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Machine Learning for First-Order Theorem Proving

Learning to Select a Good Heuristic

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Abstract

We applied two state-of-the-art machine learning techniques to the problem of selecting a good heuristic in a first-order theorem prover. Our aim was to demonstrate that sufficient information is available from simple feature measurements of a conjecture and axioms to determine a good choice of heuristic, and that the choice process can be automatically learned. Selecting from a set of 5 heuristics, the learned results are better than any single heuristic. The same results are also comparable to the prover’s own heuristic selection method, which has access to 82 heuristics including the 5 used by our method, and which required additional human expertise to guide its design. One version of our system is able to decline proof attempts. This achieves a significant reduction in total time required, while at the same time causing only a moderate reduction in the number of theorems proved. To our knowledge no earlier system has had this capability.

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References

  1. Bache, K., Lichman, M.: UCI Machine Learning Repository (2013).http://archive.ics.uci.edu/ml

  2. Baldi, P., Brunak, S., Chauvin, Y., Anderson, C.A.F., Nielsen, H.: Assessing the accuracy of prediction algorithms for classification: an overview. Bioinformatics 16(5), 412–424 (2000)

    Article  Google Scholar 

  3. Bishop, C.M.: Pattern Recognition and Machine Learning. Springer-Verlag (2006)

  4. Bridge, J.P.: Machine Learning and Automated Theorem Proving. Tech. Rep. UCAM-CL-TR-792, University of Cambridge, Computer Laboratory (2010). http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-792.pdf

  5. Chu, W., Ghahramani, Z., Falciani, F., Wild, D.L.: Biomarker discovery in microarray gene expression data with Gaussian processes. Bioinformatics 21(16), 3385–3393 (2005)

    Article  Google Scholar 

  6. Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM 5(7), 394–397 (1962). doi:10.1145/368273.368557

    Article  MathSciNet  MATH  Google Scholar 

  7. Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM 7(3), 201–215 (1960). doi:10.1145/321033.321034

    Article  MathSciNet  MATH  Google Scholar 

  8. Denzinger, J., Fuchs, M., Fuchs, M.: High performance ATP systems by combining several AI methods. In: Proceedings Fifteenth International Joint Conference on Artificial Intelligence (IJCAI) 1997, pp. 102–107. Morgan Kaufmann (1997)

  9. Denzinger, J., Fuchs, M., Goller, C., Schulz, S.: Learning from Previous Proof Experience. Technical Report AR99-4, Institut für Informatik, Technische Universität München (1999)

  10. Denzinger, J., Kronenburg, M., Schulz, S.: Discount - a distributed and learning equational prover. J. Autom. Reason. 18, 189–198 (1997). doi:10.1023/A:1005879229581

    Article  Google Scholar 

  11. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley (2000)

  12. Erkek, C.A.: Mixture of Experts Learning in Automated Theorem Proving. Master’s thesis, Bogazici University (2010)

  13. Fawcett, T.: An introduction to ROC analysis. Pattern Recogn. Lett. 27, 861–874 (2006)

    Article  Google Scholar 

  14. Fuchs, M.: Automatic selection of search-guiding heuristics for theorem proving. In: Proceedings of the 10th FLAIRS, pp. 1–5. Florida AI Research Society, Daytona Beach (1998)

  15. Fuchs, M., Fuchs, M.: Feature-based learning of search-guiding heuristics for theorem proving. AI Commun. 11(3–4), 175–189 (1998)

    Google Scholar 

  16. Goller, C.: Learning search-control heuristics for automated deduction systems with folding architecture networks. In: Proceedings European Symposium on Artificial Neural Networks. D-Facto publications (1999)

  17. Grimmett, G., Stirzaker, D.: Probability and Random Processes. Oxford University Press (2001)

  18. Guyon, I., Elisseeff, A.: An introduction to variable and feature selection. J. Mach. Learn. Res. 3, 1157–1182 (2003)

    MATH  Google Scholar 

  19. Haim, S., Walsh, T.: Online estimation of SAT solving runtime. In: Kleine Büning, H., Zhao, X. (eds.) Theory and Applications of Satisfiability Testing – SAT 2008, Lecture Notes in Computer Science, vol. 4996, pp. 133–138. Springer, Berlin (2008). doi:10.1007/978-3-540-79719-7_12

  20. Haim, S., Walsh, T.: Restart strategy selection using machine learning techniques. In: Kullmann, O. (ed.) Theory and Applications of Satisfiability Testing - SAT 2009, Lecture Notes in Computer Science, vol. 5584, pp. 312–325. Springer, Berlin (2009). doi:10.1007/978-3-642-02777-2_30

  21. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer Series in Statistics, 2nd edn. Springer (2009)

  22. He, H.: Learning from imbalanced data. IEEE Trans. Knowl. Data Eng. 21(9), 1263–1284 (2009)

    Article  Google Scholar 

  23. Hsu, C.W., Chang, C.C., Lin, C.J., et al.: A practical guide to support vector classification. Tech. rep., Department of Computer Science, National Taiwan University (2003)

  24. Huth, M., Ryan, M.: Logic in Computer Science: Modelling and Reasoning about Systems, 2nd edn. Cambridge University Press (2004)

  25. Joachims, T.: Making large-scale SVM learning practical. In: Schölkopf, B., Burges, C., Smola, A. (eds.) Advances in Kernel Methods - Support Vector Learning, chap. 11, pp. 169–184. MIT Press, Cambridge, MA (1999)

