ∅ andR φ i ↔φ j ∩T j = ∅ and, second, a message from any vertex of T i (T j ) to any vertex of T j (T i ) must follow only a path belonging to the subdigraph G D φ i ↔φ j (T i ∪R φ i ↔φ j ∪T j ) be defined in G. Let us assume that the maximum possible number μ i↔j of faulty vertices belonging to this subdigraph be defined for the subdigraph R φ i ↔φ j .…”

“…In continuation of the examples discussed in [1], the Appendix gives an example of decomposing the multicomputer system (MCS) into complexes and environments of the intercomplex exchanges that are sufficient for construction of the algorithm of intercomplex MIA in the multicomplex MCS.…”

“…Sufficient conditions for mutual informational system agreement (MISA) in the multicomplex systems performing parallel computation of a set of interacting tasks with the given fault-tolerance characteristics are presented below and in [1]. The terms, definitions and notation used below are given in [1] together with a complete problem formulation and the algorithm A1 to extract from the system the given set of disjoint complexes meeting the desired fault-tolerance characteristics.…”

“…The terms, definitions and notation used below are given in [1] together with a complete problem formulation and the algorithm A1 to extract from the system the given set of disjoint complexes meeting the desired fault-tolerance characteristics. The present paper defines the conditions for extraction of the environments of intercomplex exchanges of the consensus information that meet the given fault-tolerance characteristics and suffice to attain MISA.…”