Abstract
In standard implementations of the Grobner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the proof complexity of refutations of polynomials derived from Boolean formulas in both the Polynomial Calculus (PC) and Nullstellensatz systems. We show that the PC refutations of homogenized formulas give crucial information about the complexity of the original formulas. The minimum PC refutation degree of homogenized formulas is equal to the Nullstellensatz refutation degree of the original formulas, whereas the size of the homogenized PC refutation is equal to the size of the PC refutation for the originals. Using this relationship, we prove nearly linear (Ω(n/logn) vs. O(1)) separations between Nullstellensatz and PC degree, for a family of explicitly constructed contradictory 3CNF formulas. Previously, a Ω(n1/2) separation had been proved for equations that did not correspond to any CNF formulas, and a log n separation for equations derived from kCNF formulas.
Research partially supported by NSF grant CCR-9457782 and a scholarship from the Arizona Chapter of the ARCS Foundation.
Research supported by NSF grant CCR-9457782 and US-Israel BSF Grant 95-00238.
Research supported by NSF CCR-9734911, Sloan Research Fellowship BR-3311, and by a cooperative research grant INT-9600919/ME-103 from NSF and the MŠMT (Czech Republic), and USA-Israel-BSF Grant 97-00188
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References
M.L. Bonet, J. L. Esteban, N. Galesi, and J. Johannsen. Exponential separations between restricted resolution and cutting planes proof systems. In Proceedings from 38th FOCS, pages 638–647. 1998.
[BIK+96]_Paul W. Beame, Russell Impagliazzo, Jan Krajíček, Toniann Pitassi, and Pavel Pudlak. Lower bounds on Hilbert’s Nullstellensatz and propositional proofs. Proc. London Math. Soc, 73(3): 1–26, 1996.
[BIK+97]_S. Buss, R. Impagliazzo, J. Krajíček, P. Pudlak, A. A. Razborov, and J. Sgall. Proof complexity in algebraic systems and bounded depth Frege systems with modular counting. Computation Complexity, 6(3):256–298, 1997.
S. R. Buss and T. Pitassi. Good degree bounds on Nullstellensatz refutations of the induction principle. In Proceedings of the Eleventh Annual Conference on Computational Complexity formerly: Structure in Complexity Theory, pages 233–242, Philadelphia, PA, May 1996. IEEE.
E. Ben Sasson and A. Wigderson. Short proofs are narrow-resolution made simple. In Proceedings of 31st ACM STOC, pages 517–526. 1999.
S. R. Buss. Lower bounds on Nullstellensatz proofs via designs. In P. W. Beame and S. R. Buss, (editors), Proof Complexity and Feasible Arithmetics, DIMACS, pages 59–71. American Math. Soc, 1997.
M. Clegg, J. Edmonds, and R. Impagliazzo. Using the Grobner basis algorithm to find proofs of unsatisfiability. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pages 174–183, Philadelphia, PA, May 1996.
J. Cox, D. Little and D. O’Shea. Ideals, Varieties and Algoirthms. Springer-Verlag, 1992.
R. Celoni, W.J. Paul, and R.E. Tarjan. Space bounds for a game on graphs. Math. Systems Theory, 10:239–251, 1977.
J. L. Esteban and Jacobo Toran. Space bounds for resolution. In Proc. 16th STACS, pages 551–561. Springer-Verlag LNCS, 1999.
T. Pitassi. Algebraic propositional proof systems. In DIMACS Series in Discrete Mathematics, volume 31, pages 215–243. American Math. Soc, 1997.
R. Raz and P. McKenzie. Separation of the monotone nc hierarchy. In Proceedings of 38th IEEE Foundations of Computer Science. 1997.
A. Urquhart. Hard examples for resolution. Journal of the ACM, 34(1):209–219, 1987.
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Buresh-Oppenheim, J., Pitassi, T., Clegg, M., Impagliazzo, R. (2000). Homogenization and the Polynomial Calculus. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_78
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DOI: https://doi.org/10.1007/3-540-45022-X_78
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