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International Workshop on Logic, Language, Information, and Computation

WoLLIC 2010: Logic, Language, Information and Computation pp 34–41Cite as

Application of Logic to Integer Sequences: A Survey

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6188))

Abstract

Chomsky and Schützenberger showed in 1963 that the sequence d L (n), which counts the number of words of a given length n in a regular language L, satisfies a linear recurrence relation with constant coefficients for n, or equivalently, the generating function \(g_L(x)=\sum_{n} d_L(n) x^n\) is a rational function. In this talk we survey results concerning sequences a(n) of natural numbers which

  • satisfy linear recurrence relations over ℤ or ℤ m , and

  • have a combinatorial or logical interpretation.

We present the pioneering, but little known, work by C. Blatter and E. Specker from 1981, and its further developments, including results by I. Gessel (1984), E. Fischer (2003), and recent results by T. Kotek and the author.

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References

  1. Altschul, S.F., Madden, T.L., Schaffer, A.A., Zhang, J., Zhang, Z., Miller, W., Lipman, D.J.: Gapped blast and psi-blast: a new generation of protein database search programs. Nucleic Acids Res. 25, 3389–3402 (1997)

    Article  Google Scholar 

  2. Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Counting graph homomorphisms. In: Klazar, M., Kratochvil, J., Loebl, M., Matousek, J., Thomas, R., Valtr, P. (eds.) Topics in Discret mathematics, pp. 315–371. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  3. Birkhoff, G.D.: General theory of irregular difference equations. Acta Mathematica 54, 205–246 (1930)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bergeron, F., Labelle, G., Leroux, P.: Combinatorial Species and Tree-like Structures. Encyclopedia of Mathematics and its Applications, vol. 67. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

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

    Google Scholar 

  6. Blatter, C., Specker, E.: Modular periodicity of combinatorial sequences. Abstracts of the AMS 4, 313 (1983)

    Google Scholar 

  7. Birkhoff, G.D., Trjitzinsky, W.J.: Analytic theory of singular difference equations. Acta Mathematica 60, 1–89 (1933)

    Article  MathSciNet  Google Scholar 

  8. Chomsky, N., Schützenberger, M.P.: The algebraic theory of context free languages. In: Brafford, P., Hirschberg, D. (eds.) Computer Programming and Formal Systems, pp. 118–161. North Holland, Amsterdam (1963)

    Chapter  Google Scholar 

  9. Ebbinghaus, H.D., Flum, J.: Finite Model Theory. In: Perspectives in Mathematical Logic, Springer, Heidelberg (1995)

    Google Scholar 

  10. Everest, G., van Porten, A., Shparlinski, I., Ward, T.: Recurrence Sequences. Mathematical Surveys and Monographs, vol. 104. American Mathematical Society, Providence (2003)

    Google Scholar 

  11. Fischer, E.: The Specker-Blatter theorem does not hold for quaternary relations. Journal of Combinatorial Theory, Series A 103, 121–136 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Flajolet, P.: On congruences and continued fractions for some classical combinatorial quantities. Discrete Mathematics 41, 145–153 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fischer, E., Makowsky, J.A.: The Specker-Blatter theorem revisited. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 90–101. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  14. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  15. Gerhold, S.: On some non-holonomic sequences. Electronic Journal of Combinatorics 11, 1–7 (2004)

    Google Scholar 

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

    Google Scholar 

  17. Goulden, I.P., Jackson, D.M.: Combinatorial Enumeration. Interscience Series in Discrete Mathematics. Wiley, Chichester (1983)

    MATH  Google Scholar 

  18. Gessel, I.M., Ree, S.: Lattice paths and Faber polynomials. In: Balakrishnan, N. (ed.) Advances in combinatorial methods and applications to probability and statistics, pp. 3–14. Birkhäuser, Basel (1996)

    Google Scholar 

  19. Hemaspaandra, V.: The satanic notations: Counting classes beyond \(\sharp\mathbf{P}\) and other definitional adventures. SIGACTN: SIGACT News (ACM Special Interest Group on Automata and Computability Theory) 26 (1995)

    Google Scholar 

  20. Kotek, T., Makowsky, J.A.: Definability of combinatorial functions and their linear recurrence relations. Electronically available at arXiv:0907.5420 (2009)

    Google Scholar 

  21. Kotek, T., Makowsky, J.A.: Application of logic to generating functions: Holonomic sequences. Manuscript (2010)

    Google Scholar 

  22. Kotek, T., Makowsky, J.A.: A representation theorem for holonomic sequences. Manuscript (2010)

    Google Scholar 

  23. Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  24. Noonan, J., Zeilberger, D.: The Goulden-Jackson cluster method: Extensions, applications and implementations. J. Differ. Equations Appl. 5(4-5), 355–377 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Petkovsek, M., Wilf, H., Zeilberger, D.: A=B. AK Peters, Wellesley (1996)

    Google Scholar 

  26. 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, Rockville (1988); Reprinted in: Ernst Specker, Selecta, Birkhäuser 1990, pp. 324–350

    Google Scholar 

  27. Specker, E.: Modular counting and substitution of structures. Combinatorics, Probability and Computing 14, 203–210 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Stormo, G.D., Schneider, T.D., Gold, L., Ehrenfeucht, A.: Use of the ’perceptron’ algorithm to distinguish translational initiation sites in e. coli. Nucleic Acid Research 10, 2997–3012 (1982)

    Article  Google Scholar 

  29. Stanley, R.P.: Differentiably finite power series. European Journal of Combinatorics 1, 175–188 (1980)

    MATH  MathSciNet  Google Scholar 

  30. Zeilberger, D.: A holonomic systems approach to special functions identities. J. of Computational and Applied Mathematics 32, 321–368 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

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

    Johann A. Makowsky

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  1. Johann A. Makowsky
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Editor information

Editors and Affiliations

  1. Computer Laboratory, University of Cambridge, J.J. Thomson Avenue, CB3 0FD, Cambridge, UK

    Anuj Dawar

  2. Centro de Informática, Universidade Federal de Pernambuco, Avenida Prof Luis Freire s/n, Cidade Universitária, 50740-540, Recife, PE, Brazil

    Ruy de Queiroz

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Makowsky, J.A. (2010). Application of Logic to Integer Sequences: A Survey. In: Dawar, A., de Queiroz, R. (eds) Logic, Language, Information and Computation. WoLLIC 2010. Lecture Notes in Computer Science(), vol 6188. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13824-9_3

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  • DOI: https://doi.org/10.1007/978-3-642-13824-9_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13823-2

  • Online ISBN: 978-3-642-13824-9

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