omparison of the Complexities in Frege Proofs with Different Substitution Rules

Abstract

We compare the proof complexities in Frege systems with multiple substitution rule and with constant bounded substitution rule. We prove that any two constant bounded substitution Frege systems are polynomially equivalent both by size and by steps. Frege system with multiple substitution rule and Frege system with constant bounded substitution rule are also polynomially equivalent by size, but the first system has exponential speed-up over the second system by steps.