**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31482

##### A P-SPACE Algorithm for Groebner Bases Computation in Boolean Rings

**Authors:**
Quoc-Nam Tran

**Abstract:**

**Keywords:**
Algorithm,
Complexity,
Groebner basis,
Applications
of Computer Science.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1334139

**References:**

[1] G. S. Avrunin. Symbolic model checking using algebraic geometry. In CAV -96: Proceedings of the 8th International Conference on Computer Aided Verification, pages 26-37, London, UK, 1996. Springer-Verlag.

[2] P. Beame, R. Impagliazzo, J. Krajicek, T. Pitassi, and P. Pudlak. Lower bounds on Hilbert-s nullstellansatz and propositional proofs. In Proceddings of the 35th Annual Symposium on Foundations of Computer Science, pages 794-806, Santa Fe, NM, 1994.

[3] A. Biere, A. Cimatti, E. M. Clarke, M. Fujita, and Y. Zhu. Symbolic model checking using SAT procedures instead of BDDs. In Proceedings of Design Automation Conference (DAC-99), 1999.

[4] B. Buchberger. An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-dimensional Polynomial Ideal (in German). PhD thesis, Institute of Mathematics, Univ. Innsbruck, Innsbruck, Austria, 1965.

[5] B. Buchberger. Groebner Bases: An algorithmic method in polynomial ideal theory. In N. K. Bose, editor, Multidimensional Systems Theory, chapter 6, pages 184- 232. Reidel Publishing Company, Dodrecht, 1985.

[6] S. N. Burris and H. P. Sankappanavar. A Course in Universal Algebra. Springer-Verlag, 1981.

[7] M. Clegg, J. Edmonds, and R. Impagliazzo. Using Groebner basis algorithm to find proofs of unsatisfiability. In Proceedings of STOC-96, pages 174-183, Philadelphia, PA, USA, 1996. ACM.

[8] S. Collart, M. Kalkbrener, and D. Mall. Converting bases with the Groebner walk. Journal of Symbolic Computation, 24(3/4):465-469, 1997.

[9] L. Csanky. Fast parallel matrix conversion algorithms. SIAM J. Comput., 5:618-623, 1976.

[10] N. Dershowitz, J. Hsiang, G. Huang, and D. Kaiss. Boolean ring satisfiability.

[11] N. Dershowitz, J. Hsiang, G. Huang, and D. Kaiss. Intersection-based methods for satisfiability using boolean rings.

[12] T. D. Dub'e. The structure of polynomial ideals and Groebner bases. SIAM J. Comput., 19(4):750-773, 1990.

[13] S. Fortune and J. Wyllie. Parallelism in random access machines. In Proceedings of the 10th annual ACM symposium on Theory of Computing, pages 114-118. ACM Press, 1978.

[14] Z. Galil and V. Pan. Parallel evaluation of the determinant and of the inverse of a matrix. Inform. Proc. Lett., 30:41- 45, 1989.

[15] G. Hermann. Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Mathematische Annalen, 95:736-788, 1925.

[16] J. Hsiang. Refutational theorem proving using term rewriting systems. Artificial Intelligence, 25:255-300, 1985.

[17] G. Huang, N. Dershowitz, and J. Hsiang. Satisfiability testing using simplification in boolean rings.

[18] O. Ibarra, S. Moran, and L. Rosier. A note on the parallel complexity of computing the rank of order n matrices. Inform. Proc. Lett, 11(4):162, 1980.

[19] D. Kaiss and N. Dershowitz. Boosting satisfiability testing using boolean rings.

[20] K. Kuehnle and E. Mayr. Exponential space computation of groebner bases. In Proceedings of ISSAC. ACM, 1996.

[21] O. Kupferman, M. Vardi, and P. Wolper. An automatatheoretic approach to branching-time model checking. Journal of the ACM, 47(2):312-360, 2000.

[22] K. L. McMillan. Symbolic Model Checking. Kluwer Academic Publishers, 1993.

[23] F. Preparata and D. Sarwata. An improved parallel processor bound in fast matrix inversion. Inform. Proc. Lett., 1978.

[24] M. Sipser. Introduction to the Theory of Computation. PWS Publishing, Boston, 2nd edition, 1997.

[25] M. Sipser. Introduction to the Theory of Computation. Course Technology, 2nd edition, 2005.

[26] M. Stone. The theory of representation for boolean algebras. Trans. Amer. Math. Soc., 1936.

[27] M. Stone. Applications of the theory of boolean rings to general topology. Trans. Amer. Math. Soc., 1937.

[28] Q.-N. Tran. A fast algorithm for Groebner basis conversion and its applications. Journal of Symbolic Computation, 30:451-468, 2000.

[29] Q.-N. Tran. A new class of term orders for elimination and applications. Journal of Symbolic Computation, 42, 2007.

[30] Q.-N. Tran and M. Y. Vardi. Groebner bases computation in boolean rings for symbolic model checking. In Proceedings of IASTED 2007 Conference on Modeling and Simulation, 2007.

[31] F. Winkler. Polynomial Algorithms in Computer Algebra. Texts and Monographs in Symbolic Computation. Springer-Verlag, 1996.