On the Epistemological foundations of Logic Programming and its Extensions

One of the major roles of logic in computer science and A.I. is for knowledge representation and specification. In the context of knowledge representation, formal logic is widely praised as the tool par excellence for expressing information in a precise way and for correct and precise communication of information between different experts. In the scenario of communicating experts, the experts express their knowledge about some external problem domain in logic specifications which subsequently are communicated to other experts. In the computational logic approach to AI, computers participate in this and assist the human experts by solving computational problems using the logic specifications. The use of formal logic helps to avoid the ambiguities and lacunas that appear frequently in natural language specifications and, in a absence of computer systems that understand natural language, is the most promising way of transferring knowledge to a computer system. Obviously, when different human experts communicate their knowledge about some external problem domain via logical specifications, correct communication is only possible to the extent that they interpret the logical formulas in the same way about this problem domain. They should agree on what the formulas of the specification say about the problem domain. The way human experts interpret the logic formulas of a forreal logic in the problem world is what logicians call the declarative reading of the logic. One of the fundamental prexassumptions underlying the use of logic for knowledge representation, specification and problem solving is that a formal logic has a weU-understood and objective declarative reading. The epistemological study of a logic aims to clarify its declarative reading. A standard technique to do so is by explaining in what states of the world the formulas of the logic are true. The states of the world in which a given formula is true represent the possible states of the world. An epistemological theory that expresses when logical statements are true is what logicians also call a truth conditional semantics (e.g. (Moore 1995), p. The goal of this study is to investigate the declarative reading of logic programming (LP) and its extensions: stable logic programming, answer set programming and abductive logic programming. One of the central goals of logic programming was and still is to combine the advantages of formal declarative logic and procedural languages (Kowalski 1974). Originally, a definite logic program was seen as a classical logic Horn theory. The use of SLD-resolution induced the p~ocedural interpretation upon logic programs. When the negation as failure inference rule was introduced in Prolog systems, the view of logic programs as classical logic implications broke down. It was realized that the soundness of this rule could not be justified on the basis of the classical logic semantics of a logic program. Important efforts were made to propose alternative formal semantics for logic programming for which negation as failure is sound. This resulted in a number of different semantics of which the completion semantics (Clark 1978), the stable semantics (Gelfond & Lifschitz 1988) and the well-founded semantics (Van Gelder, Ross, & Schlipf 1991) are the most important ones. An important tool for defining these semantics was the use of embeddings of logic programming in other logics, in particular in classical logic (CL), default logic (DL) and autoepistemic logic (AEL). The question investigated here is what these formal semantics say about the declarative reading of logic programs. One view that appeared very early on is that logic programs represent definitions; recursive programs represent inductive definitions. This view is appealing in the context of many prototypical logic programs: