Frege proof system and TNC°

Abstract

A Frege proof system F is any standard system of propositional 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 y',, . . ., k such that each Vti is either an axiom of F, inferred from previous yV, and Vrv (= -+ Vi) by modus ponens or of the form q <p, where q is an atom occurring neither in p nor in any of VI' . ..., Vi 1 Such q <p , 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 [1 1], 5S. 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 P V and the soundness of EF proved in P V. In [3], 5S. R. Buss introduced the first order system S and showed that S' is essentially a conservative extension of P V. There he also introduced a second order system VI' (BD). In [23] we proved that S and VI' (BD) are isomorphic under the so-called RSUV Isomorphism. In [17], J. Krajicek proved that a proof in V,' (BD) more precisely lb -part of V11(BD) is simulated in polynomial size EF-proof and therefore that a proof in S' (more precisely Lb-part of S ) can be simulated by a polynomial size EF-proof. In Clote-Takeuti [10], we introduced a first order system TNC0 which corresponds to the computational complexity class NC'. We also introduced another first order system T0NC0 which is equivalent to TNC0. In this paper we first develop the metamathematics of F and EF in TNC0 and prove the soundness of F in TNC0. Then we show that a proof in T0NC0 is simulated by a sequence of polynomial size. We actually prove a stronger statement. Let no be the number of propositional variables in an F-proof or an EF-proof. Let f be an NC' -function defined in TNC0 such that TNC0 proves that f (a) is an F-proof (or an EF-proof) of its conclusion g(a) where a = 2n0. If we substitute