On the Complexity of Some Maltsev Conditions

Abstract

Abstract. This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete. In addition, we provide sharp bounds in terms of the size of two-generated free algebras on the number of terms needed to witness various Maltsev conditions, such as congruence distributivity.

Computational Complexity of Various Mal'cev Conditions

“…This paper generalizes two of the results in [4], first the result that whether or not a finite idempotent algebra possesses a majority term can be determined in polynomial time and second that various Mal'cev conditions, in general, require exponential time to determine satisfaction. Many of the terms used in this paper refer to concepts from Tame Congruence Theory (see [6]) though these concepts are not required in order to understand the proofs here presented.…”

“…This section proceeds along essentially the same lines as [4,Sec. 9] in proving that various Mal'cev conditions require exponential time to determine satisfaction.…”

“…We accomplish this by reducing the clone membership problem to the problem of detecting nontrivial idempotent operations in an algebra, and constructing an algebra which will satisfy the Mal'cev conditions in question exactly when it has nontrivial idempotent operations. This section generalizes [4,Sec. 9].…”

“…Clearly then A possesses a local E-special cube term on S if and only if G ∈ Sg A k (F * ) (where F * is the set of columns of F ), and by [4] this can be determined in polynomial time. Since there are only polynomially many such sets S, we can determine in polynomial time whether or not A has all local E-special cube terms for sets of size k, and by the local-global property of E this is equivalent to A's possession of a E-special cube term.…”

Recent Results on the Algebraic Approach to the CSP

“…We note that without the assumption of idempotency, Freese and Valeriote have shown [25] that to determine if the variety generated by a finite algebra omits the unary type or both the unary and affine types are both EXPTIMEcomplete problems.…”