A Tractable Class of Abduction Problems

literal: Let p be a proposition. Then p and -p are literals. clause: A clause is a disjunction of literals. Horn Clause: A Horn Clause is a clause in which there is at most one positive literal. definite clause: A definite clause is a Hom Clause in which there is exactly one positive literal. A definite clause such as can also be written as We call pi the head of the clause, and the conjunction the body of the clause. denial: A denial is a Horn clause in which there are no positive literals. A denial such as can also be written as unit clause: A unit clause is a clause which has one literal. The problem of finding a set of assumptions which explain a given proposition is in general NP-hard, even when the background theory is an acyclic Horn theory. In this paper it is shown that when the background theory is acyclic Horn and its pseudo-completion is unit refutable, there is a polynomial time algorithm for finding minimal explanations. A test for unit-refutability of clausal theories is presented, based on the topology of the connection graph of the theory.