A Logical Charaterisation of Ordered Disjunction

TL;DR

This paper provides a logical characterization of the ordered disjunction operator in LPODs, enabling direct computation of answer sets and analysis of properties like distributivity, improving understanding and implementation of preference logic programming.

Contribution

It introduces a translation of ordered disjunction into the logic of Here-and-There, facilitating direct answer set computation and property analysis of the 'x' operator.

Findings

Direct translation captures answer sets without split programs

Enables checking properties like distributivity of 'x'

Provides an alternative implementation approach

Abstract

In this paper we consider a logical treatment for the ordered disjunction operator 'x' introduced by Brewka, Niemel\"a and Syrj\"anen in their Logic Programs with Ordered Disjunctions (LPOD). LPODs are used to represent preferences in logic programming under the answer set semantics. Their semantics is defined by first translating the LPOD into a set of normal programs (called split programs) and then imposing a preference relation among the answer sets of these split programs. We concentrate on the first step and show how a suitable translation of the ordered disjunction as a derived operator into the logic of Here-and-There allows capturing the answer sets of the split programs in a direct way. We use this characterisation not only for providing an alternative implementation for LPODs, but also for checking several properties (under strongly equivalent transformations) of the 'x' operator, like for instance, its distributivity with respect to conjunction or regular disjunction. We also make a comparison to an extension proposed by K\"arger, Lopes, Olmedilla and Polleres, that combines 'x' with regular disjunction.