Relative efficency of propositional proof systems: resolution vs. cut-free LK

Resolution and cut-free LK are the most popular propositional systems used for logical automated reasoning. The question whether or not resolution and cut-free LK have the same e ciency on the system of CNF formulas has been asked and studied since 1960's [17][21]. It was shown in [8] that tree resolution has super-polynomial speed-up over (tree) cutfree LK. Naturally, the current issue is whether or not resolution and cut-free LK expressed as directed acyclic graphs (DAG) have the same efciency. In this paper, we introduce a new algorithm to eliminate atomic cuts and show that cut-free LK (DAG) polynomially simulates resolution when the input formula is expressed as a k-CNF formula. As a corollary, we show that regular resolution does not polynomially simulate cut-free LK (DAG). We also show that cut-free LK (DAG) polynomially simulates regular resolution. AMS classi cation: 03F05; 03F20