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.
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References
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)
Atserias, A., Dawar, A., Grohe, M.: Preservation under extensions on well-behaved finite structures. SIAM J. Comput. 38(4), 1364–1381 (2008)
Atserias, A., Dawar, A., Kolaitis, P.G.: On preservation under homomorphisms and unions of conjunctive queries. J. ACM 53(2), 208–237 (2006)
Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)
Chang, C.C., Keisler, H.J.: Model Theory, 3rd edn. Elsevier Science Publishers (1990)
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)
Gurevich, Y.: Toward logic tailored for computational complexity. In: Computation and Proof Theory, pp. 175–216. Springer (1984)
Libkin, L.: Elements of Finite Model Theory. Springer (2004)
Rosen, E.: Some aspects of model theory and finite structures. Bulletin of Symbolic Logic 8(3), 380–403 (2002)
Rossman, B.: Homomorphism preservation theorems. J. ACM 55(3), 15:1–15:53 (2008)
Rossman, B.: Personal Communication (2012)
Sankaran, A., Adsul, B., Madan, V., Kamath, P., Chakraborty, S.: Preservation under substructures modulo bounded cores. CoRR, abs/1205.1358 (2012)
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
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Sankaran, A., Adsul, B., Madan, V., Kamath, P., Chakraborty, S. (2012). Preservation under Substructures modulo Bounded Cores. In: Ong, L., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2012. Lecture Notes in Computer Science, vol 7456. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32621-9_22
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DOI: https://doi.org/10.1007/978-3-642-32621-9_22
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