Tractability and Learnability Arising from Algebras with Few Subpowers

Abstract

A constraint language $\Gamma$ on a finite set A has been called polynomially expressive if the number of n-ary relations expressible by $\exists\wedge$-atomic formulas over $\Gamma$ is bounded by $\exp(O(n^k))$ for some constant k. It has recently been discovered that this property is characterized by the existence of a $(k+1)$-ary polymorphism satisfying certain identities; such polymorphisms are called k-edge operations and include Mal'cev and near-unanimity operations as special cases. We prove that if $\Gamma$ is any constraint language which, for some $k>1$, has a k-edge operation as a polymorphism, then the constraint satisfaction problem for $\langle\Gamma\rangle$ (the closure of $\Gamma$ under $\exists\wedge$-atomic expressibility) is globally tractable. We also show that the set of relations definable over $\Gamma$ using quantified generalized formulas is polynomially exactly learnable using improper equivalence queries.