Sequential composition of answer set programs

TL;DR

This paper develops an algebraic framework for answer set programming by introducing sequential composition, enabling algebraic analysis of answer sets and equivalence, and proposing new classes of programs.

Contribution

It introduces the concept of sequential composition in answer set programs, linking syntax and semantics algebraically, and defines measures like index and period for cyclicality.

Findings

Algebraic characterization of answer sets and equivalence.

Introduction of index and period as measures of cyclicality.

Proposal of a new class of aperiodic answer set programs.

Abstract

This paper contributes to the mathematical foundations of logic programming by introducing and studying the sequential composition of answer set programs. On the semantic side, we show that the immediate consequence operator of a program can be represented via composition, which allows us to compute the least model semantics of Horn programs without any explicit reference to operators. As a result, we can characterize answer sets algebraically, which further provides an algebraic characterization of strong and uniform equivalence which is appealing. This bridges the conceptual gap between the syntax and semantics of an answer set program in a mathematically satisfactory way. The so-obtained algebraization of answer set programming allows us to transfer algebraic concepts into the ASP-setting which we demonstrate by introducing the index and period of an answer set program as an algebraic measure of its cyclicality. The technical part of the paper ends with a brief section introducing the algebraically inspired novel class of aperiodic answer set programs strictly containing the acyclic ones. In a broader sense, this paper is a first step towards an algebra of answer set programs and in the future we plan to lift the methods of this paper to wider classes of programs, most importantly to higher-order and disjunctive programs and extensions thereof.