380 research outputs found
On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction
The universal-algebraic approach has proved a powerful tool in the study of
the complexity of CSPs. This approach has previously been applied to the study
of CSPs with finite or (infinite) omega-categorical templates, and relies on
two facts. The first is that in finite or omega-categorical structures A, a
relation is primitive positive definable if and only if it is preserved by the
polymorphisms of A. The second is that every finite or omega-categorical
structure is homomorphically equivalent to a core structure. In this paper, we
present generalizations of these facts to infinite structures that are not
necessarily omega-categorical. (This abstract has been severely curtailed by
the space constraints of arXiv -- please read the full abstract in the
article.) Finally, we present applications of our general results to the
description and analysis of the complexity of CSPs. In particular, we give
general hardness criteria based on the absence of polymorphisms that depend on
more than one argument, and we present a polymorphism-based description of
those CSPs that are first-order definable (and therefore can be solved in
polynomial time).Comment: Extended abstract appeared at 25th Symposium on Logic in Computer
Science (LICS 2010). This version will appear in the LMCS special issue
associated with LICS 201
External definability and groups in NIP theories
We prove that many properties and invariants of definable groups in NIP
theories, such as definable amenability, G/G^{00}, etc., are preserved when
passing to the theory of the Shelah expansion by externally definable sets,
M^{ext}, of a model M. In the light of these results we continue the study of
the "definable topological dynamics" of groups in NIP theories. In particular
we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in
some new cases, including definably amenable groups in o-minimal structures.Comment: 28 pages. Introduction was expanded and some minor mistakes were
corrected. Journal of the London Mathematical Society, accepte
Fields and Fusions: Hrushovski constructions and their definable groups
An overview is given of the various expansions of fields and fusions of
strongly minimal sets obtained by means of Hrushovski's amalgamation method, as
well as a characterization of the groups definable in these structures
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