Abstract
Large formal mathematical libraries consist of millions of atomic inference steps that give rise to a corresponding number of proved statements (lemmas). Analogously to the informal mathematical practice, only a tiny fraction of such statements is named and re-used in later proofs by formal mathematicians. In this work, we suggest and implement criteria defining the estimated usefulness of the HOL Light lemmas for proving further theorems. We use these criteria to mine the large inference graph of all lemmas in the core HOL Light library, adding thousands of the best lemmas to the pool of named statements that can be re-used in later proofs. The usefulness of the new lemmas is then evaluated by comparing the performance of automated proving of the core HOL Light theorems with and without such added lemmas.
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References
Alama, J., Heskes, T., Kühlwein, D., Tsivtsivadze, E., Urban, J.: Premise selection for mathematics by corpus analysis and kernel methods. Journal of Automated Reasoning (2013), http://dx.doi.org/10.1007/s10817-013-9286-5
Alama, J., Kühlwein, D., Urban, J.: Automated and Human Proofs in General Mathematics: An Initial Comparison. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18 2012. LNCS, vol. 7180, pp. 37–45. Springer, Heidelberg (2012)
Avrachenkov, K., Dobrynin, V., Nemirovsky, D., Pham, S.K., Smirnova, E.: Pagerank based clustering of hypertext document collections. In: Myaeng, S.-H., Oard, D.W., Sebastiani, F., Chua, T.-S., Leong, M.-K. (eds.) SIGIR, pp. 873–874. ACM (2008)
Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.): ITP 2013. LNCS, vol. 7998. Springer, Heidelberg (2013)
Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. Journal of Formalized Reasoning 3(2), 153–245 (2010)
Hales, T.C.: Introduction to the Flyspeck project. In: Coquand, T., Lombardi, H., Roy, M.-F. (eds.) Dagstuhl Seminar Proceedings, vol. 05021. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl (2005)
Harrison, J.: HOL Light: A tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)
Hoder, K., Voronkov, A.: Sine qua non for large theory reasoning. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 299–314. Springer, Heidelberg (2011)
Kaliszyk, C., Krauss, A.: Scalable LCF-style proof translation. In: Blazy, et al. (eds.) [4], pp. 51–66
Kaliszyk, C., Urban, J.: Learning-assisted automated reasoning with Flyspeck. CoRR, abs/1211.7012 (2012)
Kaliszyk, C., Urban, J.: Automated reasoning service for HOL Light. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS (LNAI), vol. 7961, pp. 120–135. Springer, Heidelberg (2013)
Kaliszyk, C., Urban, J.: Stronger automation for Flyspeck by feature weighting and strategy evolution. In: Blanchette, J.C., Urban, J. (eds.) PxTP 2013. EPiC Series, vol. 14, pp. 87–95. EasyChair (2013)
Kühlwein, D., Blanchette, J.C., Kaliszyk, C., Urban, J.: MaSh: Machine learning for Sledgehammer. In: Blazy, et al. (eds.) [4], pp. 35–50
Kühlwein, D., Schulz, S., Urban, J.: E-MaLeS 1.1. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 407–413. Springer, Heidelberg (2013)
Kühlwein, D., van Laarhoven, T., Tsivtsivadze, E., Urban, J., Heskes, T.: Overview and evaluation of premise selection techniques for large theory mathematics. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 378–392. Springer, Heidelberg (2012)
McCune, W.: Prover9 and Mace4. 2005–2010, http://www.cs.unm.edu/~mccune/prover9/
Meng, J., Paulson, L.C.: Translating higher-order clauses to first-order clauses. J. Autom. Reasoning 40(1), 35–60 (2008)
Page, L., Brin, S., Motwani, R., Winograd, T.: The PageRank citation ranking: Bringing order to the Web. Technical report, Stanford Digital Library Technologies Project (1998)
Pudlák, P.: Search for faster and shorter proofs using machine generated lemmas. In: Sutcliffe, G., Schmidt, R., Schulz, S. (eds.) 3rd International Joint Conference on Automated Reasoning, Proceedings of the FLoC 2006 Workshop on Empirically Successful Computerized Reasoning. CEUR Workshop Proceedings, vol. 192, pp. 34–52 (2006)
Puzis, Y., Gao, Y., Sutcliffe, G.: Automated generation of interesting theorems. In: Sutcliffe, G., Goebel, R. (eds.) FLAIRS Conference, pp. 49–54. AAAI Press (2006)
Riazanov, A., Voronkov, A.: The design and implementation of VAMPIRE. AI Commun. 15(2-3), 91–110 (2002)
Schulz, S.: Learning search control knowledge for equational deduction. DISKI, vol. 230. Infix Akademische Verlagsgesellschaft (2000)
Schulz, S.: E - A Brainiac Theorem Prover. AI Commun. 15(2-3), 111–126 (2002)
Sutcliffe, G.: The Design and Implementation of a Compositional Competition-Cooperation Parallel ATP System. In: de Nivelle, H., Schulz, S. (eds.) Proceedings of the 2nd International Workshop on the Implementation of Logics, pp. 92–102. MPI-I-2001-2-006 in Max-Planck-Institut für Informatik. Research Report (2001)
Sutcliffe, G., Puzis, Y.: SRASS - a semantic relevance axiom selection system. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 295–310. Springer, Heidelberg (2007)
Urban, J.: MPTP - Motivation, Implementation, First Experiments. Journal of Automated Reasoning 33(3-4), 319–339 (2004)
Urban, J.: MoMM - fast interreduction and retrieval in large libraries of formalized mathematics. Int. J. on Artificial Intelligence Tools 15(1), 109–130 (2006)
Urban, J.: BliStr: The Blind Strategymaker. CoRR, abs/1301.2683 (2013)
Urban, J., Sutcliffe, G., Pudlák, P., Vyskočil, J.: MaLARea SG1 - Machine Learner for Automated Reasoning with Semantic Guidance. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 441–456. Springer, Heidelberg (2008)
Urban, J., Vyskočil, J.: Theorem proving in large formal mathematics as an emerging AI field. In: Bonacina, M.P., Stickel, M.E. (eds.) McCune Festschrift. LNCS (LNAI), vol. 7788, pp. 240–257. Springer, Heidelberg (2013)
Urban, J., Vyskočil, J., Štěpánek, P.: MaLeCoP: Machine learning connection prover. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS (LNAI), vol. 6793, pp. 263–277. Springer, Heidelberg (2011)
Veroff, R.: Using hints to increase the effectiveness of an automated reasoning program: Case studies. J. Autom. Reasoning 16(3), 223–239 (1996)
Wenzel, M., Paulson, L.C., Nipkow, T.: The Isabelle framework. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 33–38. Springer, Heidelberg (2008)
Wos, L., Overbeek, R., Lusk, E.L., Boyle, J.: Automated Reasoning: Introduction and Applications. Prentice-Hall (1984)
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Kaliszyk, C., Urban, J. (2013). Lemma Mining over HOL Light . In: McMillan, K., Middeldorp, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2013. Lecture Notes in Computer Science, vol 8312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45221-5_34
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DOI: https://doi.org/10.1007/978-3-642-45221-5_34
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