The Core of a Countably Categorical Structure

ded abstract of this article is published in the proceedings of STACS’05, LNCS 3404, Springer Verlag. A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for the classification for the computational complexity of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that the structure induced by e(S) is a core; moreover, the core is unique up to isomorphism. We prove that this result remains valid for ω-categorical structures, and prove that every ω-categorical structure has a core, which is unique up to isomorphism, and which is finite or ω-categorical. We also show that the core of an ω-categorical structure Γ is model complete, and therefore ∀∃-axiomatizable. If Γ contains all primitive positive definable relations, then the core of Γ admits quantifier elimination. We discuss consequences for constraint satisfaction with ω-categorical t...

• Type: Conference
• Year: 2005
• Where: STACS
• Authors: Manuel Bodirsky
• Keywords: Constraint Satisfaction, Constraint Satisfaction Problems, STACS 2005, Theoretical Computer Science, ω-categorical Structure