Lower bounds for cutting planes proofs with small coefficients

We consider small-weight Cutting Planes (CP* ) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds fc)r monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies baaed 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 fcn Resolution. We also prove the following two theorems : (1) Treelike CP’ proofs cannot polynomially simulate non-treelike CP* proofs. (2) Tree-like CP” proofs and Boundeddepth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the C’P* 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.

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