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Approximations, Stable Operators, WellFounded 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 wellfounded 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 wellfounded semantics
 In P
, 2006
"... We propose a solution to a longstanding problem in the foundations of wellfounded semantics (WFS) for logic programs. The problem addressed is this: which (nonmodal) logic can be considered adequate for wellfounded semantics in the sense that its minimal models (appropriately defined) coincide ..."
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Cited by 8 (2 self)
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We propose a solution to a longstanding problem in the foundations of wellfounded semantics (WFS) for logic programs. The problem addressed is this: which (nonmodal) logic can be considered adequate for wellfounded 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 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 wellfounded semantics. In addition we axiomatise the logic of HT 2models and prove that it captures the strong equivalence of theories in partial equilibrium logic.
Department of Computer Science,
"... We propose a solution to a longstanding problem in the foundations of wellfounded semantics (WFS) for logic programs. The problem addressed is this: which (nonmodal) logic can be considered adequate for wellfounded semantics in the sense that its minimal models (appropriately defined) coincide w ..."
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We propose a solution to a longstanding problem in the foundations of wellfounded semantics (WFS) for logic programs. The problem addressed is this: which (nonmodal) logic can be considered adequate for wellfounded 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 wellfounded semantics. In addition we axiomatise the logic of HT 2models and prove that it captures the strong equivalence of theories in partial equilibrium logic.
A Logic for Reasoning about WellFounded Semantics: Preliminary Report
"... Abstract. The paper presents a preliminary solution to a longstanding problem in the foundations of wellfounded semantics for logic programs. The problem addressed is this: which logic can be considered adequate for wellfounded semantics (WFS) in the sense that its minimal models (appropriately d ..."
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Abstract. The paper presents a preliminary solution to a longstanding problem in the foundations of wellfounded semantics for logic programs. The problem addressed is this: which logic can be considered adequate for wellfounded 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
On Partial and Paraconsistent Logics
, 1999
"... In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of NonContradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L 4 , the most general logic ..."
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In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of NonContradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L 4 , the most general logic under consideration, we also prove a version of the CraigLyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the right setup is chosen. Our logic L 4 has a semantics that also underlies Belnap's [4] and is related to the logic of bilattices. L 4 is in focus most of the time, but it is also shown how results obtained for L 4 can be transferred to several variants.