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Logics of Finite Hankel Rank

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Fields of Logic and Computation II

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9300))

Abstract

We discuss the Feferman-Vaught Theorem in the setting of abstract model theory for finite structures. We look at sum-like and product-like binary operations on finite structures and their Hankel matrices. We show the connection between Hankel matrices and the Feferman-Vaught Theorem. The largest logic known to satisfy a Feferman-Vaught Theorem for product-like operations is \(\mathrm {CFOL}\), first order logic with modular counting quantifiers. For sum-like operations it is \(\mathrm {CMSOL}\), the corresponding monadic second order logic. We discuss whether there are maximal logics satisfying Feferman-Vaught Theorems for finite structures.

Partially supported by a grant of the graduate school of the Technion, the National Research Network RiSE (S114), and the LogiCS doctoral program (W1255) funded by the Austrian Science Fund (FWF).

Partially supported by a grant of Technion Research Authority.

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Notes

  1. 1.

    They are nevertheless called sum-like in [40].

  2. 2.

    K. Compton and I. Gessel, [8, 19], already considered \(\tau \)-properties of finite \(\sqcup \)-index for the disjoint union of \(\tau \)-structures. In [17] this is called Gessel index. C. Blatter and E. Specker, in [3, 46], consider a substitution operation on pointed \(\tau \)-structures, \(Subst({\mathfrak A}, a, {\mathfrak B})\), where the structure \({\mathfrak B}\) is inserted into \({\mathfrak A}\) at a point a. \(Subst({\mathfrak A}, a, {\mathfrak B})\) is sum-like, and the Subst-index is called in [17] Specker index.

  3. 3.

    A similar construction was first suggested by E. Specker in conversations with the second author in 2000, cf. [40, Section 7].

  4. 4.

    This observation was suggested by T. Kotek in conversations with the second author in summer 2014.

References

  1. Barwise, J., Feferman, S.: Model-Theoretic Logics. Perspectives in Mathematical Logic. Springer, Heidelberg (1985)

    MATH  Google Scholar 

  2. Barwise, K.J.: Axioms for abstract model theory. Ann. Math. Logic 7(2), 221–265 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blatter, C., Specker, E.: Le nombre de structures finies d’une th’eorie à charactère fin, pp. 41–44. Sciences Mathématiques, Fonds Nationale de la recherche Scientifique, Bruxelles (1981)

    Google Scholar 

  4. Carlyle, J.W., Paz, A.: Realizations by stochastic finite automata. J. Comput. Syst. Sci. 5, 26–40 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chandra, A., Lewis, H., Makowsky, J.A.: Embedded implicational dependencies and their implication problem. In: (1981) ACM Symposium on the Theory of Computing, pp. 342–354. ACM (1981)

    Google Scholar 

  6. Chandra, A.K., Lewis, H.R., Makowsky, J.A.: Embedded implicational dependencies and their inference problem. In: XP1 Workshop on Database Theory (1980)

    Google Scholar 

  7. Chang, C.C., Keisler, H.J.: Model Theory. Studies in Logic, vol. 73, 3rd edn. Elsevier, Haarlem, North-Holland (1990)

    MATH  Google Scholar 

  8. Compton, K.J.: Some useful preservation theorems. J. Symb. Log. 48(2), 427–440 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Courcelle, B.: The monadic second-order logic of graphs I: recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Durand, A., Jones, N.D., Makowsky, J.A., More, M.: Fifty years of the spectrum problem: survey and new results. Bull. Symb. Logic 18(04), 505–553 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ebbinghaus, H.-D., Flum, J., Thomas, W.: Mathematical Logic. Undergraduate Texts in Mathematics, 2nd edn. Springer, Heidelberg (1994)

    Book  MATH  Google Scholar 

  12. Feferman, S.: Persistent and invariant formulas for outer extensions. Compos. Math. 20, 29–52 (1968)

    MathSciNet  MATH  Google Scholar 

  13. Feferman, S.: Two notes on abstract model theory, I: properties invariant on the range of definable relations between strcutures. Fundam. Math. 82, 153–165 (1974)

    MATH  Google Scholar 

  14. Feferman, S., Kreisel, G.: Persistent and invariant formulas relative to theories of higher order. Bull. Am. Math. Soc. 72, 480–485 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feferman, S., Vaught, R.: The first order properties of algebraic systems. Fundam. Math. 47, 57–103 (1959)

    MathSciNet  MATH  Google Scholar 

  16. Feferman, S., Vaught, R.: The first order properties of products of algebraic systems. Fundam. Math. 47, 57–103 (1959)

    MathSciNet  MATH  Google Scholar 

  17. Fischer, E., Kotek, T., Makowsky, J.A.: Application of logic to combinatorial sequences and their recurrence relations. In: Grohe, M., Makowsky, J.A. (eds.) Model Theoretic Methods in Finite Combinatorics, Contemporary Mathematics, vol. 558, pp. 1–42. American Mathematical Society (2011)

    Google Scholar 

  18. Freedman, M., Lovász, L., Schrijver, A.: Reflection positivity, rank connectivity, and homomorphisms of graphs. J. AMS 20, 37–51 (2007)

    MATH  Google Scholar 

  19. Gessel, I.: Combinatorial proofs of congruences. In: Jackson, D.M., Vanstone, S.A. (eds.) Enumeration and Design, pp. 157–197. Academic Press (1984)

    Google Scholar 

  20. Godlin, B., Kotek, T., Makowsky, J.A.: Evaluations of graph polynomials. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 183–194. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  21. Grädel, E.: The expressive power of second order horn logic. In: Jantzen, M., Choffrut, C. (eds.) STACS 1991. LNCS, vol. 480, pp. 466–477. Springer, Heidelberg (1991)

