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Minimum propositional proof length is NP-hard to linearly approximate

  • Michael Alekhnovich (a1), Sam Buss (a2), Shlomo Moran (a3) and Toniann Pitassi (a4)


We prove that the problem of determining the minimum propositional proof length is NP-hard to approximate within a factor of . These results are very robust in that they hold for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution. Horn resolution, the polynomial calculus, the sequent calculus, the cut-free sequent calculus, as well as the polynomial calculus. Our hardness of approximation results usually apply to proof length measured either by number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and reduce it to the problems of approximation of the length of proofs.



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[1]Alehknovich, M., Buss, S., Moran, S., and Pitassi, T., Minimum propositional proof length is NP-hard to linearly approximate (extended abstract), Mathematical foundations of computer science (mfcs′98), Lecture Notes in Computer Science #1450, Springer Verlag, 1998, pp. 176184.
[2]Arora, S., Babai, L., Stern, J., and Sweedyk, Z., The hardness of approximate optima in lattices, codes, and systems of linear equations, Journal of Computer and System Sciences, vol. 54 (1997), pp. 317331, Earlier version in Proceedings of the 34th Symposium on the Foundations of Computer Science, 1993, pp.724-733.
[3]Arora, S. and Lund, C., Hardness of approximations, Approximation algorithms for NP-hard problems (Hochbaum, D. S., editor), PWS Publishing Co., Boston, 1996.
[4]Beame, P., Impagliazzo, R., Krajíček, J., Pitassi, T., and Pudlák, P., Lower bounds on Hilbert's Nullstellensatz and propositional proofs, Proceedings of the London Mathematical Society, vol. 73 (1996), pp. 126.
[5]Beame, P. and Pitassi, T., Simplified and improved resolution lower bounds, 37th annual symposium on foundations of computer science, IEEE Computer Society Press, Los Alamitos, California, 1996, pp. 274282.
[6]Bellare, M., Proof checking and approximation: Towards tight results, SIGACT News, vol. 27 (1996). pp. 213. Revised version at
[7]Bellare, M., Goldwasser, S., Lund, C., and Russell, A., Efficient probabalistically checkable proofs and applications to approximation, Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, 1993, pp. 294304.
[8]Bonet, M., Gavalda, R., Domingo, C., Maciel, A., and Pitassi, T., NO feasible interpolation or automatization for bounded-depth frege systems. 1998, Manuscript.
[9]Bonet, M. L., The Lengths of Propositional Proofs and the Deduction Rule, Ph.D. thesis, U.C. Berkeley, 1991.
[10]Bonet, M. L., Pitassi, T., and Raz, R.. No feasible interpolation for TC0-Frege proofs, Proceedings of the 38th annual symposium on foundations of computer science, IEEE Computer Society, Piscataway, New Jersey, 1997, pp. 264–263.
[11]Buss, S. R., Some remarks on lengths of propositional proofs, Archive for Mathematical Logic, vol. 34 (1995), pp. 377394.
[12]Buss, S.R., On Gödel's theorems on lengths of proofs. II. Lower bounds for recognizing k symbol provability, Feasible mathematics II (Clote, P. and Remmel, J.. editors). Birkhäuser Boston, Boston, MA, 1995, pp. 5790.
[13]Clegg, M., Edmonds, J., and Impagliazzo, R.. Using the Groebner basis algorithm to find proofs of unsatisfiability, Proceedings of the twenty-eighth annual acm symposium on the theory of computing (New York), Association for Computing Machinery, 1996, pp. 174183.
[14]Cook, S. A. and Reckhow, R. A., The relative efficiency of propositional proof systems, this Journal, vol. 44 (1979), pp. 3650.
[15]Dinur, I. and Safra, S., On the hardness of approximating label cover, Technical Report TR99-015, ECCC. 1999.
[16]Goldwasser, M. H. and Motwani, R., Intractability of assembly sequencing: Unit disks in the plane, Proceedings of the Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science #1272, Springer-Verlag, 1997, pp. 307320.
[17]Goldwasser, M. H., Complexity measures for assembly sequences, International Journal of Computational Geometry & Applications, vol. 9 (1999), pp. 371417.
[18]Iwama, K., Complexity of finding short resolution proofs, Mathematical foundations of computer science 1997, Lecture Notes in Computer Science #1295, Springer-Verlag, 1997, pp. 309318.
[19]Iwama, K. and Miyano, E., Intractibility of read-once resolution, Proceedings of the Tenth Annual Conference on Structure in Complexity Theory, IEEE Computer Society, Los Alamitos, California. 1995. pp. 2936.
[20]Khanna, S., Sudan, M., and Trevisan, L., Constraint satisfaction: the approximability of minimization problems, Twelfth annual ieee conference on computational complexity, IEEE Computer Society, 1997, pp. 282296.
[21]Krajíček, J. and Pudlák, P., Some consequences of cryptographical conjectures for S21 and EF, Logic and computational complexity (Leivant, D., editor), Lecture Notes in Computer Science #960, Springer-Verlag, Berlin, 1995, pp. 210220.
[22]Lund, C. and Yannakakis, M., On the hardness of approximating minimization problems, Journal of the Association for Computing Machinery, vol. 41 (1994), pp. 960981.
[23]Raz, R. and Safra, S., A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP, Proceedings of the 29-th Annual ACM Symposium on Theory of Computing, 1997, pp. 475484.
[24]Reckhow, R. A., On the Lengths of Proofs in the Propositional Calculus, Ph.D. thesis, Department of Computer Science, University of Toronto, 1976, Technical Report #87.
[25]Statman, R., Complexity of derivations from quantifier-free Horn formulae, mechanical introduction of explicit definitions, and refinement of completeness theorems, Logic colloquium ′76 (Gandy, R. and Hyland, M., editors), North-Holland, Amsterdam, 1977, pp. 505517.


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