Space complexity in propositional calculus

We study space complexity in the framework of propositional proofs. We consider a natural model analogous to space bounded Turing machines with a read-only input tape, and such popular propositional proof systems as Resolution, Polynomial Calculus and Frege systems. We study two different space measures. The first, introduced by [5] for Resolution and extended here to other systems, is the structured measure which counts the number of clauses/monomials kept is memory simultaneously. The other is an unstructured °measure related to the number of b i t s describing the memory content. We develop lower bound techniques that enable proving tight linear lower bounds for the first measure, and tight quadratic lower bounds for the second, for large classes of tautologies (including familiar ones like the pigeonhole principle), in both Resolution and (extensions of) Polynomial Calculus. We also prove some structural results concerning the clause space for Resolution and Frege Systems.