The complexity of constraint satisfaction problems for small relation algebras

The problems that deserve an attack demonstrate it by a counterattack. Abstract Andr eka and Maddux (1994, Notre Dame Journal of Formal Logic, 35(4)) classiied the small relation algebras | those with at most 8 elements, or in other terms, at most 3 atomic relations. They showed that there are eighteen isomorphism types of small relation algebras, all representable. For each simple, small relation algebra they computed the spectrum of the algebra, namely the set of cardinalities of square representations of that relation algebra. In this paper we analyze the computational complexity of the problem of deciding the satissability of a nite set of constraints built on any small relation algebra. We give a complete classiication of the complexities of the general constraint satisfaction problem for small relation algebras. For three of the small relation algebras the constraint satisfaction problem is NP-complete, for the other fteen small relation algebras the constraint satisfaction problem has cubic (or lower) complexity. We also classify the complexity of the constraint satisfaction problem over xed nite representations of any relation algebra. If the representation has size two or less then the complexity is cubic (or lower), but if the representation is square, nite and bigger than two then the complexity is NP-complete.