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Nonuniform Deterministic Finite Automata over finite algebraic structures

Paweł M. Idziak, Piotr Kawałek, Jacek Krzaczkowski

TL;DR

This work generalizes Barrington’s Nonuniform Deterministic Finite Automata to finite algebraic structures, linking algebraic properties to circuit complexity. It provides a two-pronged set of results: a complete RP-characterization for PolEqv over finite groups under $ ext{rETH}$ and $ ext{CDH}$, and a broader framework for algebras in congruence modular varieties, yielding RP conditions for ProgCSat and CEqv based on nilpotency and supernilpotence with specific coset-structure constraints. The authors develop a robust toolkit combining Tame Congruence Theory, Mal'cev terms, and modular-circuit representations (e.g., $ extsf{AND}_d o extsf{MOD}_m o extsf{MOD}_p$ circuits) to translate program/satisfiability questions into algebraic ones, obtaining both hardness (NP-hardness of ProgCSat) and tractability (RP membership under structural hypotheses) results. The findings illuminate deep connections between universal algebra and circuit complexity, with implications for the design and analysis of nonuniform computation over algebraic domains. They also outline clear directions for extending these results beyond congruence modular varieties and for refining the role of the Constant Degree Hypothesis in these classifications.

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.

Nonuniform Deterministic Finite Automata over finite algebraic structures

TL;DR

This work generalizes Barrington’s Nonuniform Deterministic Finite Automata to finite algebraic structures, linking algebraic properties to circuit complexity. It provides a two-pronged set of results: a complete RP-characterization for PolEqv over finite groups under and , and a broader framework for algebras in congruence modular varieties, yielding RP conditions for ProgCSat and CEqv based on nilpotency and supernilpotence with specific coset-structure constraints. The authors develop a robust toolkit combining Tame Congruence Theory, Mal'cev terms, and modular-circuit representations (e.g., circuits) to translate program/satisfiability questions into algebraic ones, obtaining both hardness (NP-hardness of ProgCSat) and tractability (RP membership under structural hypotheses) results. The findings illuminate deep connections between universal algebra and circuit complexity, with implications for the design and analysis of nonuniform computation over algebraic domains. They also outline clear directions for extending these results beyond congruence modular varieties and for refining the role of the Constant Degree Hypothesis in these classifications.

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.
Paper Structure (10 sections, 19 theorems, 32 equations, 1 figure)

This paper contains 10 sections, 19 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Algebraic preliminaries
  3. Polynomial equivalence
  4. Program satisfiability
  5. Circuit equivalence
  6. Notation
  7. Useful algebraic facts
  8. ProgramCSat -- hardness
  9. ProgramCSat -- easiness
  10. Final Remarks

Key Result

Theorem 1.1

Let ${\bf{G}}$ be a finite group. Assuming rETH and CDH the problem $\operatorname{\textup{PolEqv}}$ is in RP if and only if ${\bf{G}}$ is solvable and has a nilpotent normal subgroup ${\bf{H}}$ with the quotient ${\bf{G}}/{\bf{H}}$ being also nilpotent.

Figures (1)

  • Figure :

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 3.1
  • proof
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • proof : Proof of Theorem \ref{['thm:progsat']}:
  • ...and 29 more