Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T10:58:14.143Z Has data issue: false hasContentIssue false
  • English
  • Français

THE REDUCTS OF THE HOMOGENEOUS BINARY BRANCHING C-RELATION

Published online by Cambridge University Press:  01 December 2016

MANUEL BODIRSKY ,
PETER JONSSON  and
TRUNG VAN PHAM
MANUEL BODIRSKY
Affiliation:
INSTITUT FÜR ALGEBRA TU DRESDEN 01062 DRESDEN, GERMANYE-mail: manuel.bodirsky@tu-dresden.de
PETER JONSSON
Affiliation:
DEPARTMENT OF COMPUTER AND SYSTEM SCIENCE LINKÖPINGS UNIVERSITET SE-581 83 LINKÖPING, SWEDENE-mail: peter.jonsson@liu.se
TRUNG VAN PHAM
Affiliation:
INSTITUT FÜR COMPUTERSPRACHEN THEORY AND LOGIC GROUP TECHNISCHE UNIVERSITÄT WIEN FAVORITENSTRASSE 9/E1852 A-1040 WIEN, AUSTRIAE-mail: pvtrung@logic.at
Get access Rights & Permissions [Opens in a new window]

Abstract

Let ($\rm L$;C) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of ($\rm L$;C), i.e., the structures with domain $\rm L$ that are first-order definable in ($\rm L$;C). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of ($\rm L$;C). We also study the endomorphism monoids of such reducts and show that they fall into four categories.

