Proof Guidance in PVS with Sequential Pattern Mining

  • M. Saqib NawazEmail author
  • Meng Sun
  • Philippe Fournier-Viger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11761)


The recent introduction of the big data paradigm and advancements in machine learning and deep mining techniques have made proof guidance and automation in interactive theorem provers (ITPs) an important research topic. In this paper, we provide a learning approach based on sequential pattern mining (SPM) for proof guidance in the PVS proof assistant. Proofs in a PVS theory are first abstracted to a computer-processable corpus. SPM techniques are then used on the corpus to discover frequent proof steps and proof patterns, relationships of proof steps / patterns with each other, dependency of new conjectures on already proved facts and to predict the next proof step(s). Obtained results suggest that the integration of SPM in proof assistants can be used to guide the proof process and in the development of proof tactics/strategies.


PVS Proof development process Proof corpus Frequent patterns Sequential pattern mining 



The work has been supported by the National Natural Science Foundation of China under grant no. 61772038, 61532019 and 61272160, and the Guandong Science and Technology Department (Grant no. 2018B010107004).


  1. 1.
    Alama, J., Heskes, T., Kühlwein, D., Tsivtsivadze, E., Urban, J.: Premise selection for mathematics by corpus analysis and kernel methods. J. Autom. Reasoning 52(2), 191–213 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arbab, F.: Reo: a channel-based coordination model for component composition. Math. Struct.Comput. Sci. 14(3), 329–366 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baier, C.: Probabilistic models for Reo connector circuits. J. Univ. Comput. Sci. 11(10), 1718–1748 (2005)Google Scholar
  4. 4.
    Baier, C., Wolf, V.: Stochastic reasoning about channel-based component connectors. In: Ciancarini, P., Wiklicky, H. (eds.) COORDINATION 2006. LNCS, vol. 4038, pp. 1–15. Springer, Heidelberg (2006). Scholar
  5. 5.
    Blanchette, J.C., Haslbeck, M., Matichuk, D., Nipkow, T.: Mining the archive of formal proofs. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS (LNAI), vol. 9150, pp. 3–17. Springer, Cham (2015). Scholar
  6. 6.
    Bridge, J.P., Holden, S.B., Paulson, L.C.: Machine learning for first-order theorem proving - learning to select a good heuristic. J. Autom. Reasoning 53(2), 141–172 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Duncan, H.: The use of data-mining for the automatic formation of tactics. Ph.D. thesis, University of Edinburgh, UK (2007)Google Scholar
  8. 8.
    Färber, M., Brown, C.: Internal guidance for satallax. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 349–361. Springer, Cham (2016). Scholar
  9. 9.
    Färber, M., Kaliszyk, C., Urban, J.: Monte carlo tableau proof search. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 563–579. Springer, Cham (2017). Scholar
  10. 10.
    Fournier-Viger, P., Gomariz, A., Campos, M., Thomas, R.: Fast vertical mining of sequential patterns using co-occurrence information. In: Tseng, V.S., Ho, T.B., Zhou, Z.-H., Chen, A.L.P., Kao, H.-Y. (eds.) PAKDD 2014. LNCS (LNAI), vol. 8443, pp. 40–52. Springer, Cham (2014). Scholar
  11. 11.
    Fournier-Viger, P., Gomariz, A., Gueniche, T., Mwamikazi, E., Thomas, R.: TKS: efficient mining of top-k sequential patterns. In: Motoda, H., Wu, Z., Cao, L., Zaiane, O., Yao, M., Wang, W. (eds.) ADMA 2013. LNCS (LNAI), vol. 8346, pp. 109–120. Springer, Heidelberg (2013). Scholar
  12. 12.
    Fournier-Viger, P., Gueniche, T., Zida, S., Tseng, V.S.: ERMiner: sequential rule mining using equivalence classes. In: Blockeel, H., van Leeuwen, M., Vinciotti, V. (eds.) IDA 2014. LNCS, vol. 8819, pp. 108–119. Springer, Cham (2014). Scholar
  13. 13.
    Fournier-Viger, P., et al.: The SPMF open-source data mining library version 2. In: Berendt, B., et al. (eds.) ECML PKDD 2016. LNCS (LNAI), vol. 9853, pp. 36–40. Springer, Cham (2016). Scholar
