MACE4 and SEM: A Comparison of Finite Model Generators

  • Hantao Zhang
  • Jian Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7788)


This article has three objectives: (1) Promote Mace4, a program developed by Bill McCune that searches for finite models of first-order formulas and that is the best way to remember Bill. (2) Promote the research on model generation of first-order formulas. Mace4 remains one of the best model generation programs and we need newcomers who can take over Bill’s torch, because model generation is very important to automated reasoning and has many applications. (3) Compare Mace4 with SEM in detail so that the users of these tools or new model generator developers will understand the strengths and weaknesses of both systems and take advantage from this study.


Finite models constraint propagation backtracking search 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Audemard, G., Benhamou, B.: Reasoning by Symmetry and Function Ordering in Finite Model Generation. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, p. 226. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Audemard, G., Benhamou, B., Henocque, L.: Predicting and Detecting Symmetries in FOL Finite Model Search. Journal of Automated Reasoning 36(3), 177–212 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Audemard, G., Henocque, L.: The eXtended Least Number Heuristic. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 427–442. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Audemard, G., Simon, L.: Predicting learnt clauses quality in modern SAT solver. In: Twenty-First International Joint Conference on Artificial Intelligence, IJCAI 2009 (2009)Google Scholar
  5. 5.
    Baumgartner, P., Fuchs, A., De Nivelle, H., Tinelli, C.: Computing finite models by reduction to function-free clause logic. J. of Applied Logic 7(1), 58–74 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baumgartner, P., Tinelli, C.: The Model Evolution Calculus. In: Baader, F. (ed.) CADE-19. LNCS (LNAI), vol. 2741, pp. 350–364. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Benhamou, B., Henocque, L.: A new method for finite model search in equational theories: FMSET system. Fundamenta Informaticae 39(1,2), 21–38 (1999)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bennett, F.E., Du, B., Zhang, H.: Existence of conjugate orthogonal diagonal Latin squares. J. Combin. Designs 5, 449–461 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bennett, F.E., Du, B., Zhang, H.: Existence of self-orthogonal diagonal Latin squares with a missing subsquare. Discrete Math. 261, 69–86 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bennett, F.E., Zhu, L.: Conjugate-orthogonal Latin squares and related structures. In: Dinitz, J., Stinson, D. (eds.) Contemporary Design Theory: A Collection of Surveys, pp. 41–96. Wiley, New York (1992)Google Scholar
  11. 11.
    Boy de la Tour, T.: Up-to-Isomorphism Enumeration of Finite Models - The Monadic Case. In: Bonacina, M.P., Furbach, U. (eds.) International Workshop First-Order Theorem Proving (FTP 1997). RISC-Linz Report Series No. 97-50, pp. 29–33. Schloss Hagenberg by Linz, Austria (1997)Google Scholar
  12. 12.
    Boy de la Tour, T.: Some Techniques of Isomorph-Free Search. In: Campbell, J., Roanes-Lozano, E. (eds.) AISC 2000. LNCS (LNAI), vol. 1930, pp. 240–252. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Bürckert, H.-J., Herold, A., Kapur, D., Siekmann, J.H., Stickel, M., Tepp, M., Zhang, H.: Opening the AC-unification race. J. of Automated Reasoning (4), 465–474 (1988)Google Scholar
  14. 14.
    Burris, S., Yeats, K.: The saga of the high school identities. Algebra Universalis 52, 325–342 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Caferra, R., Leitsch, A., Peltier, N.: Automated Model Building. Applied Logic Series, vol. 31. Kluwer Academic Publisher (2004)Google Scholar
  16. 16.
    Caferra, R., Zabel, N.: Extending Resolution for Model Construction. In: van Eijck, J. (ed.) JELIA 1990. LNCS, vol. 478, pp. 153–169. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  17. 17.
    Claessen, K., Sörensson, N.: New techniques that improve Mace-style finite model finding. In: Model Computation – Principles, Algorithms, Applications, CADE-19 Workshop W4, Miami, Florida, USA (2003)Google Scholar
  18. 18.
