Satisfiability of circuits and equations over finite Malcev algebras

We show that the satisfiability of circuits over finite Malcev algebra A is NP-complete or A is nilpotent. This strengthens the result from our earlier paper [Idziak and Krzaczkowski, 2018] where nilpotency has been enforced, however with the use of a stronger assumption that no homomorphic image of A has NP-complete circuits satisfiability. Our methods are moreover strong enough to extend our result of [Idziak et al., 2021] from groups to Malcev algebras. Namely we show that tractability of checking if an equation over such an algebra A has a solution enforces its nice structure: A must have a nilpotent congruence �� such that also the quotient algebra A/�� is nilpotent. Otherwise, if A has no such congruence �� then the Exponential Time Hypothesis yields a quasipolynomial lower bound. Both our results contain important steps towards a full characterization of finite algebras with tractable circuit satisfiability as well as equation satisfiability.
LIPIcs, Vol. 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022), pages 37:1-37:14