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

We define a first-order extension LK(CP) of the cutting planes proof system CP as the firstorder sequent calculus LK whose atomic formulas are CP-inequalities Ei as 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 coefficients 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, Theorem 5.1]. LK(CP) can be viewed as a two-sorted first-order theory of Z considered itself as a discretely ordered Z-module. One sort of variables are module elements, another sort are scalars. The quantification 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 quantifier 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 (defined in [8]), culminating in [20], various generalizations of CP were suggested. In particular, CP with the deduction rule. A proof in such a proof system allows to split (repeatedly) the proof into two parts, depending on whether a CP-inequality is or is not satisfied (see Example 3 in Section 2 for a formalization due to [4]). A natural question arises to extend the lower bounds for CP to these presumably stronger proof systems. It appeared to us that a convenient formalization of CP with the deduction rule is resolution working with clauses formed by CP-inequalities (see Section 3). In particular, it allows us to discuss also proofs in which the deduction formulas are not organized in a tree (as it is in [4]). We noticed that an interpolation theorem for such a proof system, and, in fact, for its first-order extension LK(CP), is an immediate consequence of a universal interpolation theorem for semantic derivations [16, Theorem 5.1] (the monotone interpolation for R(CP) needs an extra argument). Let us explain first briefly how lower bounds for CP ([4, 16, 20]) were obtained. All these lower bounds use eSective interpolation theorem. The idea of using Received September 24, 1996; revised September 5, 1997. Partially supported by the US-Czechoslovak Science and Technology Program grant #93 025, and by grant #A 101 96 02 of the AVCR. @ 1998, ASSOC;atiOn fOr SYmbOIiC LOgiC 0022-48 1 2/98/6304-0022/$2.50