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Lower bounds for modal logics

Published online by Cambridge University Press:  12 March 2014

Pavel Hrubeš
Pavel Hrubeš*
Affiliation:
Mathematical Institute, Zitna 25, 11000 Praha 1, Czech Republic, E-mail: pahrubes@centrum.cz
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Abstract

We give an exponential lower bound on number of proof-lines in the proof system K of modal logic, i.e., we give an example of K-tautologies ψ1, ψ2, … s.t. every K-proof of ψi must have a number of proof-lines exponential in terms of the size of ψi. The result extends, for the same sequence of K-tautologies, to the systems K4, Gödel–Löb's logic, S andS4. We also determine some speed-up relations between different systems of modal logic on formulas of modal-depth one.

Keywords

Proof complexitymodal logiclower boundmonotone interpolation
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 941 - 958
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Alon, N. and Boppana, R., The monotone circuit complexity of Boolean functions, Combinatorial, vol. 7 (1987), no. 1, pp. 1–22.CrossRefGoogle Scholar
[2]Buss, S. R., Polynomial size proofs of the propositional pigeonhole principle, this Journal, vol. 52 (1987), pp. 916–927.Google Scholar
[3]Karchmer, M., On proving lower bounds for circuit size, Proceedings of Structure in Complexity, 8th Annual Complexity Conference, IEEE Computer Science Pres, 1993, pp. 112–119.Google Scholar
[4]Krajíček, J., Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, this Journal, vol. 62 (1997), no. 2, pp. 457–486.Google Scholar
[5]Razborov, A. A., Lower bounds on the monotone complexity of some Boolean functions, Soviet Mathematics Doklady, vol. 31 (1985), pp. 354–357.Google Scholar