A switching lemma for small restrictions and lower bounds for k-DNF resolution

SIAM Journal on Computing

Volumn 33, Issue 5, 2004, Pages 1171-1200

  Segerlind, Nathan ^a   Buss, Sam ^b   Impagliazzo, Russell ^c  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Boolean circuit complexity; Circuit bottom fan in; k DNFs; Lower bounds; Propositional proof complexity; Random CNFs; Random restriction; Res(k); Resolution; Sipser functions; Switching lemmas; Weak pigeonhole principles

Indexed keywords

BOOLEAN CIRCUIT COMPLEXITY; LOWER BOUNDS; PROPORTIONAL PROOF COMPLEXITY; SWITCHING LEMMAS;

ALGORITHMS; FUNCTIONS; NETWORKS (CIRCUITS); POLYNOMIALS; THEOREM PROVING;

SWITCHING;

EID: 8344278749     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539703428555     Document Type: Article

References (37)

1. 1-formulae on finite structures, Ann. Pure Appl. Logic, 24 (1983), pp. 1-48.
2. A. ATSERIAS AND M. BONET, On the automatizability of resolution and related prepositional proof systems, in Proceedings of the Sixteenth International Workshop on Computer Science Logic, Lecture Notes in Comput. Sci. 2471, Springer, New York, 2002, pp. 569-583.
3. A. ATSERIAS, M. BONET, AND J. ESTEBAN, Lower bounds for the weak pigeonhole principle and random formulas beyond resolution, Inform. and Comput., 176 (2002), pp. 136-152.
4. P. BEAME, A Switching Lemma Primer, tech. report, Department of Computer Science and Engineering, University of Washington, Seattle, 1994.
5. P. BEAME, R. KARP, T. PITASSI, AND M. SAKS, The efficiency of resolution and Davis-Putnam procedures, SIAM J. Comput., 31 (2002), pp. 1048-1075.
6. P. BEAME AND T. PITASSI, Prepositional proof complexity: Past, present, and future, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 65 (1998), pp. 66-89.
7. E. BEN-SASSON, Hard examples for bounded depth Frege, in Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 563-572.
8. E. BEN-SASSON AND A. WIGDERSON, Short proofs are narrow - resolution made simple, J. ACM, 48 (2001), pp. 149-169.
9. M. BONET AND N. GALESI, A study of proof search algorithms for resolution and polynomial calculus, in Proceedings of the Fortieth Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alimitos, CA, 1999, pp. 422-431.
10. R. BOPPANA AND M. SIPSER, The complexity of finite functions, in Handbook of Theoretical Computer Science, Vol. A, J. van Leeuwen, ed., Elsevier, Amsterdam, 1990, pp. 757-804.
11. S. R. BUSS AND G. TURÁN, Resolution proofs of generalized pigeonhole principles, Theoret. Comput. Sci., 62 (1988), pp. 311-317.
12. L. CAI, J. CHEN, AND J. HÅSTAD, Circuit bottom fan-in and computational power, SIAM J. Comput., 27 (1998), pp. 341-355.
13. V. CHVÁTAL AND E. SZEMERÉDI, Many hard examples for resolution, J. ACM, 35 (1988), pp. 759-768.
14. S. COOK AND R. RECKHOW, The relative efficiency of prepositional proof systems, J. Symbolic Logic, 44 (1979), pp. 36-50.
15. S. DANTCHEV AND S. RIIS, Tree resolution proofs of the weak pigeon-hole principle, in Proceedings of the Sixteenth Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 2001, pp. 69-75.
16. P. ERDÖS AND R. RADO, Intersection theorems for finite sets, J. London Math. Soc., 35 (1960), pp. 85-90.
17. J. L. ESTEBAN, N. GALESI, AND J. MESSNER, On the complexity of resolution with bounded conjunctions, in Proceedings of the Twenty-Ninth International Colloquium on Automata, Languages and Programming, Lecture Notes in Comput. Sci. 2380, Springer, New York, 2002, pp. 220-231.
18. M. FURST, J. SAXE, AND M. SIPSER, Parity, circuits, and the polynomial-time hierarchy, Math. Systems Theory, 17 (1984), pp. 13-27.
19. A. GOERDT, Unrestricted resolution versus N-resolvtion, Theoret. Comput. Sci., 93 (1992), pp. 159-167.
20. A. HAKEN, The intractability of resolution, Theoret. Comput. Sci., 39 (1985), pp. 297-308.
21. J. HÅSTAD, Almost optimal lower bounds for small depth circuits, in Advances in Computing Research, Vol. 5, JAI Press, Greenwich, CT, 1989, pp. 143-170.
22. S. JUKNA, Extremal Combinatorics: With Applications to Computer Science, Springer, New York, 2001.
23. J. KRAJÍČEK, Bounded Arithmetic, Prepositional Logic and Complexity Theory, Cambridge University Press, Cambridge, UK, 1995.
24. J. KRAJÍČEK, On the weak pigeonhole principle, Fund. Math., 170 (2001), pp. 123-140.
25. A. MACIEL, T. PITASSI, AND A. WOODS, A new proof of the weak pigeonhole principle, J. Comput. System Sci., 64 (2002), pp. 843-872.
26. R. MOTWANI AND P. RAGHAVAN, Randomized Algorithms, Cambridge University Press, Cambridge, UK, 1995.
27. E. PALMER, Graphical Evolution: An Introduction to the Theory of Random Graphs, Wiley Interscience, Chichester, UK, 1985.
28. T. PITASSI AND R. RAZ, Regular resolution lower bounds for the weak pigeonhole principle, in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, ACM, New York, 2001, pp. 347-355.
29. P. PUDLÁK, The lengths of proofs, in Handbook of Proof Theory, S. R. Buss, ed., Elsevier North-Holland, Amsterdam, 1998, pp. 547-637.
30. R. RAZ, Resolution lower bounds for the weak pigeonhole principle, in Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 553-562.
31. A. RAZBOROV, Improved resolution lower bounds for the weak functional pigeonhole principle, Theoret. Comput. Sci., 303 (2001), pp. 233-243.
32. A. RAZBOROV, Pseudorandom generators hard for k-DNF resolution and polynomial calculus resolution. Available online from http://genesis.mi.ras.ru/ ~razborov/, 2003.
33. N. SEGERLIND, S. BUSS, AND R. IMPAGLIAZZO, A switching lemma for small restrictions and lower bounds for k-DNF resolution, in Proceedings of the Forty-Third Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 2002, pp. 604-613.
34. M. SIPSER, Borel sets and circuit complexity, in Proceedings of the Fifteenth Animal ACM Symposium on Theory of Computing, ACM, New York, 1983, pp. 61-69.
35. G. TSEITIN, On the complexity of derivations in prepositional calculus, in Studies in Constructive Mathematics and Mathematical Logic, Part II, A. O. Slisenko, ed., 1970, pp. 115-125 (in Russian).
36. A. URQUHART, Hard examples for resolution, J. ACM, 34 (1987), pp. 209-219.
37. V. VAZIRANI, Approximation Algorithms, Springer, Berlin, 2001.