Classification of Alignments Between Concepts of Formal Mathematical Systems

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10383)

Abstract

Mathematical knowledge is publicly available in dozens of different formats and languages, ranging from informal (e.g. Wikipedia) to formal corpora (e.g., Mizar). Despite an enormous amount of overlap between these corpora, only few machine-actionable connections exist. We speak of alignment if the same concept occurs in different libraries, possibly with slightly different names, notations, or formal definitions. Leveraging these alignments creates a huge potential for knowledge sharing and transfer, e.g., integrating theorem provers or reusing services across systems. Notably, even imperfect alignments, i.e. concepts that are very similar rather than identical, can often play very important roles. Specifically, in machine learning techniques for theorem proving and in automation techniques that use these, they allow learning-reasoning based automation for theorem provers to take inspiration from proofs from different formal proof libraries or semi-formal libraries even if the latter is based on a different mathematical foundation. We present a classification of alignments and design a simple format for describing alignments, as well as an infrastructure for sharing them. We propose these as a centralized standard for the community. Finally, we present an initial collection of \(\approx \)12000 alignments from the different kinds of mathematical corpora, including proof assistant libraries and semi-formal corpora as a public resource.

References

  1. [ACTZ06]
    Asperti, A., Coen, C.S., Tassi, E., Zacchiroli, S.: Crafting a proof assistant. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 18–32. Springer, Heidelberg (2007). doi:10.1007/978-3-540-74464-1_2 CrossRefGoogle Scholar
  2. [AGC+04]
    Asperti, A., Guidi, F., Coen, C.S., Tassi, E., Zacchiroli, S.: A content based mathematical search engine: Whelp. In: Filliâtre, J.-C., Paulin-Mohring, C., Werner, B. (eds.) TYPES 2004. LNCS, vol. 3839, pp. 17–32. Springer, Heidelberg (2006). doi:10.1007/11617990_2 CrossRefGoogle Scholar
  3. [BFMP11]
    Bobot, F., Filliâtre, J., Marché, C., Paskevich, A.: Why3: shepherd your herd of provers. In: Boogie 2011: First International Workshop on Intermediate Verification Languages, pp. 53–64 (2011)Google Scholar
  4. [CHK+11]
    Codescu, M., Horozal, F., Kohlhase, M., Mossakowski, T., Rabe, F.: Project abstract: logic atlas and integrator (LATIN). In: Davenport, J.H., Farmer, W.M., Urban, J., Rabe, F. (eds.) CICM 2011. LNCS, vol. 6824, pp. 289–291. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22673-1_24 CrossRefGoogle Scholar
  5. [CMK14]
    Codescu, M., Mossakowski, T., Kutz, O.: A categorical approach to ontology alignment. In: Proceedings of the 9th International Conference on Ontology Matching, pp. 1–12. CEUR-WS.org (2014)Google Scholar
  6. [Coq15]
    Coq Development Team: The Coq Proof Assistant: Reference Manual. Technical report, INRIA (2015)Google Scholar
  7. [DESTdS11]
    David, J., Euzenat, J., Scharffe, F., Trojahn dos Santos, C.: The alignment API 4.0. Semant. Web 2(1), 3–10 (2011)Google Scholar
  8. [ESC07]
    Euzenat, J., Shvaiko, P.: Ontology Matching. Springer, Heidelberg (2007)MATHGoogle Scholar
  9. [GC14]
    Ginev, D., Corneli, J.: Nnexus reloaded. In: Watt, et al. (eds.) [WDS+14], pp. 423–426Google Scholar
  10. [GK14]
    Gauthier, T., Kaliszyk, C.: Matching concepts across HOL libraries. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.) CICM 2014. LNCS, vol. 8543, pp. 267–281. Springer, Cham (2014). doi:10.1007/978-3-319-08434-3_20 CrossRefGoogle Scholar
  11. [GK15]
    Gauthier, T., Kaliszyk, C.: Sharing HOL4 and HOL Light proof knowledge. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 372–386. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48899-7_26 CrossRefGoogle Scholar
