Preservation under Substructures modulo Bounded Cores

  • Abhisekh Sankaran
  • Bharat Adsul
  • Vivek Madan
  • Pritish Kamath
  • Supratik Chakraborty
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7456)

Abstract

We investigate a model-theoretic property that generalizes the classical notion of preservation under substructures. We call this property preservation under substructures modulo bounded cores, and present a syntactic characterization via \(\Sigma_2^0\) sentences for properties of arbitrary structures definable by FO sentences. Towards a sharper characterization, we conjecture that the count of existential quantifiers in the \(\Sigma_2^0\) sentence equals the size of the smallest bounded core. We show that this conjecture holds for special fragments of FO and also over special classes of structures. We present a (not FO-definable) class of finite structures for which the conjecture fails, but for which the classical Łoś-Tarski preservation theorem holds. As a fallout of our studies, we obtain combinatorial proofs of the Łoś-Tarski theorem for some of the aforementioned cases.

Keywords

Model theory First Order logic preservation theorem 

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References

  1. 1.
    Alechina, N., Gurevich, Y.: Syntax vs. Semantics on Finite Structures. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 14–33. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  2. 2.
    Atserias, A., Dawar, A., Grohe, M.: Preservation under extensions on well-behaved finite structures. SIAM J. Comput. 38(4), 1364–1381 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Atserias, A., Dawar, A., Kolaitis, P.G.: On preservation under homomorphisms and unions of conjunctive queries. J. ACM 53(2), 208–237 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chang, C.C., Keisler, H.J.: Model Theory, 3rd edn. Elsevier Science Publishers (1990)Google Scholar
  6. 6.
    Dawar, A., Grohe, M., Kreutzer, S., Schweikardt, N.: Model Theory Makes Formulas Large. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 913–924. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Gurevich, Y.: Toward logic tailored for computational complexity. In: Computation and Proof Theory, pp. 175–216. Springer (1984)Google Scholar
  8. 8.
    Libkin, L.: Elements of Finite Model Theory. Springer (2004)Google Scholar
  9. 9.
    Rosen, E.: Some aspects of model theory and finite structures. Bulletin of Symbolic Logic 8(3), 380–403 (2002)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Rossman, B.: Homomorphism preservation theorems. J. ACM 55(3), 15:1–15:53 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rossman, B.: Personal Communication (2012)Google Scholar
  12. 12.
    Sankaran, A., Adsul, B., Madan, V., Kamath, P., Chakraborty, S.: Preservation under substructures modulo bounded cores. CoRR, abs/1205.1358 (2012)Google Scholar
  13. 13.
    Sankaran, A., Limaye, N., Sundararaman, A., Chakraborty, S.: Using preservation theorems for inexpressibility results in first order logic. Technical report (2012), http://www.cfdvs.iitb.ac.in/reports/index.php

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Abhisekh Sankaran
    • 1
  • Bharat Adsul
    • 1
  • Vivek Madan
    • 1
  • Pritish Kamath
    • 1
  • Supratik Chakraborty
    • 1
  1. 1.Indian Institute of Technology BombayIndia

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