On cascade products of answer set programs

TL;DR

This paper extends Krohn-Rhodes algebraic theory from automata to Answer Set Programming, showing that complex ASP programs can be decomposed into simple reset and standard programs using cascade products.

Contribution

It introduces a novel application of cascade products to ASP, demonstrating that all ASP programs can be represented by compositions of simple programs, thus laying groundwork for an algebraic theory of nonmonotonic reasoning.

Findings

Every ASP program can be represented as a cascade product of simple programs.

Reset and standard programs serve as fundamental building blocks in ASP.

Establishes a connection between algebraic automata theory and ASP.

Abstract

Describing complex objects by elementary ones is a common strategy in mathematics and science in general. In their seminal 1965 paper, Kenneth Krohn and John Rhodes showed that every finite deterministic automaton can be represented (or "emulated") by a cascade product of very simple automata. This led to an elegant algebraic theory of automata based on finite semigroups (Krohn-Rhodes Theory). Surprisingly, by relating logic programs and automata, we can show in this paper that the Krohn-Rhodes Theory is applicable in Answer Set Programming (ASP). More precisely, we recast the concept of a cascade product to ASP, and prove that every program can be represented by a product of very simple programs, the reset and standard programs. Roughly, this implies that the reset and standard programs are the basic building blocks of ASP with respect to the cascade product. In a broader sense, this paper is a first step towards an algebraic theory of products and networks of nonmonotonic reasoning systems based on Krohn-Rhodes Theory, aiming at important open issues in ASP and AI in general.