skip to main content
review-article
Open Access

The science of brute force

Published:24 July 2017Publication History
Skip Abstract Section

Abstract

Mathematics solves problems by pen and paper. CS helps us to go far beyond that.

References

  1. Ahmed, T., Kullmann, O., Snevily, H. On the van der Waerden numbers w(2; 3, t). Disc. Appl. Math. 174 (2014), 27--51. Google ScholarGoogle ScholarCross RefCross Ref
  2. Biere, A., Fröhlich, A. Evaluating CDCL variable scoring schemes. In SAT (Springer, 2015), 405--422. Google ScholarGoogle ScholarCross RefCross Ref
  3. Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. eds. Handbook of Satisfiability, volume 185 of FAIA. IOS Press, Amsterdam, The Netherlands, Feb. 2009.Google ScholarGoogle Scholar
  4. Buss, S. Propositional proofs in Frege and Extended Frege systems (abstract). In Computer Science---Theory and Applications (Springer, 2015), 1--6. Google ScholarGoogle ScholarCross RefCross Ref
  5. Clarke, E.M., Biere, A., Raimi, R., Zhu, Y. Bounded model checking using satisfiability solving. Formal Methods in System Design 19, 1 (2001), 7--34. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Cook, S.A. The complexity of theorem-proving procedures. In STOC (1971), 151--158. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Copty, F., Fix, L., Fraer, R., Giunchiglia, E., Kamhi, G., Tacchella, A., Vardi, M.Y. Benefits of bounded model checking at an industrial setting. In CAV (Springer, 2001), 436--453. Google ScholarGoogle ScholarCross RefCross Ref
  8. Cruz-Filipe, L., Marques-Silva, J.P., Schneider-Kamp, P. Efficient certified resolution proof checking, 2016. https://arxiv.org/abs/1610.06984.Google ScholarGoogle Scholar
  9. de Moura, L., Bjørner, N. Satisfiability modulo theories: Introduction and applications. Communications of the ACM 54, 9 (2011), 69--77. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Eén, N., Sörensson, N. Temporal induction by incremental SAT solving. Electr. Notes Theor. Comput. Sci. 89, 4 (2003), 543--560. Google ScholarGoogle ScholarCross RefCross Ref
  11. Garey, M.R., Johnson, D.S. Computers and Intractability; A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979.Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Goldberg, E.I., Novikov, Y. Verification of proofs of unsatisfiability for CNF formulas. In DATE (IEEE, 2003), 10886--10891. Google ScholarGoogle ScholarCross RefCross Ref
  13. Graham, R.L., Spencer, J.H. Ramsey theory. Scientific American 263, 1 (July 1990), 112--117. Google ScholarGoogle ScholarCross RefCross Ref
  14. Heule, M.J.H., Hunt, W.A. Jr., Wetzler, N. Trimming while checking clausal proofs. In FMCAD (IEEE, 2013), 181--188. Google ScholarGoogle ScholarCross RefCross Ref
  15. Heule, M.J.H., Kullmann, O., Marek, V.W. Solving and verifying the Boolean Pythagorean Triples problem via Cube-and-Conquer. In SAT (Springer, 2016) 228--245. Google ScholarGoogle ScholarCross RefCross Ref
  16. Heule, M.J.H., Kullmann, O., Wieringa, S., Biere, A. Cube and conquer: Guiding CDCL SAT solvers by lookaheads. In HVC (Springer, 2011), 50--65.Google ScholarGoogle Scholar
  17. Heule, M.J.H., van Maaren, H. Look-ahead based SAT solvers. In Biere et al. [3], Chapter 5, (2009), 155--184.Google ScholarGoogle Scholar
  18. Ivančić, F., Yang, Z., Ganai, M.K., Gupta, A., Ashar P. Efficient SAT-based bounded model checking for software verification. Theoretical Computer Science 404, 3 (2008), 256--274. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Karp, R.M. Reducibility among combinatorial problems. In Complexity of Computer Computations (Plenum Press, 1972), 85--103. Google ScholarGoogle ScholarCross RefCross Ref
  20. Kautz, H.A., Sabharwal, A., Selman, B. Incomplete algorithms. In Biere et al. [3], Chapter 6, (2009), 185--203.Google ScholarGoogle Scholar
  21. Konev, B., Lisitsa, A. Computer-aided proof of Erdös discrepancy properties. Artificial Intelligence 224, C (July 2015), 103--118. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Kouril, M., Paul, J.L. The van der Waerden number W(2, 6) is 1132. Experimental Mathematics 17, 1 (2008), 53--61. Google ScholarGoogle ScholarCross RefCross Ref
