Logic Programs and Three-Valued Consequence Operators

Master's thesis by Carroline Dewi Puspa Kencana Ramli

Supervisor Steffen Hölldobler
Wissensverarbeitung
– 1 August 2009
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While classical logic is considered not expressive enough to model human reasoning,

three-valued logic seems much better suited for this purpose. In [SvL08] Stenning and van Lambalgen show that their consequence operator under Fitting three-valued semantics can appropriately model human reasoning. Their operator is defined similarly to the Fitting operator which has been studied extensively. Even though their definitions and usage are very similar, it turns out that some of their properties are fundamentally different. Thus, in this thesis we deepen the knowledge about the Stenning and van Lambalgen operator, providing formal grounds for further investigation of relations between human reasoning and logic.

First we look for conditions under which both operators are continuous and when they acquire the property of being a contraction. We also introduce a level mapping characterisation of the new operator that puts it within the same framework with other three-valued semantics for logic programs, including the Fitting and well-founded semantics. Then we turn our attention to the underlying three-valued logic used to characterise these operators, dubbed the Fitting semantics. We will see that under this semantics, the model of completed program is not necessarily a model of the program itself. This happens because under Fitting semantics, the law of equivalence does not hold. We show that the Lukasiewicz semantics is a good candidate to replace Fitting semantics since it admits the law of equivalence while not changing the meaning or properties of logic programs. Further, we present the core method, a connectionist model generator for logic programs, that can easily be adapted to handle Stenning and van Lambalgen’s approach. Finally, since under the new operator negative information is difficult to express in the program, we propose a number of approaches to add this kind of expressivity to the formalism.