Archive for Mathematical Logic

, Volume 54, Issue 7–8, pp 741–767 | Cite as

On the share of closed IL formulas which are also in GL



Normal forms for wide classes of closed IL formulas were given in Čačić and Vuković (Math Commun 17:195–204, 2012). Here we quantify asymptotically, in exact numbers, how wide those classes are. As a consequence, we show that the “majority” of closed IL formulas have GL-equivalents, and by that, they have the same normal forms as GL formulas. Our approach is entirely syntactical, except for applying the results of Čačić and Vuković. As a byproduct we devise a convenient way of computing asymptotic behaviors of somewhat general classes of formulas given by their grammar rules. Its applications do not require any knowledge of the recurrence relations, generating functions, or the asymptotic enumeration methods, as all these are incorporated into two fundamental parameters.


Interpretability logic Normal forms Asymptotic enumeration 

Mathematics Subject Classification

03F05 05A16 


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  1. 1.
    Artemov S.N., Silver B.: Arithmetically complete modal theories. Six papers in logic (AMS translations). Am. Math. Soc. 2, 39–54 (1987)Google Scholar
  2. 2.
    Bender E.A., Williamson S.G.: Foundations of Combinatorics with Applications. Dover, New York (2006)Google Scholar
  3. 3.
    Bou F., Joosten J.J.: The closed fragment of IL is PSPACE hard. Electron. Notes in Theor. Comput. Sci. 278, 47–54 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Čačić, V., Vuković, M.: A note on normal forms for the closed fragment of system IL. Math. Commun. 17, 195–204 (2012), also available at
  5. 5.
    Flajolet P., Odlyzko A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240 (1990)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Flajolet P., Sedgewick R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Goris E., Joosten J.J.: A new principle in the interpretability logic of all reasonable arithmetical theories. Log. J. IGPL 19, 1–17 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hájek P., Švejdar V.: A note on the normal form of closed formulas of interpretability logic. Stud. Log. 50, 25–28 (1991)CrossRefMATHGoogle Scholar
  9. 9.
    The on-line encyclopedia of integer sequences \({^{\circledR}}\) (OEIS \({^{\circledR}}\)).
  10. 10.
    Rudin W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)MATHGoogle Scholar
  11. 11.
    Visser, A.: An overview of interpretability logic. Logic Group Preprint Series, 174, (1997)Google Scholar
  12. 12.
    Wilf H.S.: Generatingfunctionology, 3rd edn. A K Peters/CRC Press, Massachusetts/Boca Raton (2005)CrossRefGoogle Scholar
  13. 13.
    Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia

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