  14. 14.
    Fournier-Viger, P., Lin, J.C.W., Kiran, R.U., Koh, Y.S., Thomas, R.: A survey of sequential pattern mining. Data Sci. Pattern Recogn. 1(1), 54–77 (2017)Google Scholar
  15. 15.
    Gauthier, T., Kaliszyk, C.: Premise selection and external provers for HOL4. In: Proceedings of CPP 2015, pp. 48–57. ACM (2015)Google Scholar
  16. 16.
    Gauthier, T., Kaliszyk, C., Urban, J.: TacticToe: learning to reason with HOL4 tactics. In: Proceedings of LPAR 2017. EPiC Series in Computing, vol. 46, pp. 125–143 (2017)Google Scholar
  17. 17.
    Goertzel, Z., Jakubův, J., Schulz, S., Urban, J.: ProofWatch: watchlist guidance for large theories in E. In: Avigad, J., Mahboubi, A. (eds.) ITP 2018. LNCS, vol. 10895, pp. 270–288. Springer, Cham (2018). Scholar
  18. 18.
    Gueniche, T., Fournier-Viger, P., Raman, R., Tseng, V.S.: CPT+: decreasing the time/space complexity of the compact prediction tree. In: Cao, T., Lim, E.-P., Zhou, Z.-H., Ho, T.-B., Cheung, D., Motoda, H. (eds.) PAKDD 2015. LNCS (LNAI), vol. 9078, pp. 625–636. Springer, Cham (2015). Scholar
  19. 19.
    Harrison, J., Urban, J., Wiedijk, F.: History of interactive theorem proving. In: Computational Logic. Handbook of the History of Logic, vol. 9, pp. 135–214. Elsevier (2014)Google Scholar
  20. 20.
    Hasan, O., Tahar, S.: Formal verification methods. In: Encyclopedia of Information Science and Technology, 3rd edn, pp. 7162–7170. IGI Global (2015)Google Scholar
  21. 21.
    Irving, G., Szegedy, C., Alemi, A.A., Eén, N., Chollet, F., Urban, J.: Deepmath - Deep sequence models for premise selection. In: Proceedings of NIPS 2016, pp. 2243–2251. ACM (2016)Google Scholar
  22. 22.
    Kaliszyk, C., Chollet, F., Szegedy, C.: Holstep: a machine learning dataset for higher-order logic theorem proving. Proc. ICLR 2017, 1–12 (2017)Google Scholar
  23. 23.
    Kaliszyk, C., Mamane, L., Urban, J.: Machine learning of Coq proof guidance: first experiments. In: Proceedings of SCSS 2014. EPiC Series in Computing, vol. 30, pp. 27–34 (2014)Google Scholar
  24. 24.
    Kaliszyk, C., Urban, J.: FEMaLeCoP: fairly efficient machine learning connection prover. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 88–96. Springer, Heidelberg (2015). Scholar
  25. 25.
    Kaliszyk, C., Urban, J.: Hol(y)Hammer: Online ATP service for HOL light. Math. Comput. Sci. 9(1), 5–22 (2015)CrossRefGoogle Scholar
  26. 26.
    Kaliszyk, C., Urban, J., Michalewski, H., Olsák, M.: Reinforcement learning of theorem proving. Proc. NeurIPS 2018, 8836–8847 (2018)Google Scholar
  27. 27.
    Kühlwein, D., Urban, J.: MaLeS: a framework for automatic tuning of automated theorem provers. J. Autom. Reasoning 55(2), 91–116 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Loos, S.M., Irving, G., Szegedy, C., Kaliszyk, C.: Deep network guided proof search. In: Proceedings of LPAR 2017. EPiC Series in Computing, vol. 46, pp. 85–105 (2017)Google Scholar
  29. 29.
    Nawaz, M.S., Sun, M.: Reo2PVS: formal specification and verification of component connectors. In: Proceedings of SEKE 2018, pp. 391–396. KSI Research Inc. (2018)Google Scholar
  30. 30.
    Owre, S., Shankar, N., Rushby, J.M., Stringer-Calvert, D.W.J.: PVS system Guide, PVS prover Guide. PVS language reference. Technical report, SRI International, November 2001Google Scholar
  31. 31.
  32. 32.
    Russell, S.J., Norvig, P.: Artificial Intelligence - A Modern Approach, 3rd edn. Pearson Education, Upper Saddle River (2010)zbMATHGoogle Scholar
  33. 33.
    Whalen, D.. Holophrasm: a neural automated theorem prover for higher-order logic. CoRR, abs/1608.02644 2016Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2019

Authors and Affiliations

  • M. Saqib Nawaz
    • 1
    Email author
  • Meng Sun
    • 1
  • Philippe Fournier-Viger
    • 2
  1. 1.LMAM and Department of Informatics, School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.School of Humanities and Social SciencesHarbin Institute of Technology (Shenzhen)ShenzhenChina

Personalised recommendations