A lower bound for the complexity of Craig's interpolants in sentential logic

  • Daniele Mundici
Article

Abstract

For any sentenceα (in sentential logic) letdα be the delay complexity of the boolean functionfα represented byα. We prove that for infinitely manyd (and starting with somed<620) there exist valid implicationsα→β withdα,dβd such that any Craig's interpolantx has its delay complexitydχ greater thand+(1/3)·log(d/2). This is the first (non-trivial) known lower bound on the complexity of Craig's interpolants in sentential logic, whose general study may well have an impact on the central problems of computation theory.

Keywords

Mathematical Logic General Study Central Problem Computation Theory Sentential Logic 

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References

  1. [Co]
    Cook, S.A., Rechkow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Logic44, 36–50 (1979).Google Scholar
  2. [Fr]
    Friedman, H.: The complexity of explicit definitions. Adv. Math.20, 18–29 (1976).CrossRefGoogle Scholar
  3. [Mu 1]
    Mundici, D.: Robinson's consistency theorem in soft model theory. Trans. Am. Math. Soc.263, 231–241 (1981).Google Scholar
  4. [Mu 2]
    Mundici, D.: Compactness = JEP in any logic. Fund. Math.116 (to appear) (1982).Google Scholar
  5. [Mu 3]
    Mundici, D.: Complexity of Craig's interpolation. Ann. Soc. Math. Pol., Series IV: Fundamenta Informaticae V.3/2, 261–278 (1982).Google Scholar
  6. [Mu 4]
    Mundici, D.: Natural limitations of algorithmic procedures in logic. Rendiconti of the Nat. Acad. of Lincei (Rome), Serie VIII, Vol. LXIX 3/4, 101–105 (1980).Google Scholar
  7. [Mu 5]
    Mundici, D.: Irreversibility, uncertainty, relativity, and computer limitations. Il Nuovo Cimento, Europhysics Journal61B, No. 2, 297–305 (1981).Google Scholar
  8. [Mu 6]
    Mundici, D.: Duality between logics and equivalence relations. Trans. Am. Math. Soc.270, 111–129 (1982).Google Scholar
  9. [Mu 7]
    Mundici, D.: Quantifiers: an overview. In: Barwise, J., Feferman, S. (eds.): Abstract model theory and strong logics. Omega series. Berlin, Heidelberg, New York: Springer (to appear) 1983.Google Scholar
  10. [Mu 8]
    Mundici, D.: Craig's interpolation theorem in computation theory. Rendiconti of the Nat. Acad. of Lincei (Rome), Serie VIII, Vol.LXX.1, 6–11 (1981).Google Scholar
  11. [Mu 9]
    Mundici, D.: NP and Craig's interpolation theorem. In: Proceedings of the ASL Logic Colloquium '82 in Florence. Amsterdam: North-Holland (to appear) 1983.Google Scholar
  12. [Mu 10]
    Mundici, D.: Compactness, interpolation, and Friedman's third problem. Ann. Math. Logic22, 197–211 (1982).CrossRefGoogle Scholar
  13. [Mu 11]
    Mundici, D.: Interpolation, compactness, and JEP in soft model theory. Arch. math. Logik22, 61–67 (1982).Google Scholar
  14. [Sa]
    Savage, J.E.: The complexity of computing. New York: Wiley 1976.Google Scholar
  15. [Sc]
    Schnorr, C.P.: The network complexity and the Turing machine complexity of finite functions. Acta Informatica7, 95–107 (1976).CrossRefGoogle Scholar
  16. [Sm]
    Smullyan, R.M.: First-order logic. Berlin, Heidelberg, New York: Springer 1971.Google Scholar

Copyright information

© Verlag W. Kohlhammer 1983

Authors and Affiliations

  • Daniele Mundici
    • 1
  1. 1.National Research CouncilDonnini (Florence)Italy

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