    Google Scholar 

  22. Gurevich, Y.: Modest theory of short chains I. J. Symb. Logic 44, 481–490 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gurevich, Y.: Logic and the challenge of computer science. In: Börger, E. (ed) Trends in Theoretical Computer Science, Principles of Computer Science Series, Chap. 1. Computer Science Press (1988)

    Google Scholar 

  24. Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16(4), 760–778 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kotek, T., Makowsky, J.A.: Connection matrices and the definability of graph parameters. Log. Methods Comput. Sci. 10(4) (2014)

    Google Scholar 

  26. Kreisel, G.: Choice of infinitary languages by means of definability criteria; generalized recursion theory. In: Barwise, J. (ed.) The Syntax and Semantics of Infinitary Languages. LNM, vol. 72, pp. 139–151. Springer, Heidelberg (1968)

    Chapter  Google Scholar 

  27. Labai, N., Makowsky, J.A.: The Feferman-Vaught theorem and finite Hankel rank. In preparation (2015)

    Google Scholar 

  28. Labai, N., Makowsky, J.: Tropical graph parameters. In: DMTCS Proceedings of 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC), vol. 1, pp. 357–368 (2014)

    Google Scholar 

  29. Läuchli, H.: A decision procedure for the weak second order theory of linear order. In: Logic Colloquium 1966, pp. 189–197, North Holland (1968)

    Google Scholar 

  30. Lindström, P.: First order predicate logic with generalized quantifiers. Theoria 32, 186–195 (1966)

    MathSciNet  MATH  Google Scholar 

  31. Lindström, P.: On extensions of elementary logic. Theoria 35, 1–11 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lovász, L.: Connection matrics. In: Grimmet, G., McDiarmid, C. (eds.) Combinatorics, Complexity and Chance, A Tribute to Dominic Welsh, pp. 179–190. Oxford University Press (2007)

    Google Scholar 

  33. Lovász, L.: Large Networks and Graph Limits, vol. 60. Colloquium Publications, AMS, Providence (2012)

    MATH  Google Scholar 

  34. Mahr, B., Makowsky, J.A.: Characterizing specification languages which admit initial semantics. Theor. Comput. Sci. 31, 49–60 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  35. Makowsky, J.A.: \(\Delta \)-logics and generalized quantifiers. Ph.D. thesis, Department of Mathematics, ETH-Zurich, Switzerland (1974)

    Google Scholar 

  36. Makowsky, J.A.: Why horn formulas matter for computer science: initial structures and generic examples. J. Comput. Syst. Sci. 34(2/3), 266–292 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  37. Makowsky, J.A.: Model theoretic issues in theoretical computer science, part I: relational databases and abstract data types. In: Lolli, G. et al. (eds.) Logic Colloquium 1982, Studies in Logic, pp. 303–343, North Holland (1984)

    Google Scholar 

  38. Makowsky, J.A.: Compactness, embeddings and definability. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics, Perspectives in Mathematical Logic, Chap. 18. Springer, Heidelberg (1985)

    Google Scholar 

  39. Makowsky, J.A.: Invariant definability. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) KGC 1997. LNCS, vol. 1289, pp. 186–202. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  40. Makowsky, J.A.: Algorithmic uses of the Feferman-Vaught theorem. Ann. Pure Appl. Logic 126(1–3), 159–213 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Makowsky, J.A., Pnueli, Y.: Logics capturing oracle complexity classes uniformly. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  42. Makowsky, J.A., Pnueli, Y.B.: Oracles and quantifiers. In: Meinke, K., Börger, E., Gurevich, Y. (eds.) CSL 1993. LNCS, vol. 832. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  43. Makowsky, J.A., Vardi, M.: On the expressive power of data dependencies. Acta Inform. 23(3), 231–244 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ravve, E.: Model checking for various notions of products. Master’s thesis, Thesis, Department of Computer Science, Technion-Israel Institute of Technology (1995)

    Google Scholar 

  45. Shelah, S.: The monadic theory of order. Ann. Math. 102, 379–419 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  46. Specker, E.: Application of logic and combinatorics to enumeration problems. In: Börger, E. (ed.) Trends in Theoretical Computer Science, pp. 141–169. Computer Science Press (1988). (Reprinted In: Specker, E., Selecta, B., pp. 324–350 (1990))

    Google Scholar 

  47. Vardi, M.: The complexity of relational query languages. In: STOC 1982, pp. 137–146. ACM (1982)

    Google Scholar 

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Acknowledgments

We would like to thank T. Kotek for letting us use his example, and for valuable discussions.

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Authors and Affiliations

  1. Department of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel

    Nadia Labai & Johann A. Makowsky

  2. Department of Informatics, Vienna University of Technology, Vienna, Austria

    Nadia Labai

Authors
  1. Nadia Labai
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  2. Johann A. Makowsky
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Correspondence to Nadia Labai .

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Editors and Affiliations

  1. Steklov Mathematical Institute, Moscow, Russia

    Lev D. Beklemishev

  2. University of Michigan, Ann Arbor, Michigan, USA

    Andreas Blass

  3. Tel Aviv University, Tel Aviv, Israel

    Nachum Dershowitz

  4. Universität des Saarlandes, Saarbrücken, Germany

    Bernd Finkbeiner

  5. Microsoft Research, REDMOND, Washington, USA

    Wolfram Schulte

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Labai, N., Makowsky, J.A. (2015). Logics of Finite Hankel Rank. In: Beklemishev, L., Blass, A., Dershowitz, N., Finkbeiner, B., Schulte, W. (eds) Fields of Logic and Computation II. Lecture Notes in Computer Science(), vol 9300. Springer, Cham. https://doi.org/10.1007/978-3-319-23534-9_14

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  • DOI: https://doi.org/10.1007/978-3-319-23534-9_14

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