In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.

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...

This is a review of some of the definitions of the concept of a stable model that have been proposed in the literature. These definitions are equivalent to each other, at least when applied to traditional Prologstyle programs, but there are reasons why each of them is valuable and interesting. A new characterization of stable models can suggest an alternative picture of the intuitive meaning of logic programs; or it can lead to new algorithms for generating stable models; or it can work better than others when we turn to generalizations of the traditional syntax that are important from the perspective of answer set programming; or it can be more convenient for use in proofs; or it can be interesting simply because it demonstrates a relationship between seemingly unrelated ideas.

We compare two recent extensions of the answer set (stable model) semantics of logic programs. One of them, due to Lifschitz, Tang and Turner, allows the bodies and heads of rules to contain nested expressions. The other, due to Niemela and Simons, uses weight constraints. We show that there is a simple, modular translation from the language of weight constraints into the language of nested expressions that preserves the program's answer sets. This translation can be used to study equivalent transformations of logic programs written in the input language of the answer set programming system SMODELS. Keywords: answer sets, cardinality constraints, SMODELS, stable models, weight constraints. 1

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Abstract. In this paper we introduce an extension of Equilibrium Logic (a logical characterisation of the Answer Set Semantics for logic programs) consisting in the inclusion of modal temporal operators, as those used in Linear Temporal Logic. As a result, we obtain a very expressive formalism that allows nonmonotonic reasoning for temporal domains. To show an example of its utility, we present a translation of a language for reasoning about actions into this formalism. 1

EXTENDED VERSION Abstract. In this paper we propose an extension of Answer Set Programming (ASP) [1], and in particular, of its most general logical counterpart, Quantified Equilibrium Logic (QEL) [2], to deal with partial functions. Although the treatment of equality in QEL can be established in different ways, we first analyse the choice of decidable equality with complete functions and Herbrand models, recently proposed in the literature [3]. We argue that this choice yields some counterintuitive effects from a logic programming and knowledge representation point of view. We then propose a variant called QEL = F where the set of functions is partitioned into partial and Herbrand functions (we also call constructors). In the rest of the paper, we show a direct connection to Scott’s Logic of Existence [4] and present a practical application, proposing an extension of normal logic programs to deal with partial functions and equality, so that they can be translated into function-free normal programs, being possible in this way to compute their answer sets with any standard ASP solver. 1

Abstract. Over the years, the stable-model semantics has gained a position of the correct (two-valued) interpretation of default negation in programs. However, for programs with aggregates (constraints), the stable-model semantics, in its broadly accepted generalization stemming from the work by Pearce, Ferraris and Lifschitz, has a competitor: the semantics proposed by Faber, Leone and Pfeifer, which seems to be essentially different. Our goal is to explain the relationship between the two semantics. Pearce, Ferraris and Lifschitz’s extension of the stable-model semantics is best viewed in the setting of arbitrary propositional theories. We propose here an extension of the Faber-Leone-Pfeifer semantics, or FLP semantics, for short, to the full propositional language, which reveals both common threads and differences between the FLP and stable-model semantics. We use our characterizations of FLP-stable models to derive corresponding results on strong equivalence and on normal forms of theories under the FLP semantics. We apply a similar approach to define supported models for arbitrary propositional theories, and to study their properties. 1

This note provides background information and references to the tutorial on recent research developments in logic programming inspired by need of knowledge representation.

Abstract. This paper analyses the idea of strong equivalence for transition systems represented as logic programs under the Answer Set Programming (ASP) paradigm. To check strong equivalence, we use a linear temporal extension of Equilibrium Logic (a logical characterisation of ASP) and its monotonic basis, the intermediate logic of Here-and-There (HT). Trivially, equivalence in this temporal extension of HT provides a sufficient condition for temporal strong equivalence and, as we show in the paper, it can be transformed into a provability test into the standard Linear Temporal Logic (LTL), something that can be automatically checked using any of the LTL available provers. The paper shows an example of the potential utility of this method by detecting some redundant rules in a simple actions reasoning scenario. 1