The Complexity of Propositional Proofs with the Substitution Rule Alasdair Urquhart * University of Toronto Abstract We prove that for sufficiently large N , there are tautologies of size O(N ) that require proofs containing Ω(N ) lines in axiomatic systems of propo- sitional logic based on axioms and the rule of substitution for single vari- ables. These tautologies have proofs with O(log 2 N ) lines in systems with the multiple substitution rule. 1 Introduction In his survey paper [1] on the lengths of propositional proofs, Buss mentions the possibility that the rule allowing multiple substitution for variables in a tautology may allow some speedup over the single-variable substitution rule in the context of propositional proof systems. In this note, we verify this conjecture by showing that the lower bounds on the number of lines proved for Frege systems (systems based on schematic axioms and rules) can be extended in a fairly simple way to Frege systems with the single-variable substitution rule, showing a speedup of the multiple-variable rule over the single-variable rule. The best lower bounds for the number of lines and symbols in Frege systems are all based on a simple idea of counting the number of “active formulas” in a given inference, and then observing that in the proof of a tautology in which all of the complex subformulas are essential, every such subformula must appear somewhere in the proof as an active formula. In a sequel to Buss’s paper [4], the present author showed an Ω(N/ log N ) lower bound to the number of lines in proofs of tautologies of size O(N ) in S-Frege systems (Frege systems with the substitution rule); the proof was not based on the active subformula idea, but rather on a counting argument using tautologies that encode incompressible binary strings. This proof differs from the earlier “active formula” proof in not providing any explicit examples of hard tautologies. * The author gratefully acknowledges the support of the National Sciences and Engineering Research Council of Canada. 1