## Journal of Symbolic Logic

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

Jan Krajicek

#### Abstract

We define a first-order extension LK(CP) of the cutting planes proof system CP as the first-order sequent calculus LK whose atomic formulas are CP-inequalities $\sum_i a_i \cdot x_i \geq b$ (x$_i$'s variables, a$_i$'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).

#### Article information

Source
J. Symbolic Logic, Volume 63, Issue 4 (1998), 1582-1596.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745649

Mathematical Reviews number (MathSciNet)
MR1665783

Zentralblatt MATH identifier
0930.03081

JSTOR