Circuit equivalence in 2-nilpotent algebras

TL;DR

This paper investigates the computational complexity of circuit equivalence in 2-nilpotent algebras, establishing that the problem is solvable in polynomial time for these algebras, which are a specific class within congruence modular varieties.

Contribution

It provides the first polynomial-time algorithm for circuit equivalence in 2-nilpotent algebras, filling a key gap in the classification of algebraic circuit problems.

Findings

Circuit equivalence in 2-nilpotent algebras is in P.

Addresses an open case in the classification of circuit equivalence complexity.

Advances understanding of algebraic circuit problems in universal algebra.

Abstract

The circuit equivalence problem of a finite algebra $\mathbf A$ is the computational problem of deciding whether two circuits over $\mathbf A$ define the same function or not. This problem not just generalises the equivalence problem for Boolean circuits, but is also of high interest in universal algebra, as it models the problems of checking identities in $\mathbf A$. In this paper we discuss the complexity for algebras from congruence modular varieties. A partial classification was already given by Idziak and Krzaczkowski, leaving essentially only a gap for nilpotent but not supernilpotent algebras. We start a systematic study of this open case, proving that the circuit equivalence problem is in P for $2$-nilpotent such algebras.