Most classical results in circuit complexity theory concern circuits over the Boolean domain. Besides their simplicity and the ease of comparing different languages, the actual architecture of computers is also an important motivating factor. On the other hand, by restricting attention to Boolean circuits, we lose sight of the much richer landscape of circuits over larger domains. Our goal is to bridge these two worlds: to use deep algebraic tools to obtain results in computational complexity theory, including circuit complexity, and to apply results from computational complexity to gain a better understanding of the structure of finite algebras. In this paper, we propose a unifying algebraic framework which we believe will help achieve this goal. Our work is inspired by branching programs and nonuniform deterministic automata introduced by Barrington, as well as by their generalization proposed by Idziak et al. We begin our investigation by studying the languages recognized by natural classes of algebraic structures. In particular, we characterize language classes recognized by circuits over simple algebras and over algebras from congruence modular varieties.

翻译：电路复杂性理论中的大多数经典结果涉及布尔域上的电路。除了其简单性和便于比较不同语言之外，计算机的实际架构也是一个重要的激励因素。另一方面，若仅局限于布尔电路，我们便会忽略更丰富领域上电路的广阔图景。我们的目标是架起这两个世界之间的桥梁：利用深刻的代数工具来获得计算复杂性理论（包括电路复杂性）中的结果，并应用计算复杂性的成果来加深对有限代数结构的理解。在本文中，我们提出了一个统一的代数框架，相信这将有助于实现这一目标。我们的工作受到Barrington引入的分支程序和非均匀确定性自动机，以及Idziak等人提出的其推广形式的启发。我们首先研究由自然类别的代数结构所识别的语言，特别地，我们刻画了由简单代数上的电路和同余模簇中的代数上的电路所识别的语言类。