The cutting plane proof system with bounded degree of falsity
The cutting plane proof system for proving the unsatisfiability of propositional formulas in conjunctive normalform is based on a natural representation of formulas as systems of integer inequalities. We define a restriction of this system, the cutting plane system with bounded degree of falsity, and show the results: This system p-simulates resolution and has polynomial size proofs for the pigeonhole formulas. The formulas from [ 9] only have superpolynomially long proofs in the system. Our system is the only known system with provably superpolynomial proof size, but polynomial size proofs for the pigeonhole formulas.
Unable to display preview. Download preview PDF.
- M. Ajtai, The complexity of the pigeonhole principle, Proceedings of the 29th Symposium on Foundations of Computer Science (1988).Google Scholar
- S. Buss, Polynomial size proofs of the propositional pigeonhole principle, Journ. Symb. Logic 52 (1987) pp.916–927.Google Scholar
- C.-L. Chang, R.C.-T. Lee, Symbolic Logic and mechanical theorem proving, Academic Press (1973).Google Scholar
- P. Clote, Bounded arithmetic and computational complexity, Proceedings Structures in Complexity (1990) pp. 186–199.Google Scholar
- W. Cook, C.R. Coullard, G. Turan, On the complexity of cutting plane proofs, Discr. Appl. Math. 18 (1987) pp. 25–38.Google Scholar
- S. A. Cook, R.A. Reckhow, The relative efficiency of propositional proof systems, Journ. Sym. Logic 44 (1979) pp. 36–50.Google Scholar
- A. Haken, The intractability of resolution, Theor. Comp. Sci. 39 (1985) pp. 297–308.Google Scholar
- A. Goerdt, Cutting plane versus Frege proof systems, CSL (1990) LNCS 533, pp. 174–194.Google Scholar
- A. Urquhart, Hard examples for resolution, JACM 34 (1) (1987) pp. 209–219.Google Scholar
- R.A. Smullyan, First-order Logic, Springer Verlag (1968).Google Scholar
- V. Chvatal, Probabilistic methods in graph theory, Annals of Operations Research 1 (1984) pp. 171–182.Google Scholar