Pearce, David


Revisiting Explicit Negation in Answer Set Programming

arXiv.org Artificial Intelligence

A common feature in Answer Set Programming is the use of a seco nd negation, stronger than default negation and sometimes called explicit, strong or classica l negation. This explicit negation is normally used in front of atoms, rather than allowing its use as a regular op erator. In this paper we consider the arbitrary combination of explicit negation with nested expressions, as those defined by Lifschitz, Tang and Turner. We extend the concept of reduct for this new syntax and then pr ove that it can be captured by an extension of Equilibrium Logic with this second negation. We study som e properties of this variant and compare to the already known combination of Equilibrium Logic with Nel son's strong negation.


Functional ASP with Intensional Sets: Application to Gelfond-Zhang Aggregates

arXiv.org Artificial Intelligence

In this paper, we propose a variant of Answer Set Programming (ASP) with evaluable functions that extends their application to sets of objects, something that allows a fully logical treatment of aggregates. Formally, we start from the syntax of First Order Logic with equality and the semantics of Quantified Equilibrium Logic with evaluable functions (QELF). Then, we proceed to incorporate a new kind of logical term, intensional set (a construct commonly used to denote the set of objects characterised by a given formula), and to extend QELF semantics for this new type of expression. In our extended approach, intensional sets can be arbitrarily used as predicate or function arguments or even nested inside other intensional sets, just as regular first-order logical terms. As a result, aggregates can be naturally formed by the application of some evaluable function (count, sum, maximum, etc) to a set of objects expressed as an intensional set. This approach has several advantages. First, while other semantics for aggregates depend on some syntactic transformation (either via a reduct or a formula translation), the QELF interpretation treats them as regular evaluable functions, providing a compositional semantics and avoiding any kind of syntactic restriction. Second, aggregates can be explicitly defined now within the logical language by the simple addition of formulas that fix their meaning in terms of multiple applications of some (commutative and associative) binary operation. For instance, we can use recursive rules to define sum in terms of integer addition. Last, but not least, we prove that the semantics we obtain for aggregates coincides with the one defined by Gelfond and Zhang for the Alog language, when we restrict to that syntactic fragment. (Under consideration for acceptance in TPLP)


FQHT: The Logic of Stable Models for Logic Programs with Intensional Functions

AAAI Conferences

We study a logical system FQHT that is appropriate for reasoning about nonmonotonic theories with intensional functions as treated in the approach of Bartholomew and Lee (2012). We provide a logical semantics, a Gentzen style proof theory and establish completeness results. The adequacy of the approach is demonstrated by showing that it captures the Bartholemew/Lee semantics and satisfies a strong equivalence property.


Farinas del Cerro

AAAI Conferences

We study a logical system FQHT that is appropriate for reasoning about nonmonotonic theories with intensional functions as treated in the approach of Bartholomew and Lee (2012). We provide a logical semantics, a Gentzen style proof theory and establish completeness results. The adequacy of the approach is demonstrated by showing that it captures the Bartholemew/Lee semantics and satisfies a strong equivalence property.


An Approach to Minimal Belief Via Objective Belief

AAAI Conferences

As a doxastic counterpart to epistemic logic based on S5 we study the modal logic KSD that can be viewed as an approach to modelling a kind of objective and fair belief. We apply KSD to the problem of minimal belief and develop an alterna- tive approach to nonmonotonic modal logic using a weaker concept of expansion. This corresponds to a certain minimal kind of KSD model and yields a new type of nonmonotonic doxastic reasonin


Pearce

AAAI Conferences

As a doxastic counterpart to epistemic logic based on S5 we study the modal logic KSD that can be viewed as an approach to modelling a kind of objective and fair belief. We apply KSD to the problem of minimal belief and develop an alterna- tive approach to nonmonotonic modal logic using a weaker concept of expansion.


Interpolation in Equilibrium Logic and Answer Set Programming: the Propositional Case

arXiv.org Artificial Intelligence

Interpolation is an important property of classical and many non classical logics that has been shown to have interesting applications in computer science and AI. Here we study the Interpolation Property for the propositional version of the non-monotonic system of equilibrium logic, establishing weaker or stronger forms of interpolation depending on the precise interpretation of the inference relation. These results also yield a form of interpolation for ground logic programs under the answer sets semantics. For disjunctive logic programs we also study the property of uniform interpolation that is closely related to the concept of variable forgetting.


Characterising equilibrium logic and nested logic programs: Reductions and complexity

arXiv.org Artificial Intelligence

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 polynomial 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. We provide reductions not only for decision problems, but also for the central semantical concepts of equilibrium logic and nested logic programs. In particular, our encodings map a given decision problem into some formula such that the latter is valid precisely in case the former holds. The basic 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.