Discretely ordered modules as a first-order extension of the cutting planes proof system

Journal of Symbolic Logic

Volumn 63, Issue 4, 1998, Pages 1582-1596

  Krajíček, Jan ^a,b  

^a INSTITUTE OF COMPUTER SCIENCE   (Czech Republic)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0040689280     PISSN: 00224812     EISSN: None     Source Type: Journal    
DOI: 10.2307/2586668     Document Type: Article

References (26)

1. N. ALON and R. BOPPANA, The monotone circuit complexity of Boolean functions, Combinatorica, vol. 7(1) (1987), pp. 1-22.
2. A.E. ANDREEV, On a method for obtaining lower bounds for the complexity of individual monotone functions, Doklady AN SSSR, vol. 282(5) (1985), pp. 1033-1037.
3. L. BABAI, A. GÁL, J. KOLLÁR, L. RÓNYAI, T. SZABÓ, and A. WIGDERSON, Extremal bipartite graphs and superpolynomial lower bounds for monotone span programs, Proceedings of the 28th annual ACM Symposium on theory of computing, 1996, pp. 603-611.
4. M. L. BONET, T. PITASSI, and R. RAZ, Lower bounds for cutting planes proofs with small coefficients, Proceedings of the 27th annual ACM Symposium on theory of computing, 1995, pp. 575-589.
5. S. BUSS, R. IMPAGLIAZZO, J. KRAJÍČEK, P. PUDLÁK, A.A. RAZBOROV, and J. SGALL, Proof complexity in algebraic systems and constant depth Frege systems with modular counting, Computational Complexity, vol. 6 (1996/1997), pp. 256-298.
6. M. CLEGG, J. EDMONDS, and R. IMPAGLIAZZO, Using the Groebner basis algorithm to find proofs of unsatisfiability, Proceedings of the 28th annual ACM Symp. on theory of computing, ACM Press, 1996, pp. 174-183.
7. S.A. COOK and A.R. RECKHOW, The relative efficiency of propositional proof systems, this JOURNAL, vol. 44(1) (1979), pp. 36-50.
8. W. COOK, C. R. COULLARD, and G. TURÁN, On the complexity of cutting plane proofs, Discrete Applied Mathematics, vol. 18 (1987), pp. 25-38.
9. M.J. FISHER and M.O. RABIN, Superexponential complexity of Presburger's arithmetic, SIAM-AMS Proceedings, vol. 7 (1974), pp. 27-41.
10. A. HAKEN and S.A. COOK, An exponential lower bound for the size of monotone real circuits, Journal of Computer and System Science (1995), to appear.
11. R. IMPAGLIAZZO and T. PITASSI, Interpolation for generalized cutting planes proof systems, a report in the CZ-US email seminar written by T. Pitassi, 1996.
12. M. KARCHMER and A. WIGDERSON, Monotone circuits for connectivity require super-logarithmic depth, Proceedings of the 20th annual ACM Symposium on theory of computing, ACM Press, 1988, pp. 539-550.
13. J. KRAJÍČEK, Lower bounds to the size of constant-depth propositional proofs, this JOURNAL, vol. 59(1) (1994), pp. 73-86.
14. _, Bounded arithmetic, propositional logic and complexity theory, Cambridge University Press, 1995.
15. _, Interpolation by a game, Mathematical Logic Quarterly (1996), to appear.
16. _, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, this JOURNAL, vol. 62(2) (1997), pp. 457-486.
17. 2 and EF, Logic and computational complexity (D. Leivant, editor), Lecture Notes in Computer Science, vol. 960. Springer-Verlag, 1995, Proceedings of the meeting held in Indianapolis, October 1994, pp. 210-220.
18. H.W. LENSTRA, Integer programming with a fixed number of variables, Mathematics of Operations Research, vol. 8 (1983), pp. 538-548.
19. L. LOVÁSZ, Algorithmic theory of numbers, graphs and convexity, CBMS-NSF Regional conferences series in applied mathematics, Society for Industrial and Applied mathematics, Philadelphia, 1986.
20. P. PUDLÁK, Lower bounds for resolution and cutting planes proofs and monotone computations, this JOURNAL, vol. 62 (1997), pp. 605-633.
21. P. PUDLÁK and J. SGALL, Algebraic models of computation, and interpolation for algebraic proof systems Proof complexity and feasible arithmetic (S. R. Buss, editor), Lecture Notes in Computer Science, DIMACS series in Discrete Mathematics and Theoretical Computer Science, vol. 39, Springer-Verlag, 1998, pp. 279-295.
22. A.A. RAZBOROV, Lower bounds on the monotone complexity of some Boolean functions, Soviet Mathematics, Doklady, vol. 31 (1985), pp. 354-357.
23. _, Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic, Izvestiya of the R.A.N., vol. 59(1) (1995), pp. 201-224.
24. A. ROSENBLOOM, Monotone real circuits are more powerful than monotone boolean circuits, Information Processing Letters, vol. 61 (1997), pp. 161-164.
25. G. TAKEUTI, Proof theory, North-Holland, 1975.
26. L. VAN DEN DRIES and J. HOLLY, Quantifier elimination for modules with scalar variables, Annals of Pure and Applied Logic, vol. 57 (1992), pp. 161-179.