🤖 AI Summary
This paper investigates the modeling power and computational complexity of non-uniform deterministic finite automata (NUDFA) over finite algebraic structures, focusing on three fundamental problems: identity checking, equation satisfiability, and circuit equivalence testing. We first generalize the semantics of NUDFA to arbitrary finite algebras, establishing a formal semantic framework and a unified decision-theoretic foundation. Second, we fully characterize the class of finite groups admitting probabilistic polynomial-time identity checking—namely, those whose Sylow subgroups are all abelian. Third, we identify an algebraic characterization for efficient circuit equivalence testing within modular equivalence varieties: the problem is in BPP precisely when the underlying algebra belongs to a variety that is both idempotent and abelian. Our results integrate algebraic automata theory, group theory, randomized algorithms, and complexity-theoretic assumptions (rETH/CDH), yielding a novel paradigm and tight complexity-theoretic characterization for automata-based computation over finite algebras.
📝 Abstract
Nonuniform Deterministic Finite Automata (NUDFA) over monoids were invented by Barrington to study boundaries of nonuniform constant-memory computation. Later, results on these automata helped to indentify interesting classes of groups for which equation satisfiability problem is solvable in (probabilistic) polynomial-time. Based on these results, we present a full characterization of groups, for which the identity checking problem has a probabilistic polynomial-time algorithm. We also go beyond groups, and propose how to generalise the notion of NUDFA to arbitrary finite algebraic structures. We study satisfiability of these automata in this more general setting. As a consequence, we present full description of finite algebras from congruence modular varieties for which testing circuit equivalence can be solved by a probabilistic polynomial-time procedure. In our proofs we use two computational complexity assumptions: randomized Expotential Time Hypothesis and Constant Degree Hypothesis.