    Dénes, J., Keedwell, A.D.: Latin squares and their applications. Academic Press, New York (1974)zbMATHGoogle Scholar
  19. 19.
    Du, B.: Self-orthogonal diagonal Latin square with missing subsquare. JCMCC 37, 193–203 (2001)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Een, N., Svrensson, N.: Minisat: A SAT solver with conflict-clause minimization. In: SAT 2005 (2005) (poster paper)Google Scholar
  21. 21.
    Fujita, M., Slaney, J., Bennett, F.: Automatic generation of some results in finite algebra. In: Proc. Int’l Joint Conf. on Artificial Intelligence (IJCAI 1993), pp. 52–57 (1993)Google Scholar
  22. 22.
    Galton, A.: Note on a lemma of Ladkin. Journal of Logic and Computation 6(1), 1–4 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hasegawa, R., Koshimura, M., Fujita, H.: MGTP: A Parallel Theorem Prover Based on Lazy Model Generation. In: Kapur, D. (ed.) CADE-11. LNCS, vol. 607, pp. 776–780. Springer, Heidelberg (1992)Google Scholar
  24. 24.
    Huang, Z., Zhang, H., Zhang, J.: Improving first-order model searching by propositional reasoning and lemma learning. In: The Seventh International Conference on Theory and Applications of Satisfiability Testing (SAT 2004), Vancouver, BC, Canada (May 2004)Google Scholar
  25. 25.
    Jia, X., Zhang, J.: A Powerful Technique to Eliminate Isomorphism in Finite Model Search. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 318–331. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    Kapur, D., Zhang, H.: An Overview of RRL: Rewrite Rule Laboratory. In: Dershowitz, N. (ed.) RTA 1989. LNCS, vol. 355, pp. 513–529. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  27. 27.
    Kim, S., Zhang, H.: ModGen: Theorem proving by model generation. In: Proc. of National Conference of American Association on Artificial Intelligence (AAAI 1994), Seattle, WA, pp. 162–167. MIT Press (1994)Google Scholar
  28. 28.
    Kumar, V.: Algorithms for constraint satisfaction problems: A survey. AI Magazine 13(1), 32–44 (1992)Google Scholar
  29. 29.
    Kunen, K.: The structure of conjugacy closed loops. Transactions of the American Mathematical Society 352(6), 2889–2911 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Leech, J.: Skew lattices in rings. Algebra Universalis 26, 48–72 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Manthey, R., Bry, F.: SATCHMO: A Theorem Prover Implemented in Prolog. In: Lusk, E., Overbeek, R. (eds.) CADE 1988. LNCS, vol. 310, pp. 415–434. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  32. 32.
    McCune, W.: Experiments with discrimination tree indexing and path indexing for term retrieval. J. of Automated Reasoning 9(2), 147–167 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    McCune, W.: MACE 2.0 Reference Manual and Guide. Technical Memorandum No. 249, ANL/MCS-TM-249, Argonne National Lab, Argonne, IL, USA (1994),
  34. 34.
    McCune, W.: Otter 3.3 Reference Manual, Technical Memorandum No. 263, Argonne National Laboratory, Argonne, IL, USA (August 2003),
  35. 35.
    McCune, W.: Mace4 reference manual and guide, Technical Memorandum No. 264, Argonne National Laboratory, Argonne, IL, USA (August 2003),
  36. 36.
    McCune, W.: Library for Automated Deduction Research (2009),
  37. 37.
    McCune, W.: Solution of the Robbins problem. J. of Automated Reasoning 19(3), 263–276 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    McCune, W., Henschen, L.J.: Experiments with semantic paramodulation. J. of Automated Reasoning 1(3), 231–261 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. J. ACM 27(2), 356–364 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Pichler, R.: Algorithms on Atomic Representations of Herbrand Models. In: Dix, J., Fariñas del Cerro, L., Furbach, U. (eds.) JELIA 1998. LNCS (LNAI), vol. 1489, pp. 199–215. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  41. 41.
    Slaney, J.: Finder: Finite Domain Enumerator. In: Bundy, A. (ed.) CADE-12. LNCS, vol. 814, pp. 798–801. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  42. 42.
