Random Horn Formulas and Propagation Connectivity for Directed Hypergraphs

Horn formulas are a subclass of CNF expressions, where every clause contains at most one unnegated variable. This class is tractable in the sense that many problems that are hard for CNF expressions in general are polynomially solvable for Horn formulas (such as satisfiability and equivalence). It is partly for this reason that Horn formulas are of basic importance in artificial intelligence and other areas. Random Horn formulas have been studied in [DBC01, DV06, Ist02, LMST09, MIDV07]. A Horn formula is definite if it consists of clauses containing exactly one unnegated variable. We consider definite Horn formulas with clauses of size 3, i.e., with clauses of the form (ā ∨ b̄ ∨ c), which can also be written as a, b → c. Here a and b form the body of the clause and c is the head of the clause. Implication between a definite Horn formula φ and a definite Horn clause C can be decided by forward chaining: mark variables in the body of C and, while there is a clause in φ with all its body variables marked, mark its head variable as well. Then C is implied by φ iff its head gets marked. We consider random definite Horn formulas with clauses of size 3 over n variables, where every clause is included with probability p. It follows directly from the results of [LMST09] that p = (2 lnn)/n is a threshold probability for the following property: every pair of variables implies every other variable (see also [DBC01] for a related result). In this paper we consider the property that some pair of variables implies every other variable. This property is closely related to the property of propagation connectivity for 3-uniform undirected hypergraphs, introduced recently by Berke and Onsjö [BO09a]. They consider a marking process like forward