Lower Bounds for Cutting Planes Proofs with Small Coe cients

We consider small-weight Cutting Planes (CP) proofs; that is, Cutting Planes (CP) proofs with coeecients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems : (1) Tree-like CP proofs cannot polynomially simulate non-tree-like CP proofs. (2) Tree-like CP proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP proof system. In particular, they work for CP with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.