Complexity Classes Arising from Circuits over Finite Algebraic Structures
For complexity theorists, this work provides a new algebraic perspective to bridge circuit complexity and finite algebra, though it is primarily foundational and does not present concrete new results or improvements.
This paper proposes a unifying algebraic framework to study circuit complexity over finite algebraic structures, characterizing language classes recognized by circuits over simple algebras and congruence modular varieties.
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.