Complexity Classes Arising from Circuits over Finite Algebraic Structures

2026-04-23

AI Summary

Traditional circuit complexity theory is confined to the Boolean domain and struggles to characterize computational power over more general algebraic structures. This work develops a unified algebraic framework that algebraizes Barrington’s branching programs and the model of Idziak et al., enabling a systematic study of language classes recognized by circuits over finite algebras—particularly simple algebras and those belonging to congruence-modular varieties. By integrating universal algebra, congruence-modularity theory, non-uniform automata, and circuit complexity analysis, the paper provides the first complete characterization of the language classes captured by such algebraic circuits. This establishes a bidirectional correspondence between algebraic structure and computational complexity, laying a foundational theoretical basis for algebraic models of computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Applications of cryptography

Abstract

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.

Problem

• circuit complexity
• finite algebraic structures
• Boolean circuits
• computational complexity
• congruence modular varieties

Innovation

• algebraic circuits
• circuit complexity
• finite algebras
• congruence modular varieties
• branching programs