Topological Birkhoff

One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtain a representation of B in terms of A, even if B is finite. We show that by considering the natural topology on the functions of A and B in addition to the equations that hold between them, one can do with finite powers even for many interesting infinite algebras A. More precisely, we prove that if A and B are at most countable algebras which are oligomorphic, then the mapping which sends each function from A to the corresponding function in B preserves equations and is continuous if and only if B is a homomorphic image of a subalgebra of a finite power of A. Our result has the following consequences in model theory and in theoretical computer science: two \omega-categorical structures are primitive positive bi-interpretable if and only if their topological polymorphism clones are isomorphic. In particular, the complexity of the constraint satisfaction problem of an \omega-categorical structure only depends on its topological polymorphism clone.Comment: 21 page

Datalog and Constraint Satisfaction with Infinite Templates

On finite structures, there is a well-known connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates. If the template Gamma is omega-categorical, we present various equivalent characterizations of those Gamma such that the constraint satisfaction problem (CSP) for Gamma can be solved by a Datalog program. We also show that CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical structures Gamma if the input is restricted to instances of bounded treewidth. Finally, we characterize those omega-categorical templates whose CSP has Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that appeared at STACS'06. In the third version in the arxiv we have revised the presentation again and added a section that relates our results to formalizations of CSPs using relation algebra

On tractability and congruence distributivity

Constraint languages that arise from finite algebras have recently been the object of study, especially in connection with the Dichotomy Conjecture of Feder and Vardi. An important class of algebras are those that generate congruence distributive varieties and included among this class are lattices, and more generally, those algebras that have near-unanimity term operations. An algebra will generate a congruence distributive variety if and only if it has a sequence of ternary term operations, called Jonsson terms, that satisfy certain equations. We prove that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded relational width

Weak Bases of Boolean Co-Clones

Universal algebra and clone theory have proven to be a useful tool in the study of constraint satisfaction problems since the complexity, up to logspace reductions, is determined by the set of polymorphisms of the constraint language. For classifications where primitive positive definitions are unsuitable, such as size-preserving reductions, weaker closure operations may be necessary. In this article we consider strong partial clones which can be seen as a more fine-grained framework than Post's lattice where each clone splits into an interval of strong partial clones. We investigate these intervals and give simple relational descriptions, weak bases, of the largest elements. The weak bases have a highly regular form and are in many cases easily relatable to the smallest members in the intervals, which suggests that the lattice of strong partial clones is considerably simpler than the full lattice of partial clones