On the Integration of Connectionist and Logic-Based Systems

It is a long-standing and important problem to integrate logic-based systems and connectionist systems. In brief, this problem is concerned with how each of these two paradigms interacts with the other and how each complements the other: how one may give a logical interpretation of neural networks, how one may interpret connectionism within a logical framework, and how one may combine the advantages of each within a single integrated system. In this paper, the computation and approximate computation by neural networks of semantic operators TP determined by logic programs P is studied; the converse of this problem, namely, the extraction of logic programs from given neural networks is also briefly considered. The foundations of the relevant notions employed in this problem are revisited and clarified and new definitions are presented which avoid embedding spaces of interpretations in the real line. In particular, such definitions are formulated relating to (1) pointwise and uniform approximation of TP, and (2) approximation and computation of (least) fixed points of TP. There are related notions of approximation and convergence of neural networks, and related notions of approximation and convergence of programs and these are discussed briefly, although the focus here is on (1) and (2). Necessary and sufficient conditions for uniform approximation of TP by neural networks are given in terms of continuity. Finally, the class of programs for which these methods can be employed to compute fixed points is greatly extended from the rather small class of acyclic programs to the (computationally adequate) class of all definite programs.