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.

翻译：代数A上的电路可满足性问题CSAT(A)是判定A上的方程（由两个电路编码）是否有解的问题。尽管有限代数上方程组的求解要么属于P类，要么是NP完全的，但对于CSAT(A)尚未有类似的二分法结果。事实上，Idziak、Kawalek和Krzaczkowski构造了幂零Mal'tsev代数A的实例，在假设ETH及电路理论中一个未决猜想成立的前提下，这些实例上的CSAT(A)问题可在拟多项式时间内求解，而非多项式时间。电路等价问题CEQV(A)同样具有这一性质。本文将其结果推广至所有Fitting长度>2的幂零Mal'tsev代数。这不仅推进了对同余模簇中代数上CSAT（及CEQV）复杂性进行分类的研究项目，且我们相信所开发的工具在幂零代数研究中具有独立的价值。