The Journal of Symbolic Logic

Research Article

Lower bounds for cutting planes proofs with small coefficients

Maria Boneta1, Toniann Pitassia2 and Ran Raza3

a1 Department de Lenguajes y Systemas Informaticos, Edificio e Pau Gargallo 5, 08028 Barcelona, Spain, E-mail:

a2 Department of Computer Science, University of Arizona, Tucson, AZ 85721, USA, E-mail:

a3 Department of Applied Math, Weizmann Institute, Rehovat 76100, Israel, E-mail:


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 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.

(Received July 10 1995)

(Revised November 22 1995)