## Journal of Symbolic Logic

### Lower Bounds for Cutting Planes Proofs with Small Coefficients

#### Abstract

We consider small-weight Cutting Planes $(\mathrm{CP}^\ast)$ proofs; that is, Cutting Planes (CP) proofs with coefficients up to $\operatorname{Poly}(n)$. We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of $\mathrm{CP}^\ast$ 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 $\mathrm{CP}^\ast$ proofs cannot polynomially simulate non-tree-like $\mathrm{CP}^\ast$ proofs. (2) Tree-like $(\mathrm{CP}^\ast$ proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the $\mathrm{CP}^\ast$ proof system. In particular, they work for $\mathrm{CP}^\ast$ 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.

#### Article information

Source
J. Symbolic Logic, Volume 62, Issue 3 (1997), 708-728.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745294

Mathematical Reviews number (MathSciNet)
MR1472120

Zentralblatt MATH identifier
0889.03050

JSTOR