Functional ASP with Intensional Sets: Application to Gelfond-Zhang Aggregates. Issue 3 (10th August 2018)

• Record Type: Journal Article
• Authors: CABALAR, PEDRO; FANDINNO, JORGE; FARIÑAS DEL CERRO, LUIS; PEARCE, DAVID
• Editors: Dal Palu, Alessandro; Tarau, Paul
• Is Part Of: Theory and practice of logic programming. Volume 18:Issue 3/4(2018)
• Page Start: 390
• Page End: 405
• Publication Date: 2018-08-10
• Subjects: Answer Set Programming, -- Equilibrium Logic, -- Partial Functions, -- Aggregates; Logic programming -- Periodicals; Artificial intelligence -- Computer programs -- Periodicals; Constraint programming (Computer science) -- Periodicals; 005.115
• DOI: 10.1017/S1471068418000169
• Languages: English
• ISSNs: 1471-0684

Abstract

Abstract: 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 ( ${\rm QEL}^=_{\cal F}$ ). 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 ${\rm QEL}^=_{\cal F}$ 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 ${\rm QEL}^=_{\cal F}$ 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 definesum 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 ${\cal A}\mathit{log}$ language, when we restrict to that syntactic fragment.