CSAT and CEQV for nilpotent Maltsev algebras of Fitting length > 2

TL;DR

This paper extends the understanding of the computational complexity of circuit satisfaction and equivalence problems for a broad class of nilpotent Maltsev algebras, showing they are solvable in quasipolynomial time under certain assumptions.

Contribution

It generalizes previous results to all nilpotent Maltsev algebras with Fitting length greater than 2, advancing the classification of their computational complexity.

Findings

CSAT(A) and CEQV(A) are solvable in quasipolynomial time under ETH and a circuit theory conjecture.

The results apply to all nilpotent Maltsev algebras of Fitting length > 2.

New tools developed are of independent interest for studying nilpotent algebras.

Abstract

The circuit satisfaction problem CSAT(A) of an algebra A is the problem of deciding whether an equation over A (encoded by two circuits) has a solution or not. While solving systems of equations over finite algebras is either in P or NP-complete, no such dichotomy result is known for CSAT(A). In fact, Idziak, Kawalek and Krzaczkowski constructed examples of nilpotent Maltsev algebras A, for which, under the assumption of ETH and an open conjecture in circuit theory, CSAT(A) can be solved in quasipolynomial, but not polynomial time. The same is true for the circuit equivalence problem CEQV(A). In this paper we generalize their result to all nilpotent Maltsev algebras of Fitting length >2. This not only advances the project of classifying the complexity of CSAT (and CEQV) for algebras from congruence modular varieties, but we also believe that the tools we developed are of independent interest in the study of nilpotent algebras.