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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
For simplicity in the remaining part of the paper we will not give complete HTTP links, but rather use single keyword abbreviations. Complete names of logics and modules are given in the online service.
- 2.
References
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
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
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)
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
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)
Coq Development Team: The Coq Proof Assistant: Reference Manual. Technical report, INRIA (2015)
David, J., Euzenat, J., Scharffe, F., Trojahn dos Santos, C.: The alignment API 4.0. Semant. Web 2(1), 3–10 (2011)
Euzenat, J., Shvaiko, P.: Ontology Matching. Springer, Heidelberg (2007)
Ginev, D., Corneli, J.: Nnexus reloaded. In: Watt, et al. (eds.) [WDS+14], pp. 423–426
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
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
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)
Hales, T.C., et al.: A formal proof of the Kepler conjecture. CoRR, abs/1501.02155 (2015)
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)
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
Kaliszyk, C., Rabe, F.: Towards knowledge management for HOL light. In: Watt, et al. (eds.) [WDS+14], pp. 357–372
Kohlhase, M., Rabe, F.: QED reloaded: towards a pluralistic formal library of mathematical knowledge. J. Formalized Reason. 9(1), 201–234 (2016)
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
Kaliszyk, C., Urban, J.: HOL(y)Hammer: online ATP service for HOL light. Math. Comput. Sci. 9(1), 5–22 (2015)
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
Müller, D., Gauthier, T., Kaliszyk, C., Kohlhase, M., Rabe, F.: Classification of alignments between concepts of formal mathematical systems. Technical report (2017)
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
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
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
Public repository for alignments. https://gl.mathhub.info/alignments/Public
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
Rabe, F., Kohlhase, M.: A scalable module system. Inf. Comput. 230(1), 1–54 (2013)
Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds.): CICM 2014. LNCS, vol. 8543. Springer, Cham (2014)
Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)
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
Acknowledgements
We were supported by the German Science Foundation (DFG) under grants KO 2428/13-1 and RA-1872/3-1, the Austrian Science Fund (FWF) grant P26201, and the ERC starting grant no. 714034 SMART.
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Müller, D., Gauthier, T., Kaliszyk, C., Kohlhase, M., Rabe, F. (2017). Classification of Alignments Between Concepts of Formal Mathematical Systems. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds) Intelligent Computer Mathematics. CICM 2017. Lecture Notes in Computer Science(), vol 10383. Springer, Cham. https://doi.org/10.1007/978-3-319-62075-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-62075-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62074-9
Online ISBN: 978-3-319-62075-6
eBook Packages: Computer ScienceComputer Science (R0)