  26. Kadioglu, S., Malitsky, Y., Sabharwal, A., Samulowitz, H., Sellmann, M.: Algorithm selection and scheduling. In: Lee, J. (ed.) Principles and Practice of Constraint Programming – CP 2011, Lecture Notes in Computer Science, vol. 6876, pp. 454–469. Springer, Berlin (2011). doi:10.1007/978-3-642-23786-7_35

  27. Kohavi, R.: A study of cross-validation and bootstrap for accuracy estimation and model selection. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI’95), vol. 2, pp. 1137–1143. Morgan Kaufmann (1995)

  28. Lanckriet, G.R.G., Bie, T.D., Cristianini, N., Jordan, M.I., Noble, W.S.: A statistical framework for genomic data fusion. Bioinformatics 20(16), 2626–2635 (2004)

    Article  Google Scholar 

  29. Luenberger, D.G.: Linear and Nonlinear Programming. Kluwer (2003)

  30. McCune, W.: Prover9 and Mace4 (2005–2010). http://www.cs.unm.edu/~mccune/prover9/

  31. Mercer, J.: Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. R. Soc. Lond. 209, 415–446 (1909)

    Article  MATH  Google Scholar 

  32. Mitchell, T.: Machine Learning. McGraw Hill (1997)

  33. Morik, K., Brockhausen, P., Joachims, T.: Combining statistical learning with a knowledge-based approach – a case study in intensive care monitoring. In: International Conference on Machine Learning (ICML), pp. 268–277. Bled, Slowenien (1999)

  34. Nudelman, E., Leyton-Brown, K., Hoos, H., Devkar, A., Shoham, Y.: Understanding random SAT: Beyond the clauses-to-variables ratio. In: Wallace, M. (ed.) Principles and Practice of Constraint Programming – CP 2004, Lecture Notes in Computer Science, vol. 3258, pp. 438–452. Springer, Berlin (2004). doi:10.1007/978-3-540-30201-8_33

  35. Pilkington, N.C.V., Trotter, M.W.B., Holden, S.B.: Multiple kernel learning for drug discovery. Mol. Inform. 31(3–4), 313–322 (2012)

    Article  Google Scholar 

  36. Rasmussen, C.E., Williams, C.KI.: Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA (2006)

  37. Rosenblatt, F.: Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms. Spartan Books (1962)

  38. Samulowitz, H., Memisevic, R.: Learning to solve QBF. In: Proceedings of the 22nd National Conference on Artificial Intelligence - AAAI’07, vol. 1, pp. 255–260. AAAI Press (2007). http://dl.acm.org/citation.cfm?id=1619645.1619686

  39. Schulz, S.: Learning Search Control Knowledge for Equational Deduction. No. 230 in DISKI. Akademische Verlagsgesellschaft Aka GmbH Berlin (2000)

  40. Schulz, S.: E – a brainiac theorem prover. AI Commun. 15(2/3), 111–126 (2002)

    MATH  Google Scholar 

  41. Shawe-Taylor, J., Cristianini, N.: Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, Cambridge (2000)

  42. Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge (2004)

  43. Sutcliffe, G.: The TPTP problem library and associated infrastructure: the FOF and CNF parts, v3.5.0. J. Autom. Reason. 43(4), 337–362 (2009)

    Article  MATH  Google Scholar 

  44. Ting, K.M.: An instance-weighted method to induce cost-sensitive trees. IEEE Trans. Knowl. Data Eng. 14(3), 659–665 (2002)

    Article  Google Scholar 

  45. Urban, J.: MaLARea: a metasystem for automated reasoning in large theories. In: Urban, J., Sutcliffe, G., Schulz, S. (eds.) Proceedings of the CADE-21 Workshop on Empirically Successful Automated Reasoning in Large Theories, no. 257 in CEUR Workshop Proceedings, pp. 45–58 (2007)

  46. Williams, C.KI., Barber, D.: Bayesian classification with Gaussian processes. IEEE Trans Pattern. Anal. Mach. Intell. 20(12), 1342–1351 (1998)

    Article  Google Scholar 

  47. Xu, L., Hutter, F., Hoos, H., Leyton-Brown, K.: SATzilla: Portfolio-based algorithm selection for SAT. J. Artif. Intell. Res. 32, 565–606 (2008)

    MATH  Google Scholar 

  48. Xu, L., Hutter, F., Hoos, H., Leyton-Brown, K.: Features for SAT (2012). Available at www.cs.ubc.ca/labs/beta/Projects/SATzilla/

  49. Xu, L., Hutter, F., Shen, J., Hoos, H., Leyton-Brown, K.: Satzilla2012: improved algorithm slection based on cost-sensitive classification models. In: Balint, A., Belov, A., Diepold, D., Gerber, S., Järvisalo, M., Sinz, C. (eds.) Proceedings of SAT Challange 2012: Solver and Benchmark Descriptions, Department of Computer Science Series of Publications B, vol. B-2012-2, pp. 57–58. University of Helsinki (2012)

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Authors and Affiliations

  1. Computer Laboratory, University of Cambridge, William Gates Building, 15 JJ Thomson Avenue, Cambridge, CB3 0FD, UK

    James P. Bridge, Sean B. Holden & Lawrence C. Paulson

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  1. James P. Bridge
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  2. Sean B. Holden
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  3. Lawrence C. Paulson
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Correspondence to Sean B. Holden.

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Bridge, J.P., Holden, S.B. & Paulson, L.C. Machine Learning for First-Order Theorem Proving. J Autom Reasoning 53, 141–172 (2014). https://doi.org/10.1007/s10817-014-9301-5

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