This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of "here-and-there". We show that on logic programs equilibrium logic coincides with the inference operation associated with the stable model and answer set semantics of Gelfond and Lifschitz. We thereby obtain a very simple characterisation of answer set semantics as a form of minimal model reasoning in N2, while equilibrium logic itself provides a natural generalisation of this semantics to arbitrary theories. We discuss briefly some consequences and applications of this result. 1 Introduction By contrast with the minimal model style of reasoning characteristic of several approaches to the semantics of logic programs, the stable model semantics of Gelfond and Lifschitz [8] was, from the outset, much closer in spirit to the styles of reasoning found in othe...

Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.

We define a logical semantics called back-and-forth, applicable to normal and disjunctive datalog programs as well as to programs possessing a second, explicit or `strong' negation operator. We show that on normal programs it is equivalent to the well-founded semantics (WFS), and that on disjunctive programs it is equivalent to the P-stable semantics of Eiter, Leone and Sacc`a, hence to Przymusinski's 3valued stable semantics. The main advantage is that it is characterised by simple conditions on models in a well-known nonclassical logic and therefore provides a better insight into the nature of partial stable models from a logical standpoint. It also suggests why the P-stable models are a natural generalisation of WFS to the disjunctive case. On extended programs with strong negation, the back-and-forth semantics is apparently new, differing from answer sets, from WSFX and from the static semantics. Keywords: stable models, P-stable models, disjunctive programs, intermediate logics,...

ABSTRACT We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a self-dual logic-- constructible duality. We develop a self-dual model by considering an interval of worlds in an intuitionistic Kripke model. The duality arises through how we judge truth and falsity. Truth is judged forward in the Kripke model, as in intuitionistic logic, while falsity is judged backwards. We develop a self-dual algebra such that every point in the algebra is representable by some formula in the logic. This algebra arises as an instantiation of a Heyting algebra into several categorical constructions. In particular, we show that this algebra is an instantiation of the Chu construction applied to a Heyting algebra, the second Dialectica construction applied to a Heyting algebra, and as an algebra for the study of recursion and corecursion. Thus the algebra provides a common base for these constructions, and suggests itself as an important part of any constructive logical treatment of duality. Implicit programming is suggested as a new paradigm for computing with constructible duality as its formal system. We show that all the operators that have computable least fixed points are definable explicitly and all operators with computable optimal fixed points are definable implicitly within constructible duality. Implicit programming adds a novel definitional mechanism that allows functions to be defined implicitly. This new programming feature is especially useful for programming with co-recursively defined data-types such as circular lists. iii DEDICATION To my cousin Jordan Lackey (1963-1995) whose courage with AIDS was an inspiration. iv

Generalising the ideas of [10] we define a simple extension of the notion of unfounded set, called assumption set, that applies to disjunctive logic programs with strong negation. We show that assumption-free interpretations of such extended logic programs coincide with equilibrium models in the sense of [13] and hence with the answer sets of [3, 4].

Equilibrium logic is an approach to nonmonotonic reasoning that extends the stable model and answer-set semantics for logic programs. In particular, it includes the general case of nested logic programs, where arbitrary Boolean combinations are permitted in heads and bodies of rules, as special kinds of theories. In this paper, we present efficient reductions of the main reasoning tasks associated with equilibrium logic and nested logic programs into quantified propositional logic, an extension of classical propositional logic where quantifications over atomic formulas are permitted. Thus, quantified propositional logic is a fragment of second-order logic, and its formulas are usually referred to as quantified Boolean formulas (QBFs). We provide reductions not only for decision problems, but also for the central semantical objects of equilibrium logic and nested logic programs. In particular, our encodings map a given reasoning task into some QBF such that the latter is valid precisely in case the former holds. The reasoning tasks we deal with here are the consistency problem, brave reasoning, and skeptical reasoning. Additionally, we also provide encodings for testing equivalence of theories or programs under different notions of equivalence, viz. ordinary, strong, and uniform equivalence. For all considered reasoning tasks, we analyse their computational complexity and give strict complexity bounds. Hereby, our encodings yield upper

Abstract. For a given semantics, two nonmonotonic theories Π1 and Π2 can be said to be equivalent if they have the same intended models and strongly (resp., uniformly) equivalent if for any Σ, Π1∪Σ and Π2∪Σ are equivalent, where Σ is a set of sentences (resp., literals). In the general case, no restrictions are placed on the language (signature) of Σ. Relativised notions of strong and uniform equivalence are obtained by requiring that Σ belongs to a specified sublanguage L of the theories Π1 and Π2. For normal and disjunctive logic programs under stablemodel semantics, relativised strong and uniform equivalence have been defined and characterised in previous work by Woltran. Here, we extend these concepts to nonmonotonic theories in equilibrium logic and discuss applications in the context of prediction and explanation. 1

ing from the nature of the states involved, we can specify the change that an atomic program a effects by means of a two place transition relation R a on a set of states. This perspective gives rise to the study of so-called transition systems. The most general style of reasoning about programs and transition system is found in propositional dynamic logics (Pratt [ Pratt, 1976 ] [ Pratt, 1980 ] , Harel [ Harel, 1984 ] ) and in algebras of processes (Hennessy [ Hennessy, 1988 ] ). Processes and transition systems are studied from the perspective of modal logic in Stirling [ Stirling, 1987 ] and Van Benthem and Bergstra [ Benthem and Bergstra, 1993 ] . Dynamic semantics can be put to use to stipulate relational denotations for propositions. In this perspective, a state of information is a set of possible worlds, and a program updates a state of information by removing the worlds incompatible with the new information. Thus, the semantics of language is defined in terms of its potential to...

It is known that paraconsistent logical systems are more appropriate for inconsistency-tolerant and uncertainty reasoning than other types of logical systems. In this paper, a paraconsistent computation tree logic, PCTL, is obtained by adding paraconsistent negation to the standard computation tree logic CTL. PCTL can be used to appropriately formalize inconsistency-tolerant temporal reasoning. A theorem for embedding PCTL into CTL is proved. The validity, satisfiability, and model-checking problems of PCTL are shown to be decidable. The embedding and decidability results indicate that we can reuse the existing CTL-based algorithms for validity, satisfiability, and model-checking. An illustrative example of medical reasoning involving the use of PCTL is presented.

Abstract It is known that paraconsistent logical systems are more appropriate for inconsistency-tolerant and uncertainty reasoning than other types of logical systems. In this paper, a paraconsistent computation tree logic, PCTL, is obtained by adding paraconsistent negation to the standard computation tree logic CTL. PCTL can be used to appropriately formalize inconsistency-tolerant temporal reasoning. A theorem for embedding PCTL into CTL is proved. The validity, satisfiability, and model-checking problems of PCTL are shown to be decidable. The embedding and decidability results indicate that we can reuse the existing CTL-based algorithms for validity, satisfiability, and model-checking. An illustrative example of medical reasoning involving the use of PCTL is presented.