  12. [GKU16]
    Gauthier, T., Kaliszyk, C., Urban, J.: Initial experiments with statistical conjecturing over large formal corpora. In: Kohlhase, A., et al. (eds.) Work in Progress at CICM 2016. CEUR, vol. 1785, pp. 219–228. CEUR-WS.org (2016)Google Scholar
  13. [H+15]
    Hales, T.C., et al.: A formal proof of the Kepler conjecture. CoRR, abs/1501.02155 (2015)Google Scholar
  14. [Hur09]
    Hurd, J.: OpenTheory: package management for higher order logic theories. In: Reis, G.D., Théry, L. (eds.) Programming Languages for Mechanized Mathematics Systems, pp. 31–37. ACM (2009)Google Scholar
  15. [KK13]
    Kaliszyk, C., Krauss, A.: Scalable LCF-style proof translation. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 51–66. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39634-2_7 CrossRefGoogle Scholar
  16. [KR14]
    Kaliszyk, C., Rabe, F.: Towards knowledge management for HOL light. In: Watt, et al. (eds.) [WDS+14], pp. 357–372Google Scholar
  17. [KR16]
    Kohlhase, M., Rabe, F.: QED reloaded: towards a pluralistic formal library of mathematical knowledge. J. Formalized Reason. 9(1), 201–234 (2016)MathSciNetGoogle Scholar
  18. [KS10]
    Krauss, A., Schropp, A.: A mechanized translation from higher-order logic to set theory. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 323–338. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14052-5_23 CrossRefGoogle Scholar
  19. [KU15]
    Kaliszyk, C., Urban, J.: HOL(y)Hammer: online ATP service for HOL light. Math. Comput. Sci. 9(1), 5–22 (2015)CrossRefMATHGoogle Scholar
  20. [KW10]
    Keller, C., Werner, B.: Importing HOL light into Coq. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 307–322. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14052-5_22 CrossRefGoogle Scholar
  21. [MGK+17]
    Müller, D., Gauthier, T., Kaliszyk, C., Kohlhase, M., Rabe, F.: Classification of alignments between concepts of formal mathematical systems. Technical report (2017)Google Scholar
  22. [NSM01]
    Naumov, P., Stehr, M.-O., Meseguer, J.: The HOL/NuPRL proof translator – A practical approach to formal interoperability. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 329–345. Springer, Heidelberg (2001). doi:10.1007/3-540-44755-5_23 CrossRefGoogle Scholar
  23. [ORS92]
    Owre, S., Rushby, J.M., Shankar, N.: PVS: a prototype verification system. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 748–752. Springer, Heidelberg (1992). doi:10.1007/3-540-55602-8_217 Google Scholar
  24. [OS06]
    Obua, S., Skalberg, S.: Importing HOL into Isabelle/HOL. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS, vol. 4130, pp. 298–302. Springer, Heidelberg (2006). doi:10.1007/11814771_27 CrossRefGoogle Scholar
  25. [PRA]
    Public repository for alignments. https://gl.mathhub.info/alignments/Public
  26. [Rab13]
    Rabe, F.: The MMT API: a generic MKM system. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS, vol. 7961, pp. 339–343. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39320-4_25 CrossRefGoogle Scholar
  27. [RK13]
    Rabe, F., Kohlhase, M.: A scalable module system. Inf. Comput. 230(1), 1–54 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. [WDS+14]
    Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.): CICM 2014. LNCS, vol. 8543. Springer, Cham (2014)Google Scholar
  29. [Wie06]
    Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)Google Scholar
  30. [WPN08]
    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). doi:10.1007/978-3-540-71067-7_7

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.FAU Erlangen-NürnbergErlangenGermany
  2. 2.University of InnsbruckInnsbruckAustria
  3. 3.Jacobs UniversityBremenGermany

Personalised recommendations