Abstract
In this paper we present MathCheck2, a tool which combines sophisticated search procedures of current SAT solvers with domain specific knowledge provided by algorithms implemented in computer algebra systems (CAS). MathCheck2 is aimed to finitely verify or to find counterexamples to mathematical conjectures, building on our previous work on the MathCheck system. Using MathCheck2 we validated the Hadamard conjecture from design theory for matrices up to rank 136 and a few additional ranks up to 156. Also, we provide an independent verification of the claim that Williamson matrices of order 35 do not exist, and demonstrate for the first time that 35 is the smallest number with this property. Finally, we provided more than 160 matrices to the Magma Hadamard database that are not equivalent to any matrices previously included in that database.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
From Doron Zeilberger’s talk at the Fields institute in Toronto, December 2015 (http://www.fields.utoronto.ca/video-archive/static/2015/12/379-5401/mergedvideo.ogv, minute 44).
- 2.
For more information on this format, please refer to http://www.satcompetition.org/2009/format-benchmarks2009.html.
References
Ábrahám, E.: Building bridges between symbolic computation and satisfiability checking. In: Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, pp. 1–6. ACM, New York (2015)
Armand, M., Faure, G., Grégoire, B., Keller, C., Théry, L., Wener, B.: Verifying SAT and SMT in CoQ for a fully automated decision procedure. In: PSATTT 2011: International Workshop on Proof-Search in Axiomatic Theories and Type Theories, pp. 11–25 (2011)
Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symbolic Comput. 24(3), 235–265 (1997)
Bouton, T., Caminha B. de Oliveira, D., Déharbe, D., Fontaine, P.: veriT: an open, trustable and efficient SMT-Solver. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 151–156. Springer, Heidelberg (2009)
Char, B.W., Fee, G.J., Geddes, K.O., Gonnet, G.H., Monagan, M.B.: A tutorial introduction to Maple. J. Symbolic Comput. 2(2), 179–200 (1986)
Colbourn, C.J., Dinitz, J.H. (eds.): Handbook of Combinatorial Designs. Discrete Mathematics and its Applications (Boca Raton), 2nd edn. Chapman & Hall/CRC, Boca Raton (2007)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 2nd edn. Springer, New York (1992)
Decker, W., Greuel, G.M., Pfister, G., Schönemann, H.: Singular 4-0-2 – A computer algebra system for polynomial computations (2015). http://www.singular.uni-kl.de
Hadamard, J.: Résolution d’une question relative aux déterminants. Bull. Sci. Math. 17(1), 240–246 (1893)
Hearn, A.: Reduce user’s manual, version 3.8 (2004)
Hedayat, A., Wallis, W.: Hadamard matrices and their applications. Ann. Stat. 6(6), 1184–1238 (1978)
Hnich, B., Prestwich, S.D., Selensky, E., Smith, B.M.: Constraint models for the covering test problem. Constraints 11(2), 199–219 (2006)
Holzmann, W.H., Kharaghani, H., Tayfeh-Rezaie, B.: Williamson matrices up to order 59. Des. Codes Crypt. 46(3), 343–352 (2008)
Junges, S., Loup, U., Corzilius, F., Ábrahám, E.: On Gröbner bases in the context of satisfiability-modulo-theories solving over the real numbers. In: Muntean, T., Poulakis, D., Rolland, R. (eds.) CAI 2013. LNCS, vol. 8080, pp. 186–198. Springer, Heidelberg (2013)
Konev, B., Lisitsa, A.: A SAT attack on the Erdős discrepancy conjecture. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 219–226. Springer, Heidelberg (2014)
Kotsireas, I.S.: Algorithms and metaheuristics for combinatorial matrices. In: Handbook of Combinatorial Optimization, pp. 283–309. Springer, New York (2013)
Liang, J.H., Ganesh, V., Poupart, P., Czarnecki, K.: Learning rate based branching heuristic for SAT solvers. In: Creignou, N., Le Berre, D., Le Berre, D., Le Berre, D., Le Berre, D., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 123–140. Springer, Heidelberg (2016). doi:10.1007/978-3-319-40970-2_9
Liang, J.H., Ganesh, V., Poupart, P., Czarnecki, K.: Exponential recency weighted average branching heuristic for SAT solvers. In: Proceedings of AAAI 2016 (2016)
Marques-Silva, J.P., Sakallah, K., et al.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)
Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of the 38th Annual Design Automation Conference, pp. 530–535. ACM, New York (2001)
de Moura, L., Kong, S., Avigad, J., van Doorn, F., von Raumer, J.: The lean theorem prover (system description). In: Felty, P.A., Middeldorp, A. (eds.) CADE-25. LNCS, vol. 9195, pp. 378–388. Springer, Switzerland (2015)
Muller, D.E.: Application of Boolean Algebra to Switching Circuit Design and to Error Detection. Electron. Comput. Trans. IRE Prof. Group Electron. Comput. EC-3(3), 6–12 (1954)
Nadel, A., Ryvchin, V.: Efficient SAT solving under assumptions. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 242–255. Springer, Heidelberg (2012)
Đoković, D.Ž.: Williamson matrices of order \(4n\) for \(n = 33\), \(35\), \(39\). Discrete Math. 115(1), 267–271 (1993)
Đoković, D.Ž., Kotsireas, I.S.: Compression of periodic complementary sequences and applications. Des. Codes Crypt. 74(2), 365–377 (2015)
Paley, R.E.: On orthogonal matrices. J. Math. Phys. 12(1), 311–320 (1933)
Prestwich, S.D., Hnich, B., Simonis, H., Rossi, R., Tarim, S.A.: Partial symmetry breaking by local search in the group. Constraints 17(2), 148–171 (2012)
Reed, I.: A class of multiple-error-correcting codes and the decoding scheme. Trans. IRE Prof. Group Inf. Theory 4(4), 38–49 (1954)
Riel, J.: nsoks: A Maple script for writing \(n\) as a sum of \(k\) squares
Seberry, J.: Library of Williamson Matrices. http://www.uow.edu.au/~jennie/WILLIAMSON/williamson.html
Sloane, N.: Library of Hadamard Matrices. http://neilsloane.com/hadamard/
Sylvester, J.J.: Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton’s rule, ornamental tile-work, and the theory of numbers. London Edinb. Dublin Philos. Mag. J. Sci. 34(232), 461–475 (1867)
SC\({{}^{2}}\): Satisfiability checking and symbolic computation. http://www.sc-square.org/
The Sage Developers: Sage Mathematics Software (Version 7.0) (2016). http://www.sagemath.org
Walsh, J.L.: A closed set of normal orthogonal functions. Am. J. Math. 45(1), 5–24 (1923)
Williamson, J.: Hadamard’s determinant theorem and the sum of four squares. Duke Math. J 11(1), 65–81 (1944)
Wolfram, S.: The Mathematica Book, version 4. Cambridge University Press (1999)
Zulkoski, E., Ganesh, V., Czarnecki, K.: MathCheck: a math assistant via a combination of computer algebra systems and SAT solvers. In: Felty, P.A., Middeldorp, A. (eds.) CADE-25. LNCS, vol. 9195, pp. 607–622. Springer, Switzerland (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
Bright, C., Ganesh, V., Heinle, A., Kotsireas, I., Nejati, S., Czarnecki, K. (2016). MathCheck2: A SAT+CAS Verifier for Combinatorial Conjectures. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-45641-6_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-45640-9
Online ISBN: 978-3-319-45641-6
eBook Packages: Computer ScienceComputer Science (R0)