Counting solutions of equations over two-element algebras

Solving equations is one of the most important problems in computer science. Apart from the problem of existence of solutions of equations we may consider the problem of a number of solutions of equations. Such a problem is much more difficult than the decision one. This paper presents a complete classification of the complexity of the problem of counting solutions of equations over any fixed two-element algebra. It is shown that the complexity of such problems depends only on the clone of term operations of the algebra and for any fixed two-element algebra such a problem is either in FP or #Pcomplete

The complexity of problems connected with two-element algebras

This paper presents a complete classification of the complexity of the SAT and equivalence problems for two-element algebras. Cases of terms and polynomials are considered. We show that for any fixed two-element algebra the considered SAT problems are either in P or NP-complete and the equivalence problems are either in P or coNP-complete. We show that the complexity of the considered problems, parametrized by an algebra, are determined by the clone of term operations of the algebra and does not depend on generating functions for the clone

Expressive Power, Satisfiability and Equivalence of Circuits over Nilpotent Algebras

Satisfiability of Boolean circuits is NP-complete in general but becomes polynomial time when restricted for example either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is connected with solving equations over finite algebras. This in turn is one of the oldest and well-known mathematical problems which for centuries was the driving force of research in algebra. Let us only mention Galois theory, Gaussian elimination or Diophantine Equations. The last problem has been shown to be undecidable, however in finite realms such problems are obviously decidable in nondeterministic polynomial time. A project of characterizing finite algebras m A with polynomial time algorithms deciding satisfiability of circuits over m A has been undertaken in [Pawel M. Idziak and Jacek Krzaczkowski, 2018]. Unfortunately that paper leaves a gap for nilpotent but not supernilpotent algebras. In this paper we discuss possible attacks on filling this gap

The Complexity of Problems Connected with Two-element Algebras

This paper presents a complete classification of the complexity of the SAT and equivalence problems for two-element algebras. Cases of terms and polynomials are considered. We show that for any fixed two-element algebra the considered SAT problems are either in P or NP-complete and the equivalence problems are either in P or coNP-complete. We show that the complexity of the considered problems, parametrized by an algebra, are determined by the clone of term operations of the algebra and does not depend on generating functions for the clone

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

Term Satisfiability Problem for Two-Element Algebras is in QL or is NQL-Complete

We show that the term satisfiability problem for finite algebras is in NQL. Moreover we present a complete classification of the computational complexity of the term satisfiability problem for two-element algebras. We show that for any fixed twoelement algebra the problem is either in QL or NQL-complete. We show that the complexity of the considered problem, parameterized by an algebra, is determined by the clone of term operations of the algebra

Even faster algorithms for CSATOver supernilpotent algebras

Recently, a few papers considering the polynomial equation satisfiability problem and the circuitsatisfiability problem over finite supernilpotent algebras from so called congruence modular varietieswere published. All the algorithms considered in these papers are quite similar and rely on checkinga not too big set of potential solutions. Two of these algorithms achieving the lowest time complexityup to now, were presented in [1] (algorithm working for finite supernilpotent algebras) and in [5](algorithm working in the group case). In this paper we show a deterministic algorithm of the sametype solving the considered problems for finite supernilpotent algebras which has lower computationalcomplexity than the algorithm presented in [1] and in most cases even lower than the group casealgorithm from [5]. We also present a linear time Monte Carlo algorithm solving the same problem.This, together with the algorithm for nilpotent but not supernilpotent algebras presented in [17], isthe very first attempt to solving the circuit satisfiability problem using probabilistic algorithms