Describing complex objects by elementary ones is a common strategy in mathematicsand science in general. In their seminal 1965 paper, Kenneth Krohn and JohnRhodes 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, wecan show in this paper that the Krohn-Rhodes Theory is applicable inAnswer Set Programming (ASP). More precisely, we recast the concept of a cascadeproduct to ASP, and prove that every program can be represented by a product ofvery simple programs, the reset and standard programs. Roughly, this impliesthat the reset and standard programs are the basic building blocks of ASP withrespect to the cascade product. In a broader sense, this paper is a first steptowards an algebraic theory of products and networks of nonmonotonic reasoningsystems based on Krohn-Rhodes Theory, aiming at important open issues in ASPand AI in general.
