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.

翻译：用基本组件描述复杂对象是数学及一般科学中的常见策略。在1965年的开创性论文中，Kenneth Krohn和John Rhodes证明，每个有限确定性自动机均可由极其简单的自动机构成的级联积表示（或“模拟”），这催生了基于有限半群的优雅自动机代数理论（Krohn-Rhodes理论）。令人惊讶的是，通过关联逻辑程序与自动机，本文证明Krohn-Rhodes理论可应用于答案集编程（ASP）。具体而言，我们将级联积概念重构到ASP中，并证明每个程序均可由极简程序（即重置程序和标准程序）的乘积表示。大致而言，这意味着相对于级联积而言，重置程序和标准程序是ASP的基本构建模块。更广泛意义上，本文是基于Krohn-Rhodes理论的非单调推理系统乘积与网络代数理论研究的初步探索，旨在解决ASP及人工智能领域的重要开放性问题。