  23. Kullmann, O. Fundaments of branching heuristics. In Biere et al. [3], Chapter 7, (2009), 205--244.Google ScholarGoogle Scholar
  24. Lam, C.W.H. The search for a finite projective plane of order 10. The American Mathematical Monthly 98, 4 (April 1991), 305--318. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Lamb, E. Maths proof smashes size record: Supercomputer produces a 200-terabyte proof---but is it really mathematics? Nature 534 (June 2016), 17--18. Google ScholarGoogle ScholarCross RefCross Ref
  26. Landman, B.M., Robertson, A. Ramsey Theory on the Integers, volume 24 of Student mathematical library. American Mathematical Society, Providence, RI, 2003.Google ScholarGoogle Scholar
  27. Lauria, M., Pudlák, P., Rödl, V., Thapen, N. The complexity of proving that a graph is Ramsey. In ICALP (Springer, 2013), 684--695. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Marques-Silva, J.P., Lynce, I., Malik, S. Conflict-driven clause learning SAT solvers. In Biere et al. [3], Chapter 4, (2009), 131--153.Google ScholarGoogle Scholar
  29. McGuire, G., Tugemann, B., Civario, G. There is no 16-clue Sudoku: Solving the Sudoku minimum number of clues problem via hitting set enumeration. Experimental Mathematics 23, 2 (2014), 190--217. Google ScholarGoogle ScholarCross RefCross Ref
  30. Myers, K.J. Computational advances in Rado numbers. PhD thesis, Rutgers University, New Brunswick, NJ, 2015.Google ScholarGoogle Scholar
  31. Oe, D., Stump, A., Oliver, C., Clancy, K. versat: A verified modern SAT solver. In VMCAI (Springer 2012) 363--378.Google ScholarGoogle Scholar
  32. Radziszowski, S.P. Small Ramsey numbers. The Electronic Journal of Combinatorics (January 2014), Dynamic Surveys DS1, Revision 14.Google ScholarGoogle Scholar
  33. Ramsey, F.P. On a problem of formal logic. Proceedings of the London Mathematical Society 30 (1930), 264--286. Google ScholarGoogle ScholarCross RefCross Ref
  34. Schur, I. Über die Kongruenz xm + ym = zm (mod p). Jahresbericht der Deutschen Mathematiker Vereinigung 25 (1917), 114--116.Google ScholarGoogle Scholar
  35. Tao, T. The Erdös discrepancy problem. Discrete Analysis 1 (February 2016), 29. Google ScholarGoogle ScholarCross RefCross Ref
  36. Voronkov, A. AVATAR: The architecture for first-order theorem provers. In CAV (Springer, 2014) 696--710. Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Wilson, R. Four Colors Suffice: How the Map Problem Was Solved. Princeton University Press, Princeton, NJ, revised edition, 2013.Google ScholarGoogle Scholar

Index Terms

  1. The science of brute force

          Recommendations

          Comments

          Login options

          Check if you have access through your login credentials or your institution to get full access on this article.

          Sign in

          Full Access

          • Published in

            cover image Communications of the ACM
            Communications of the ACM  Volume 60, Issue 8
            August 2017
            92 pages
            ISSN:0001-0782
            EISSN:1557-7317
            DOI:10.1145/3127343
            Issue’s Table of Contents

            Copyright © 2017 ACM

            Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

            Publisher

            Association for Computing Machinery

            New York, NY, United States

            Publication History

            • Published: 24 July 2017

            Permissions

            Request permissions about this article.

            Request Permissions

            Check for updates

            Qualifiers

            • review-article
            • Popular
            • Refereed

          PDF Format

          View or Download as a PDF file.

          PDF

          eReader

          View online with eReader.

          eReader

          HTML Format

          View this article in HTML Format .

          View HTML Format