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 的 CSAT (A) 电路满意度问题是决定 A (由两条电路编码的) 方程式是否具有解决办法的问题。 虽然在 P 或 NP 完全 中解决有限代数的方程式系统, 但 CSAT (A) 却不知道这种二分法结果。 事实上, Idziak、 Kawalek 和 Krzaczkowski 构建了无能的 Maltsev 代数 A 的例子, 根据ETH 的假设和电路理论的开源预测, CSAT (A) 可以在准极学中解决,而不是在多边时间中解决。 CEQV (A) 的电路等同问题也是如此。 在本文中,我们将其结果概括为所有无能的 Maltsev 代数 > 2 。 这不仅推进了CSAT (和CEQV) 的复杂度分类项目, 我们还相信我们开发的工具在Qent ALG 模型研究中具有独立的兴趣。