Cutting plane and Frege proofs

The cutting plane refutation system CP for propositional logic is an extension of resolution and is based on showing the non-existence of solutions for families of integer linear inequalities. We deene the system CP + , a modiication of the cutting plane system, and show that CP + can polynomially simulate Frege systems F. In 8], it is shown that F polynomially simulates CP + , so both systems are polynomially equivalent. To establish this result, propositional formulas are represented in a natural manner, and an eeective version of cut elimination is proved for the system CP +. Additionally, an alternative proof is given which directly translates F proofs into CP +. Thus, within a polynomial factor, one can simulate classical propositional logic proofs using the cut rule by refutation-style proofs involving linear inequalities with the ceiling operation. Since there are polynomial size cutting plane CP proofs for many elementary combinatorial principles (pigeonhole principle, k-equipartition principle), we propose propositional versions of the Paris-Harrington theorem , Kanamori-McAloon theorem, and variants as possible candidates for combinatorial tautologies which may require exponential size cutting plane and Frege proofs.