Keywords

03C0703C9820B2708A3505C5503C05omega-categoricityfirst-order reductstree-like structuresC-relationhomogeneous structuresmodel-completenessendomorphism monoids
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 4 , December 2016 , pp. 1255 - 1297
Copyright
Copyright © The Association for Symbolic Logic 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adeleke, S. A. and Macpherson, D., Classification of infinite primitive Jordan permutation groups . Proceedings of the London Mathematical Society, vol. 72 (1996), no. 1, pp. 63–123.Google Scholar
Adeleke, S. A. and Neumann, P. M., Primitive permutation groups with primitive Jordan sets . Proceedings of the London Mathematical Society, vol. 53 (1994), no. 2, pp. 209–229.CrossRefGoogle Scholar
Adeleke, S. A. and Neumann, P. M., Relations Related to Betweenness: Their Structure and Automorphisms, Memoirs of the American Mathematical Society, vol. 623, American Mathematical Society, Providence, RI, 1998.Google Scholar
Aho, A., Sagiv, Y., Szymanski, T., and Ullman, J., Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions . SIAM Journal on Computing, vol. 10 (1981), no. 3, pp. 405–421.CrossRefGoogle Scholar
Bhattacharjee, M., Macpherson, D., Möller, R. G., and Neumann, P. M., Notes on Infinite Permutation Groups, Springer Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, 1998.CrossRefGoogle Scholar
Bodirsky, M., Complexity classification in infinite-domain constraint satisfaction. Empowerment of Memory to Direct Research, Diderot University, 2012, available at arXiv:1201.0856 (In French).Google Scholar
Bodirsky, M., Ramsey classes: Examples and constructions , Surveys in Combinatorics (Czumaj, A., Georgakopoulos, A., Král, D., Lozin, V., and Pikhurko, O., editors), London Mathematical Society Lecture Note Series, vol. 424, Cambridge University Press, Cambridge, 2015, pp. 1–48.Google Scholar
Bodirsky, M., Chen, H., and Pinsker, M., The reducts of equality up to primitive positive interdefinability, this Journal, vol. 75 (2010), no. 4, pp. 1249–1292.Google Scholar
Bodirsky, M., Hils, M., and Martin, B., On the scope of the universal-algebraic approach to constraint satisfaction , Proceedings of the Annual Symposium on Logic in Computer Science (LICS) (Jouannaud, J-P., editor), IEEE Computer Society, Los Alamitos, CA, 2010, pp. 90–99.Google Scholar
Bodirsky, M., Jonsson, P., and Pham, T. V., The complexity of phylogeny constraint satisfaction, Proceedings of the 33rd International Symposium on Theoretical Aspects of Computer Science (STACS-2016) , Orléans, France, Feb, 2016.Google Scholar
Bodirsky, M. and Kára, J., The complexity of equality constraint languages . Theory of Computing Systems, vol. 3 (2008), no. 2, pp. 136–158. A conference version appeared in the proceedings of Computer Science Russia (CSR’06).CrossRefGoogle Scholar
Bodirsky, M. and Kára, J., The complexity of temporal constraint satisfaction problems . Journal of the Association for Computing Machinery, vol. 57 (2009), no. 2, pp. 1–41. An extended abstract appeared in the Proceedings of the Symposium on Theory of Computing (STOC’08).Google Scholar
Bodirsky, M. and Mueller, J. K., Rooted phylogeny problems . Logical Methods in Computer Science, vol. 7 (2011), no. 4, pp. 165–173. An extended abstract appeared in the proceedings of ICDT’10.Google Scholar
Bodirsky, M. and Pinsker, M., Reducts of Ramsey structures , Model Theoretic Methods in Finite Combinatorics (Grohe, M., and Makowsky, J. A., editors), AMS Contemporary Mathematics, vol. 558, American Mathematical Society, Providence, RI, 2011, pp. 489–519.Google Scholar
Bodirsky, M. and Pinsker, M., Minimal functions on the random graph . Israel Journal of Mathematics, vol. 200 (2014), no. 1, pp. 251–296.Google Scholar
Bodirsky, M. and Pinsker, M., Schaefer’s theorem for graphs . Journal of the Association for Computing Machinery, vol. 62 (2015), no. 3, Article no. 19, pp. 52. A conference version appeared in the Proceedings of STOC 2011, pp. 655–664.Google Scholar
Bodirsky, M., Pinsker, M., and Pongrácz, A., The 42 reducts of the random ordered graph . Journal of the London Mathematical Society, vol. 111 (2015), no. 3, pp. 591–632. Preprint available at arXiv:1309.2165.CrossRefGoogle Scholar
Bodirsky, M., Pinsker, M., and Tsankov, T., Decidability of definability, this Journal, vol. 78 (2013), no. 4, pp. 1036–1054. A conference version appeared in the Proceedings of LICS 2011.Google Scholar
Bodirsky, M. and Wrona, M., Equivalence constraint satisfaction problems , Proceedings of Computer Science Logic (Cégielski, P. and Durand, A., editors), LIPICS, vol. 16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Wadern, 2012, pp. 122–136.Google Scholar
Bryant, D., Building trees, hunting for trees, and comparing trees , Ph.D. thesis, University of Canterbury, 1997.Google Scholar
Bryant, D. and Steel, M., Extension operations on sets of leaf-labelled trees . Advances in Applied Mathematics, vol. 16 (1995), pp. 425–453.Google Scholar
Cameron, P. J., Transitivity of permutation groups on unordered sets . Mathematische Zeitschrift, vol. 148 (1976), pp. 127–139.Google Scholar
Cameron, P. J., Some treelike objects . Quarterly Journal of Mathematics, Oxford Second Series, vol. 38 (1987), no. 150, pp. 155–183.Google Scholar
Cameron, P. J., Oligomorphic Permutation Groups, Cambridge University Press, Cambridge, 1990.Google Scholar
Cantor, G., Über unendliche, lineare Punktmannigfaltigkeiten. Mathematische Annalen, vol. 23 (1884), pp. 453–488.CrossRefGoogle Scholar
Deuber, W., A generalization of Ramsey’s theorem for regular trees . Journal of Combinatorial Theory, Series B, vol. 18 (1975), pp. 18–23.Google Scholar
Frasnay, C., Quelques problèmes conbinatoires concernant les ordres totaux et les relations monomorphes . Annales de l’Institut Fourier (Grenoble), vol. 15 (1965), pp. 415–524.Google Scholar
Haskell, D. and Macpherson, D., Cell decompositions of C-minimal structures . Annals of Pure and Applied Logic, vol. 66 (1994), pp. 113–162.Google Scholar
Henzinger, M., King, V., and Warnow, T., Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology . Proceedings of the 7th Symposium on Discrete Algorithms (SODA’96) (Tardos, É., Editor), Association for Computing Machinery (ACM), and Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996, pp. 333–340.Google Scholar
Herwig, B., Macpherson, H. D., Martin, G., Nurtazin, A., and Truss, J. K., On ℵ0 -categorical weakly o-minimal structures. Annals of Pure and Applied Logic, vol. 101 (2000), no. 1, pp. 65–93.CrossRefGoogle Scholar
Hodges, W., A Shorter Model Theory, Cambridge University Press, Cambridge, 1997.Google Scholar
Junker, M. and Ziegler, M., The 116 reducts of $\left( {Q, < ,a} \right)$ , this Journal, vol. 74 (2008), no. 3, pp. 861–884.Google Scholar
Kechris, A., Pestov, V., and Todorcevic, S., Fraissé limits, Ramsey theory, and topological dynamics of automorphism groups . Geometric and Functional Analysis, vol. 15 (2005), no. 1, pp. 106–189.CrossRefGoogle Scholar
Macpherson, D., A survey of Jordan groups , Automorphisms of First-order Structures (Kaye, R. and Macpherson, D., editors), Oxford University Press, New York, NY, 1994, pp. 73–110.Google Scholar
Macpherson, D., A survey of Homogeneous Structures . Discrete Mathematics, vol. 311 (2011), no. 15, pp. 1599–1634.Google Scholar
Macpherson, D. and Steinhorn, C., On variants of o-minimality . Annals of Pure and Applied Logic, vol. 79 (1996), no. 2, pp. 165–209.Google Scholar
Milliken, K. R., A Ramsey theorem for trees . Journal of Combinatorial Theory, Series A, vol. 26 (1979), no. 3, pp. 215–237.Google Scholar
Nešetřil, J., Ramsey classes and homogeneous structures . Combinatorics, Probability & Computing, vol. 14 (2005), no. 1–2, pp. 171–189.Google Scholar
Ng, M. P., Steel, M., and Wormald, N. C., The difficulty of constructing a leaf-labelled tree including or avoiding given subtrees . Discrete Applied Mathematics, vol. 98 (2000), pp. 227–235.Google Scholar
Pach, P. P., Pinsker, M., Pluhár, G., Pongrácz, A., and Szabó, C., Reducts of the random partial order . Advances in Mathematics, vol. 267 (2014), pp. 94–120.CrossRefGoogle Scholar
Ramsey, F. P., On a problem of formal logic . Proceedings of the London Mathematical Society (2), vol. 30 (1930), no. 1, pp. 264–286.Google Scholar
Steel, M., The complexity of reconstructing trees from qualitative charaters and subtrees . Journal of Classification, vol. 9 (1992), pp. 91–116.Google Scholar
Thomas, S., Reducts of the random graph, this Journal, vol. 56 (1991), no. 1, pp. 176–181.Google Scholar