    Slaney, J., Fujita, M., Stickel, M.: Automated reasoning and exhaustive search: Quasigroup existence problems. Computers & Math. with Appl. 29(2), 115–132 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Slaney, J., Lusk, E.L., McCune, W.: SCOTT: Semantically Constrained Otter (System Description). In: Bundy, A. (ed.) CADE-12. LNCS, vol. 814, pp. 764–768. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  44. 44.
    Spinks, M.: On middle distributivity for Skew lattices. Semigroup Forum 61(3), 341–345 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stein, S.K.: On the foundations of quasigroups. Trans. Amer. Math. Soc. 85, 228–256 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Tammet, T.: Using Resolution for Deciding Solvable Classes and Building Finite Models. In: Barzdins, J., Bjorner, D. (eds.) Baltic Computer Science. LNCS, vol. 502, pp. 33–64. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  47. 47.
    Tammet, T.: Finite model building: improvements and comparisons. In: Model Computation – Principles, Algorithms, Applications, CADE-19 Workshop W4, Miami, Florida, USA (2003)Google Scholar
  48. 48.
    Zhang, H.: Sato: An Efficient Propositional Prover. In: McCune, W. (ed.) CADE-14. LNCS, vol. 1249, pp. 272–275. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  49. 49.
    Zhang, H.: Specifying Latin squares in propositional logic. In: Veroff, R. (ed.) Automated Reasoning and Its Applications, Essays in Honor of Larry Wos. MIT Press (1997)Google Scholar
  50. 50.
    Zhang, H.: Combinatorial designs by SAT solvers. In: Biere, A., Heule, M., Van Haaren, H., Walsh, T. (eds.) Handbook of Satisfiability, ch. 17. IOS Press (2009)Google Scholar
  51. 51.
    Zhang, H., Stickel, M.: Implementing the Davis-Putnam method. J. of Automated Reasoning 24, 277–296 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Zhang, J.: Problems on the Generation of Finite Models. In: Bundy, A. (ed.) CADE-12. LNCS, vol. 814, pp. 753–757. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  53. 53.
    Zhang, J.: Constructing finite algebras with Falcon. J. of Automated Reasoning 17(1), 1–22 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Zhang, J.: On the relational translation method for propositional modal logics. Technical Report ISCAS-LCS-96-12, Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences (December 1996)Google Scholar
  55. 55.
    Zhang, J.: Showing the independence of an axiom for temporal intervals by model generation. Association for Automated Reasoning Newsletter, No. 40 (1998),
  56. 56.
    Zhang, J.: System Description: MCS: Model-Based Conjecture Searching. In: Ganzinger, H. (ed.) CADE-16. LNCS (LNAI), vol. 1632, pp. 393–397. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  57. 57.
    Zhang, J.: Test problem and Perl scripts for finite model searching. Association for Automated Reasoning Newsletter, No. 47 (April 2000),
  58. 58.
    Zhang, J.: Computer Search for Counterexamples to Wilkie’s Identity. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 441–451. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  59. 59.
    Zhang, J., Zhang, H.: SEM: a system for enumerating models. In: Proc. 14th Int’l Joint Conf. on Artif. Intel. (IJCAI), pp. 298–303 (1995)Google Scholar
  60. 60.
    Zhang, J., Zhang, H.: Constraint Propagation in Model Generation. In: Montanari, U., Rossi, F. (eds.) CP 1995. LNCS, vol. 976, pp. 398–414. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  61. 61.
    Zhang, J., Zhang, H.: Extending Finite Model Searching with Congruence Closure Computation. In: Buchberger, B., Campbell, J. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 94–102. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  62. 62.
    Zhang, X.: Incomplete perfect Mendelsohn designs with block size four. Discrete Mathematics 254, 565–597 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Hantao Zhang
    • 1
  • Jian Zhang
    • 2
  1. 1.Department of Computer ScienceThe University of IowaIowa CityU.S.A.
  2. 2.State Key Laboratory of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingChina

Personalised recommendations