Truth versus information in logic programming

Abstract

AbstractThe semantics of logic programs was originally described in terms of two-valued logic. Soon, however, it was realised that three-valued logic had some natural advantages, as it provides distinct values not only for truth and falsehood but also for “undefined”. The three-valued semantics proposed by Fitting (Fitting, M. 1985. A Kripke–Kleene semantics for logic programs. Journal of Logic Programming 2, 4, 295–312) and Kunen (Kunen, K. 1987. Negation in logic programming. Journal of Logic Programming 4, 4, 289–308) are closely related to what is computed by a logic program, the third truth value being associated with non-termination. A different three-valued semantics, proposed by Naish, shared much with those of Fitting and Kunen but incorporated allowances for programmer intent, the third truth value being associated with underspecification. Naish used an (apparently) novel “arrow” operator to relate the intended meaning of left and right sides of predicate definitions. In this paper we suggest that the additional truth values of Fitting/Kunen and Naish are best viewed as duals. We use Belnap's four-valued logic (Belnap, N. D. 1977. A useful four-valued logic. In Modern Uses of Multiple-Valued Logic, J. M. Dunn and G. Epstein, Eds. D. Reidel, Dordrecht, Netherlands, 8–37), also used elsewhere by Fitting, to unify the two three-valued approaches. The truth values are arranged in a bilattice, which supports the classical ordering on truth values as well as the “information ordering”. We note that the “arrow” operator of Naish (and our four-valued extension) is essentially the information ordering, whereas the classical arrow denotes the truth ordering. This allows us to shed new light on many aspects of logic programming, including program analysis, type and mode systems, declarative debugging and the relationships between specifications and programs, and successive execution states of a program.