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Truth versus information in logic programming

Published online by Cambridge University Press:  03 June 2013

LEE NAISH  and
HARALD SØNDERGAARD
LEE NAISH
Affiliation:
Department of Computing and Information Systems, The University of Melbourne, Victoria 3010, Australia (e-mail: [email protected], [email protected])
HARALD SØNDERGAARD
Affiliation:
Department of Computing and Information Systems, The University of Melbourne, Victoria 3010, Australia (e-mail: [email protected], [email protected])
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Abstract

The 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.

Keywords

declarative debugginginformation orderintended interpretationlogic program specificationmany-valued logicmodesprogram analysisspecification semantics
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 14 , Issue 6 , November 2014 , pp. 803 - 840
Copyright
Copyright © Cambridge University Press 2013 

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