1,161 research outputs found
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 there is a polynomial time algorithm that
decides if an equation over 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
Intermediate problems in modular circuits satisfiability
In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite
algebras had been introduced and applied to sketch P versus NP-complete
borderline for circuits satisfiability over algebras from congruence modular
varieties. However the problem for nilpotent (which had not been shown to be
NP-hard) but not supernilpotent algebras (which had been shown to be polynomial
time) remained open.
In this paper we provide a broad class of examples, lying in this grey area,
and show that, under the Exponential Time Hypothesis and Strong Exponential
Size Hypothesis (saying that Boolean circuits need exponentially many modular
counting gates to produce boolean conjunctions of any arity), satisfiability
over these algebras have intermediate complexity between and , where measures how much a nilpotent algebra
fails to be supernilpotent. We also sketch how these examples could be used as
paradigms to fill the nilpotent versus supernilpotent gap in general.
Our examples are striking in view of the natural strong connections between
circuits satisfiability and Constraint Satisfaction Problem for which the
dichotomy had been shown by Bulatov and Zhuk
Complex Algebras of Arithmetic
An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a
sequence of arithmetic and logical operations to be performed on sets of
natural numbers. Arithmetic circuits can also be viewed as the elements of the
smallest subalgebra of the complex algebra of the semiring of natural numbers.
In the present paper, we investigate the algebraic structure of complex
algebras of natural numbers, and make some observations regarding the
complexity of various theories of such algebras
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
Admissibility in Finitely Generated Quasivarieties
Checking the admissibility of quasiequations in a finitely generated (i.e.,
generated by a finite set of finite algebras) quasivariety Q amounts to
checking validity in a suitable finite free algebra of the quasivariety, and is
therefore decidable. However, since free algebras may be large even for small
sets of small algebras and very few generators, this naive method for checking
admissibility in \Q is not computationally feasible. In this paper,
algorithms are introduced that generate a minimal (with respect to a multiset
well-ordering on their cardinalities) finite set of algebras such that the
validity of a quasiequation in this set corresponds to admissibility of the
quasiequation in Q. In particular, structural completeness (validity and
admissibility coincide) and almost structural completeness (validity and
admissibility coincide for quasiequations with unifiable premises) can be
checked. The algorithms are illustrated with a selection of well-known finitely
generated quasivarieties, and adapted to handle also admissibility of rules in
finite-valued logics
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
Mace4 Reference Manual and Guide
Mace4 is a program that searches for finite models of first-order formulas.
For a given domain size, all instances of the formulas over the domain are
constructed. The result is a set of ground clauses with equality. Then, a
decision procedure based on ground equational rewriting is applied. If
satisfiability is detected, one or more models are printed. Mace4 is a useful
complement to first-order theorem provers, with the prover searching for proofs
and Mace4 looking for countermodels, and it is useful for work on finite
algebras. Mace4 performs better on equational problems than did our previous
model-searching program Mace2.Comment: 17 page
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