In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by first-order formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic log-time reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for low-level circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].

This paper of Tseitin is a landmark as the first to give non-trivial lower bounds for propositional proofs; although it pre-dates the first papers on

The Boolean formula value problem is in alternating log time and, more generally, parenthesis context-free languages are in alternating log time. The evaluation of reverse Polish notation Boolean formulas is also in alternating log time. These results are optimal since the Boolean formula value problem is complete for alternating log time under deterministic log time reductions. Consequently, it is also complete for alternating log time under AC reductions.

A new approach to Buss’s NC¹ algorithm [Proc. 19thACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 1987, pp. 123-131] for evaluation of Boolean formulas is presented. This problem is shown to be complete for NC¹ over AC¬ reductions. This approach is then used to solve the more general problem of evaluating arithmetic formulas by using arithmetic circuits.

It is generally conjectured that there is an exponential separation between Frege and extended Frege systems. This paper reviews and introduces some candidates for families of combinatorial tautologies for which Frege proofs might need to be superpolynomially longer than extended Frege proofs. Surprisingly, we conclude that no particularly good or convincing examples are known. The examples of combinatorial tautologies that we consider seem to give at most a quasipolynomial speed-up of extended Frege proofs over Frege proofs, with the sole possible exception of tautologies based on a theorem of Frankl.

We present a simplified proof that Brent/Spira restructuring of Boolean formulas can be improved to allow a Boolean formula of size n to be transformed into an equivalent log depth formula of size O(n

. In this paper we introduce Gentzen-style quantied propositional proof systems L i for the theories R i 2 . We formalize the systems L i within the bounded arithmetic theory R 1 2 and we show that for i 1, R i 2 can prove the validity of a sequent derived by an L i -proof. This statement is formally called i-RFN(L i ). We show if R i 2 ` 8xA(x) where A 2 b i , then for each integer n there is a translation of the formula A into quantied propositional logic such that R i 2 proves there is an L i -proof of this translated formula. Using the proofs of these two facts we show that L i is in some sense the strongest system for which R i 2 can prove i-RFN and we show for i j 2 that the 8 b j -consequences of R i 2 are nitely axiomatized. 1. Introduction Propositional proof systems and bounded arithmetic are closely connected. Cook [10] introduced the equational arithmetic theory PV of polynomial time computable functions and showed PV co...