Simulating Cutting Plane Proofs with Restricted Degree of Falsity by Resolution

Abstract

Goerdt [Goe91] considered a weakened version of the Cutting Plane proof system with a restriction on the degree of falsity of intermediate inequalities. (The degree of falsity of an inequality written in the form ∑ a _i x _i + ∑ b _i (1 − x _i ) ≥ A, a _i ,b _i  ≥ 0 is its constant term A.) He proved a superpolynomial lower bound on the proof length of Tseitin-Urquhart tautologies when the degree of falsity is bounded by $\frac{n}{log^2 n+1}$ (n is the number of variables).

In this paper we show that if the degree of falsity of a Cutting Planes proof Π is bounded by d(n) ≤ n/2, this proof can be easily transformed into a resolution proof of length at most |∏| · (d(n)ⁿ− 1)64^d(n). Therefore, an exponential bound on the proof length of Tseitin-Urquhart tautologies in this system for d(n) ≤ cn for an appropriate constant c > 0 follows immediately from Urquhart’s lower bound for resolution proofs [Urq87].