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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

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