Cutting planes, connectivity, and threshold logic

Originating from work in operations research the cutting plane refutation system CP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial size CP proofs are given for the undirected s-t connectivity principle. The subsystems CPq of CP , for q ≥ 2, are shown to be polynomially equivalent to CP , thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem for CP2-proofs and thereby for arbitrary CP -proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extension CPLE, introduced in [9] and there shown to psimulate Frege systems, is proved to be polynomially equivalent to Frege systems. Lastly, since linear inequalities are related to threshold gates, we introduce a new threshold logic and prove a completeness theorem. Mathematics Subject Classification: 03B05, 03F07, 03F20, 90C10.