Abstract
A relational structure is a core, if all endomorphisms are embeddings. This notion is important for the classification of 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 again ω-categorical. We thus reduced the classification of the complexity of constraint satisfaction problems with ω-categorical templates to the classifiaction of constraint satisfaction problems where the templates are ω-categorical cores. If Γ contains all primitive positive definable relations, then the core of Γ admits quantifier elimination. We discuss further consequences for constraint satisfaction with ω-categorical templates.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Achlioptas, D.: The complexity of G-free colourability. Discrete Mathematics 165, 21–30 (1997)
Adeleke, S., Neumann, P.M.: Structure of partially ordered sets with transitive automorphism groups. AMS Memoir 57(334) (1985)
Aho, A., Sagiv, Y., Szymanski, T., Ullman, J.: Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM Journal on Computing 10(3), 405–421 (1981)
Allen, J.F.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(11), 832–843 (1983)
Bauslaugh, B.: Core-like properties of infinite graphs and structures. Disc. Math. 138(1), 101–111 (1995)
Bauslaugh, B.: Cores and compactness of infinite directed graphs. Journal of Combinatorial Theory, Series B 68(2), 255–276 (1996)
Bodirsky, M.: Constraint satisfaction with infinite domains. Dissertation an der Humboldt-Universität zu Berlin (2004)
Bodirsky, M., Kutz, M.: Pure dominance constraints. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 287–298. Springer, Heidelberg (2002)
Bodirsky, M., Nešetřil, J.: Constraint satisfaction with countable homogeneous templates. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 44–57. Springer, Heidelberg (2003)
Bulatov, A.: Tractable conservative constraint satisfaction problems. In: Proceedings of LICS 2003, pp. 321–330 (2003)
Bulatov, A., Krokhin, A., Jeavons, P.G.: Classifying the complexity of constraints using finite algebras (2003) (submitted)
Cameron, P.J.: Oligomorphic Permutation Groups. Cambridge Univ. Press, Cambridge (1990)
Cameron, P.J.: The random graph. In: Graham, R.L., Nešetřil, J. (eds.) The Mathematics of Paul Erdõs (1996)
Chang, Keisler: Model theory. Princeton University Press, Princeton (1977)
Cherlin, G.: The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments. AMS Memoir 131(621) (January 1998)
Cornell, T.: On determining the consistency of partial descriptions of trees. In: Proceedings of the ACL, pp. 163–170 (1994)
Droste, M.: Structure of partially ordered sets with transitive automorphism groups. AMS Memoir 57(334) (1985)
Feder, T., Vardi, M.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM Journal on Computing 28, 57–104 (1999)
Garey, M., Johnson, D.: A guide to NP-completeness. CSLI Press (1978)
Gusfield, D.: Algorithms on strings, trees, and sequences. Computer Science and Computational Biology. Cambridge University Press, New York (1997)
Hell, P., Nesetril, J.: The core of a graph. Discrete Math. 109, 117–126 (1992)
Hell, P., Nešetřil, J.: On the complexity of H-coloring. Journal of Combinatorial Theory, Series B 48, 92–110 (1990)
Hodges, W.: A shorter model theory. Cambridge University Press, Cambridge (1997)
Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. JACM 44(4), 527–548 (1997)
Jeavons, P., Jonsson, P., Krokhin, A.A.: Reasoning about temporal relations: The tractable subalgebras of Allen’s interval algebra. JACM 50(5), 591–640 (2003)
Lachlan, A.H.: Stable finitely homogeneous structures: A survey. In: Algebraic Model Theory. NATO ASI Series, vol. 496, pp. 145–159 (1996)
Nebel, B., Bürckert, H.-J.: Reasoning about temporal relations: A maximal tractable subclass of Allen’s interval algebra. JACM 42(1), 43–66 (1995)
Schaeffer, T.J.: The complexity of satisfiability problems. In: Proceedings of STOC 1978, pp. 216–226 (1978)
Steel, M.: The complexity of reconstructing trees from qualitative charaters and subtrees. Journal of Classification 9, 91–116 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bodirsky, M. (2005). The Core of a Countably Categorical Structure. In: Diekert, V., Durand, B. (eds) STACS 2005. STACS 2005. Lecture Notes in Computer Science, vol 3404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31856-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-31856-9_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24998-6
Online ISBN: 978-3-540-31856-9
eBook Packages: Computer ScienceComputer Science (R0)