Representation Theorem for Hypergraph Satisfiability

Given a set of propositions, one can consider its inconsistency hypergraph. Then the safisfiability of sets of clauses with respect to that hypergraph (see [2], [4]) turns out to be the usual satisfiability. The problem is which hypergraphs can be obtained from sets of formulas as inconsistency hypergraphs. In the present paper it is shown that this can be done for all hypergraphs with countably many vertices and pairwise incomparable edges. Then, some applications in searching NP-completeness of certain combinatorial problems are shown. 1. Preliminaries. Let us recall some definitions and facts which can also be found in Cowen [1] and Kolany [2]. A hypergraph is a structure G = (V, E), where V is a set and E is a family of nonempty subsets of V . The elements of V will be called vertices, and the elements of the set E , edges of the hypergraph G. Sets of vertices will sometimes be called clauses. A hypergraph is compact iff every edge contains a finite one. A hypergraph is locally finite iff every vertex belongs to a finite number of edges only. Notice that a graph is a hypergraph with at most two-element edges. Here, the vertices of a fixed hypergraph G will be interpreted as some elementary propositions and the edges of G will be inconsistent sets of them. This interpretation leads to the following generalization of satisfiability of families of disjunctions. Definition 1.1. A set of vertices σ satisfies a family of clauses A (wrt. G) iff 1. σ does not contain any edge, 2. σ meets all clauses of A, that is σ ∩A 6= ∅, for all A in A. If some σ satisfies the family A, we say that A is satisfiable wrt. G, or G-satisfiable, for brevity.