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Approximations, Stable Operators, Well-Founded Fixpoints And Applications In Nonmonotonic Reasoning
, 2000
"... In this paper we develop an algebraic framework for studying semantics of nonmonotonic logics. Our approach is formulated in the language of lattices, bilattices, operators and fixpoints. The goal is to describe fixpoints of an operator O defined on a lattice. The key intuition is that of an approxi ..."
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Cited by 24 (10 self)
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In this paper we develop an algebraic framework for studying semantics of nonmonotonic logics. Our approach is formulated in the language of lattices, bilattices, operators and fixpoints. The goal is to describe fixpoints of an operator O defined on a lattice. The key intuition is that of an approximation, a pair (x, y) of lattice elements which can be viewed as an approximation to each lattice element z such that x z y. The key notion is that of an approximating operator, a monotone operator on the bilattice of approximations whose fixpoints approximate the fixpoints of the operator O. The main contribution of the paper is an algebraic construction which assigns a certain operator, called the stable operator, to every approximating operator on a bilattice of approximations. This construction leads to an abstract version of the well-founded semantics. In the paper we show that our theory offers a unified framework for semantic studies of logic programming, default logic and autoepistemic logic.
Logical foundations of well-founded semantics
- In P
, 2006
"... We propose a solution to a long-standing problem in the foun-dations of well-founded semantics (WFS) for logic programs. The problem addressed is this: which (non-modal) logic can be considered adequate for well-founded semantics in the sense that its minimal models (appropriately defined) coin-cide ..."
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Cited by 8 (2 self)
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We propose a solution to a long-standing problem in the foun-dations of well-founded semantics (WFS) for logic programs. The problem addressed is this: which (non-modal) logic can be considered adequate for well-founded semantics in the sense that its minimal models (appropriately defined) coin-cide with the partial stable models of a logic program? We approach this problem by identifying the HT 2 frames pre-viously proposed by Cabalar to capture WFS as structures of a kind used by Došen to characterise a family of logics weaker than intuitionistic and minimal logic. We define a notion of minimal, total HT 2 model which we call partial equilibrium model. Since for normal logic programs these models coincide with partial stable models, we propose the resulting partial equilibrium logic as a logical foundation for well-founded semantics. In addition we axiomatise the logic of HT 2-models and prove that it captures the strong equiva-lence of theories in partial equilibrium logic.
A causal logic of logic programming
- Proc. Ninth Conference on Principles of Knowledge Representation and Reasoning, KR’04
, 2004
"... The causal logic from (Bochman 2003b) is shown to provide a natural logical basis for logic programming. More exactly, it is argued that any logic program can be seen as a causal theory satisfying the Negation As Default principle (alias Closed World Assumption). Moreover, unlike well-known translat ..."
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Cited by 6 (2 self)
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The causal logic from (Bochman 2003b) is shown to provide a natural logical basis for logic programming. More exactly, it is argued that any logic program can be seen as a causal theory satisfying the Negation As Default principle (alias Closed World Assumption). Moreover, unlike well-known translations of logic programs to other nonmonotonic formalisms, the established correspondence between logic programs and causal theories is bidirectional in the sense that, for an appropriate causal logic, any causal theory is reducible to a logic program. The correspondence is shown to hold for logic programs of a most general kind involving disjunctions and default negations in heads of the rules. It is shown also to be adequate for a broad range of logic programming semantics, including stable, supported and partial stable models. The results strongly suggest that the causal logic can serve as a (long missing) logic of logic programming.
A Logical Foundation for Logic Programming I: Biconsequence Relations and Nonmonotonic Completion
- Journal of Logic Programming
, 1998
"... We suggest a general logical formalism for Logic Programming based on a four-valued inference. We show that it forms a proper setting for representing logic programs with negation as failure of a most general kind and for describing logics and semantics that characterize their behavior. In this ..."
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Cited by 5 (0 self)
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We suggest a general logical formalism for Logic Programming based on a four-valued inference. We show that it forms a proper setting for representing logic programs with negation as failure of a most general kind and for describing logics and semantics that characterize their behavior. In this way we also extend the connection between Logic and Logic Programming beyond positive programs. In addition, the suggested formalism will allow us to see a reasoning about logic programs as a most simple kind of nonmonotonic reasoning in general. Keywords. Foundations of logic programming, negation as failure, semantics for logic programs, nonmonotonic reasoning. 1
Department of Computer Science,
"... We propose a solution to a long-standing problem in the foundations of well-founded semantics (WFS) for logic programs. The problem addressed is this: which (non-modal) logic can be considered adequate for well-founded semantics in the sense that its minimal models (appropriately defined) coincide w ..."
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We propose a solution to a long-standing problem in the foundations of well-founded semantics (WFS) for logic programs. The problem addressed is this: which (non-modal) logic can be considered adequate for well-founded semantics in the sense that its minimal models (appropriately defined) coincide with the partial stable models of a logic program? We approach this problem by identifying the HT 2 frames previously proposed by Cabalar to capture WFS as structures of a kind used by Doˇsen to characterise a family of logics weaker than intuitionistic and minimal logic. We define a notion of minimal, total HT 2 model which we call partial equilibrium model. Since for normal logic programs these models coincide with partial stable models, we propose the resulting partial equilibrium logic as a logical foundation for well-founded semantics. In addition we axiomatise the logic of HT 2-models and prove that it captures the strong equivalence of theories in partial equilibrium logic.
A Logic for Reasoning about Well-Founded Semantics: Preliminary Report
"... Abstract. The paper presents a preliminary solution to a long-standing problem in the foundations of well-founded semantics for logic programs. The problem addressed is this: which logic can be considered adequate for well-founded semantics (WFS) in the sense that its minimal models (appropriately d ..."
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Abstract. The paper presents a preliminary solution to a long-standing problem in the foundations of well-founded semantics for logic programs. The problem addressed is this: which logic can be considered adequate for well-founded semantics (WFS) in the sense that its minimal models (appropriately defined) coincide with the partial stable models of a logic program? We approach this problem by identifying the HT 2 frames previously proposed by Cabalar [4] to capture WFS as structures of a kind used by Dosen [5] to characterise a family of logics weaker than intuitionistic and minimal logic. We identify partial stable models as minimal models in this semantics and we axiomatise the resulting logic. 1
