Discretely Ordered Modules as a First-Order Extension of The Cutting Planes Proof System

We deene a rst-order extension LK(CP) of the cutting planes proof system CP as the rst-order sequent calculus LK whose atomic formulas are CP-inequalities P i ai xi b (xi's variables, ai's and b constants). We prove an interpolation theorem for LK(CP) yielding as a corollary a conditional lower bound for LK(CP)-proofs. For a subsystem R(CP) of LK(CP), essentially resolution working with clauses formed by CP-inequalities, we prove a monotone interpolation theorem obtaining thus an unconditional lower bound (depending on the maximum size of coeecients in proofs and on the maximum number of CP-inequalities in clauses). We also give an interpolation theorem for polynomial calculus working with sparse polynomials. The proof relies on a universal interpolation theorem for semantic derivations 16, Thm. 5.1]. LK(CP) can be viewed as a two-sorted rst-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module elements, another sort are scalars. The quantiication is allowed only over the former sort. We shall give a construction of a theory LK(M) for any discretely ordered module M (e.g., LK(Z) extends LK(CP)). The interpolation theorem generalizes to these theories obtained from discretely ordered Z-modules. We shall also discuss a connection to quantiier elimination for such theories. We formulate a communication complexity problem whose (suitable) solution would allow to improve the monotone interpolation theorem and the lower bound for R(CP). After lower bounds for the cutting planes proof system CP (deened in 8]), culminating in 21], various generalizations of CP were suggested. In particular, CP with the deduction rule. A proof in such a proof system allows to split