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 ixi + ∑ bi(1 - xi) ≥ A, ai, bi ≥ 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 n/log2 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 |Π|·( d(n)-1n)64d(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]. © Springer-Verlag Berlin Heidelberg 2005.