Satisfiability in multi-valued circuits

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted 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 strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time.Comment: 50 page

O polinomima u algebrama Maljceva

We establish several properties of higher commutators, which were introduced by A. Bulatov, in congruence permutable varieties. We use these commutators to prove that the clone of polynomial functions of a finite Mal’cev algebra whose congruence lattice is of height at most 2, can be described by a finite set of relations. For a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates congruence modular variety, we are able to show that the property of affine completeness is decidable. Moreover, polynomial equivalence problem has polynomial complexity in the length of the input polynomials.Ustanovljavamo osobine viˇsih komutatora, koje je uveo A. Bulatov, u kongruencijki permutabilnim varijetetima. Te komutatore koristimo da bi dokazali da se klon polinomijalnih funkcija konaˇcne Maljcevljeve algebre ˇcija je mreˇza kongruencija visine najviˇse dva moˇze opisati konaˇcnim skupom relacija. Za konaˇcne nilpotentne algebre konaˇcnog tipa koje su proizvod algebri koje imaju red stepena prostog broja i koje generiˇsu kongruencijki modularan varijetet pokazu-jemo da je osobina afine kompletnosti odluˇciva. Takod¯e, pokazujemo za istu klasu da problem polinomijalne ekvivalencije ima polinomnu sloˇzenost u zavisnosti od duˇzine unetih polinomijalnih terma