Abstract
For any sentenceα (in sentential logic) letd α be the delay complexity of the boolean functionf α represented byα. We prove that for infinitely manyd (and starting with somed<620) there exist valid implicationsα→β withd α,d β≦d such that any Craig's interpolantx has its delay complexityd χ 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.
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References
Cook, S.A., Rechkow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Logic44, 36–50 (1979).
Friedman, H.: The complexity of explicit definitions. Adv. Math.20, 18–29 (1976).
Mundici, D.: Robinson's consistency theorem in soft model theory. Trans. Am. Math. Soc.263, 231–241 (1981).
Mundici, D.: Compactness = JEP in any logic. Fund. Math.116 (to appear) (1982).
Mundici, D.: Complexity of Craig's interpolation. Ann. Soc. Math. Pol., Series IV: Fundamenta Informaticae V.3/2, 261–278 (1982).
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).
Mundici, D.: Irreversibility, uncertainty, relativity, and computer limitations. Il Nuovo Cimento, Europhysics Journal61B, No. 2, 297–305 (1981).
Mundici, D.: Duality between logics and equivalence relations. Trans. Am. Math. Soc.270, 111–129 (1982).
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.
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).
Mundici, D.: NP and Craig's interpolation theorem. In: Proceedings of the ASL Logic Colloquium '82 in Florence. Amsterdam: North-Holland (to appear) 1983.
Mundici, D.: Compactness, interpolation, and Friedman's third problem. Ann. Math. Logic22, 197–211 (1982).
Mundici, D.: Interpolation, compactness, and JEP in soft model theory. Arch. math. Logik22, 61–67 (1982).
Savage, J.E.: The complexity of computing. New York: Wiley 1976.
Schnorr, C.P.: The network complexity and the Turing machine complexity of finite functions. Acta Informatica7, 95–107 (1976).
Smullyan, R.M.: First-order logic. Berlin, Heidelberg, New York: Springer 1971.
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Mundici, D. A lower bound for the complexity of Craig's interpolants in sentential logic. Arch math Logik 23, 27–36 (1983). https://doi.org/10.1007/BF02023010
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DOI: https://doi.org/10.1007/BF02023010