Nonuniform Deterministic Finite Automata over finite algebraic structures

TL;DR

This paper characterizes groups where identity checking for Nonuniform Deterministic Finite Automata (NUDFA) is in probabilistic polynomial time and extends the concept to arbitrary finite algebraic structures, analyzing their satisfiability.

Contribution

It provides a full characterization of groups with polynomial-time identity checking and generalizes NUDFA to finite algebraic structures, analyzing circuit equivalence complexity.

Findings

Characterization of groups with polynomial-time identity checking

Extension of NUDFA to finite algebraic structures

Description of algebras with efficient circuit equivalence testing

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.