 
Summary: SPACE COMPLEXITY IN PROPOSITIONAL CALCULUS #
MICHAEL ALEKHNOVICH + , ELI BENSASSON # , ALEXANDER A. RAZBOROV § , AND
AVI WIGDERSON #
SIAM J. COMPUT. c
# 2002 Society for Industrial and Applied Mathematics
Vol. 31, No. 4, pp. 11841211
Abstract. We study space complexity in the framework of propositional proofs. We consider a
natural model analogous to Turing machines with a readonly input tape and such popular proposi
tional proof systems as resolution, polynomial calculus, and Frege systems. We propose two di#erent
space measures, corresponding to the maximal number of bits, and clauses/monomials that need to be
kept in the memory simultaneously. We prove a number of lower and upper bounds in these models,
as well as some structural results concerning the clause space for resolution and Frege systems.
Key words. proof complexity, resolution, Frege, polynomial calculus
AMS subject classification. 03F20
PII. S0097539700366735
1. Introduction. Complexity of propositional proofs plays as important a role
in the theory of feasible proofs as the role played by the complexity of Boolean circuits
in the theory of e#cient computations. It is also well recognized that there exists a
very productive crossfertilization of techniques between the two fields. Partly because
of this similarity, most of the research in the proofcomplexity area concentrated on
