Negation by default and unstratifiable logic programs

AbstractThe default approach to the theory of logic programs (and deductive databases) is based on the interpretation of negation by default rules. Default logic is a well-suited formalism to express the Closed World Assumption and to define the declarative semantics of stratifiable logic programs. The case of disjunctive consequences in rules is treated. General logic programs may not have a meaning with respect to default semantics. The contribution of the paper is to exhibit an interesting class of programs having a default semantics, called effectively stratifiable programs. This time, disjunctive consequences are not considered. Effective stratification is a weaker constraint than stratification, local stratification and weak stratification. Besides enlarging the class of stratifiable logic programs, the paper contributes to provide a constructive definition of well-founded models of logic programs. The class of effectively stratifiable logic programs matches the class of programs having a total well-founded model and in general, the default semantics extends the well-founded semantics

Knowledge Compilation of Logic Programs Using Approximation Fixpoint Theory

To appear in Theory and Practice of Logic Programming (TPLP), Proceedings of ICLP 2015 Recent advances in knowledge compilation introduced techniques to compile \emph{positive} logic programs into propositional logic, essentially exploiting the constructive nature of the least fixpoint computation. This approach has several advantages over existing approaches: it maintains logical equivalence, does not require (expensive) loop-breaking preprocessing or the introduction of auxiliary variables, and significantly outperforms existing algorithms. Unfortunately, this technique is limited to \emph{negation-free} programs. In this paper, we show how to extend it to general logic programs under the well-founded semantics. We develop our work in approximation fixpoint theory, an algebraical framework that unifies semantics of different logics. As such, our algebraical results are also applicable to autoepistemic logic, default logic and abstract dialectical frameworks

Nonmonotonic consequences in default domain theory

Default domain theory is a framework for representing and reasoning about commonsense knowledge. Although this theory is motivated by ideas in Reiter’s work on default logic, it is in some sense a dual framework. We make Reiter’s default extension operator into a constructive method of building models, not theories. Domain theory, which is a well established tool for representing partial information in the semantics of programming languages, is adopted as the basis for constructing partial models. This paper considers some of the laws of nonmonotonic consequence, due to Gabbay and to Kraus, Lehmann, and Magidor, in the light of default domain theory. We remark that in some cases Gabbay’s law of cautious monotony is open to question. We consider an axiomatization of the nonmonotonic consequence relation on prime open sets in the Scott topology – the natural logic – of a domain, which omits this law. We prove a representation theorem showing that such relations are in one to one correspondence with the consequence relations determined by extensions in Scott domains augmented with default sets. This means that defaults are very expressive: they can, in a sense, represent any reasonable nonmonotonic entailment. Results about what kind of defaults determine cautious monotony are also discussed. In particular, we show that the property of unique extensions guarantees cautious monotony, and we give several classes of default structures which determine unique extensions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41772/1/10472_2004_Article_325432.pd

Ultimate approximations in nonmonotonic knowledge representation systems

We study fixpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order approximations by means of a precision ordering. We show that each lattice operator O has a unique most precise or ultimate approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We apply our theory to logic programming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their standard counterparts We argue that ultimate semantics for logic programming have attractive epistemological properties and that, while in general they are computationally more complex than the standard semantics, for many classes of theories, their complexity is no worse.Comment: This paper was published in Principles of Knowledge Representation and Reasoning, Proceedings of the Eighth International Conference (KR2002

Logic Programming with Default, Weak and Strict Negations

This paper treats logic programming with three kinds of negation: default, weak and strict negations. A 3-valued logic model theory is discussed for logic programs with three kinds of negation. The procedure is constructed for negations so that a soundness of the procedure is guaranteed in terms of 3-valued logic model theory.Comment: 14 pages, to appear in Theory and Practice of Logic Programming (TPLP

Automated Synthesis of Tableau Calculi

This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules that can then be used to reason within the logic. The method guarantees that the generated rules form a calculus which is sound and constructively complete. If the logic can be shown to admit finite filtration with respect to a well-defined first-order semantics then adding a general blocking mechanism provides a terminating tableau calculus. The process of generating tableau rules can be completely automated and produces, together with the blocking mechanism, an automated procedure for generating tableau decision procedures. For illustration we show the workability of the approach for a description logic with transitive roles and propositional intuitionistic logic.Comment: 32 page