Cutting planes is a proof system in which lines are linear inequalities. It has two main variants: syntactic cutting planes, in which specific derivation rules are given, and semantic cutting planes, in which any bounded fan-in derivation which is semantically correct (for all zero-one assignments to variables) is allowed. Only the syntactic version is a Cook–Reckhow proof system, since verifying a semantic cutting planes proof is coNP-complete.
Extending earlier work of Pudlák, we give an exponential lower bounds for semantic cutting planes.
We also show that semantic cutting planes is exponentially stronger than syntactic cutting planes, and exhibit two contradictory lines which take exponentially long to refute in syntactic cutting planes.
This work is a combination of two earlier preprints: a preprint of Pavel Hrubeš proving the exponential lower bound for semantic cutting planes, and a preprint of Massimo Lauria and myself proving the exponential separation between semantic and syntactic cutting planes.