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Frege proof system and TNC°

Published online by Cambridge University Press:  12 March 2014

Gaisi Takeuti
Gaisi Takeuti*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801, USA, E-mail: [email protected]
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Extract

A Frege proof system F is any standard system of prepositional calculus, e.g., a Hilbert style system based on finitely many axiom schemes and inference rules. An Extended Frege system EF is obtained from F as follows. An EF-sequence is a sequence of formulas ψ1, …, ψκ such that eachψi is either an axiom of F, inferred from previous ψu and ψv (= ψu → ψi) by modus ponens or of the form q ↔ φ, where q is an atom occurring neither in φ nor in any of ψ1,…,ψi−1. Such q ↔ φ, is called an extension axiom and q a new extension atom. An EF-proof is any EF-sequence whose last formula does not contain any extension atom. In [12], S. A. Cook and R. Reckhow proved that the pigeonhole principle PHP has a simple polynomial size EF-proof and conjectured that PHP does not admit polynomial size F-proof. In [5], S. R. Buss refuted this conjecture by furnishing polynomial size F-proof for PHP. Since then the important separation problem of polynomial size F and polynomial size EF has not shown any progress.

In [11], S. A. Cook introduced the system PV, a free variable equational logic whose provable functional equalities are ‘polynomial time verifiable’ and showed that the metamathematics of F and EF can be developed in PV and the soundness of EF proved in PV. In [3], S. R. Buss introduced the first order system and showed that is essentially a conservative extension of PV. There he also introduced a second order system (BD).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 2 , June 1998 , pp. 709 - 738
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1] Barrington, D. A. Mix, Immerman, N., and Straubing, H., On uniform with NCl , Structure in complexity theory, third annual conference, IEEE Computer Society Press, 1988, to appear in Journal of Computer and System Sciences, pp. 47–59.10.1109/SCT.1988.5262Google Scholar
[2] Buss, S. R., Algorithms for Boolean formula evaluation and for tree-contraction, In proof theory, complexity and arithmetic (Clote, P. and Krajíček, J., editors), Oxford University Press, pp. 95–115.Google Scholar
[3] Buss, S. R., Bounded arithmetic, Bibliopolis, Napoli, 1986.Google Scholar
[4] Buss, S. R., The Boolean formula value problem is in ALOGTIME, Proceedings of the 19th annual ACM symposium on theory of computing, 1987, pp. 123–131.Google Scholar
[5] Buss, S. R., Polynomial size proofs of the propositional pigeonhole principle, this Journal, vol. 52 (1987), pp. 66–92.Google Scholar
[6] Buss, S. R., Propositional consistency proofs, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 3–29.Google Scholar
[7] Buss, S. R., Cook, S. A., Gupta, A., and Ramachandran, V., An optimal parallel algorithm for formula evaluation, SIAM Journal on Computing, vol. 21 (1992), pp. 755–780.10.1137/0221046Google Scholar
[8] Clote, P., Sequential, machine-independent characterizations of the parallel complexity classes ALOGTIME, ACk, NCk and NC, Feasible mathematics (Buss, S. R. and Scott, P., editors), Birkhäuser, 1990, pp. 49–70.Google Scholar
[9] Clote, P., ALOGTIME and a conjecture of S. A. Cook, Annals of Mathematics and Artificial Intelligence, vol. 6 (1992), pp. 57–106, extended abstract in proceedings of IEEE Logic in Computer Science, Philadelphia, 06 1990.Google Scholar
[10] Clote, P. and Takeuti, G., First order bounded arithmetic and small boolean circuit complexity class, Feasible mathematics II (Clote, P. and Remmel, J., editors), Birkhäuser, 1995, pp. 154–218.Google Scholar
[11] Cook, S. A., Feasibly constructive proofs and the propositional calculus, Proceedings of the 7th ACM symposium on the theory of computation, 1975, pp. 83–97.Google Scholar
[12] Cook, S. A. and Reckhow, R., The relative efficiency of propositional proof systems, this Journal, vol. 44 (1977), pp. 36–50.Google Scholar
[13] Dowd, M., Propositional representation of arithmetical proofs, Ph.D. dissertation , University of Toronto, 04 1979, Department of Computer Science Technical Report 132/79.Google Scholar
[14] Karchmer, M., Communication complexity: A new approach to circuit depth, MIT Press, 1989.Google Scholar
[15] Karchmer, M. and Wigderson, A., Monotone circuits for connectivity require sur-logarithmic depth, Proceedings of the 20th annual ACM symposium on theory of computation, ACM Press, 1988, pp. 539–550.Google Scholar
[16] Krajíček, J., Bounded arithmetic, propositional logic and complexity theory, Cambridge University Press, to appear.Google Scholar
[17] Krajíček, J., On Frege and extended Frege proof systems, Feasible mathematics II (Clote, P. and Remmel, J., editors), Birkhäser, 1995, pp. 284–319.10.1007/978-1-4612-2566-9_10Google Scholar
[18] Krajíček, J. and Publák, P., Propositional proof systems, the consistency of first order theories and the complexity of computations, this Journal, vol. 54 (1989), pp. 1063–1079.Google Scholar
[19] Raz, R. and Wigderson, A., Monotone circuits for matching require linear depth, Proceedings of the 22nd annual ACM symposium on theory of computing, 1990, pp. 287–292.Google Scholar
[20] Razborov, A. A., Lower bounds on the monotone complexity of some Boolean functions, Doklady AkademiiNauk SSSR, vol. 281 (1985), no. 4, pp. 798–801, English translation in Soviet Math Doklady, vol. 31 (1985), pp. 345–357.Google Scholar
[21] Reckhow, R. A., On the lengths of proofs in the propositional calculus, Ph.D. thesis , University of Toronto, 1976, Department of Computer Science, Technical Report 87.Google Scholar
[22] Takeuti, G., and (BD), Archive for Mathematical Logic, vol. 29 (1990), pp. 149–169.10.1007/BF01621092Google Scholar
[23] Takeuti, G., RSUV isomorphisms, Arithmetic, proof theory and computational complexity (Clote, P. and Krajíček, J., editors), Oxford University Press, 1993, pp. 364–386.Google Scholar
[24] Takeuti, G., RSUV isomorphisms for TAC i , TNC i and TLS, Archive for Mathematical Logic, vol. 33 (1995), pp. 427–453.Google Scholar