A family of extensions of SLDNF-resolution for normal abductive programs is presented. The main difference between our approach and existing procedures is the treatment of non-ground abductive goals. A completion semantics is given and the soundness and completeness of the procedures has been proven. The research presented here, provides a simple framework of abductive procedures, in which a number of parameters can be set, in order to fit the abduction procedure to the application under consideration.

Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.

We present a formal theory on the semantics of logic programs and abductive logic programs with first order integrity constraints. The theory provides an elegant, uniform formalisation for the three most widely accepted families of semantics: completion semantics, stable semantics and wellfounded semantics. The theory is based on the notion of a justification, which is a mathematical object describing, given an interpretation, how the truth value of a literal can be justified on the basis of the program. We identify the three different notions of justifications underlying the three types of semantics. In addition, we defend an alternative declarative reading of logic programming, different from the current predominant view of logic programming as a form of defeasible logic. Logic programs are interpreted as sets of definitions of predicates. The framework is suited to evaluate the extent to which this intuition is supported by the three classes of semantics. 1 Introduction. At this mo...