An Arbitrary First Order Theory Can Be Represented by a Program: A Theorem

Abstract

How can we represent knowledge inside a computer? For formalized knowledge, classical logic seems to be the most adequate tool. Classical logic is behind all formalisms of classical mathematics, and behind many formalisms used in Artificial Intelligence. There is only one serious problem with classical logic: due to the famous Godel's theorem, classical logic is algorithmically undecidable; as a result, when the knowledge is represented in the form of logical statements, it is very difficult to check whether, based on this statement, a given query is true or not. To make knowledge representations more algorithmic, a special field of logic programming was invented. An important portion of logic programming is algorithmically decidable. To cover knowledge that cannot be represented in this portion, several extensions of the decidable fragments have been proposed. In the spirit of logic programming, these extensions are usually introduced in such a way that even if a general algorithm is not available, good heuristic methods exist. It is important to check whether the already proposed extensions are sufficient, or further extensions is necessary. In the present paper, we show that one particular extension, namely, logic programming with classical negation, introduced by M. Gelfond and V. Lifschitz, can represent (in some reasonable sense) an arbitrary first order logical theory.