Classification of Alignments Between Concepts of Formal Mathematical Systems
- 6 Citations
- 679 Downloads
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.
Notes
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.
References
- [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
- [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
- [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
- [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
- [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
- [Coq15]Coq Development Team: The Coq Proof Assistant: Reference Manual. Technical report, INRIA (2015)Google Scholar
- [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
- [ESC07]Euzenat, J., Shvaiko, P.: Ontology Matching. Springer, Heidelberg (2007)zbMATHGoogle Scholar
- [GC14]Ginev, D., Corneli, J.: Nnexus reloaded. In: Watt, et al. (eds.) [WDS+14], pp. 423–426Google Scholar
- [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
- [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
- [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
- [H+15]Hales, T.C., et al.: A formal proof of the Kepler conjecture. CoRR, abs/1501.02155 (2015)Google Scholar
- [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
- [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
- [KR14]Kaliszyk, C., Rabe, F.: Towards knowledge management for HOL light. In: Watt, et al. (eds.) [WDS+14], pp. 357–372Google Scholar
- [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
- [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
- [KU15]Kaliszyk, C., Urban, J.: HOL(y)Hammer: online ATP service for HOL light. Math. Comput. Sci. 9(1), 5–22 (2015)CrossRefzbMATHGoogle Scholar
- [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
- [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
- [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
- [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
- [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
- [PRA]Public repository for alignments. https://gl.mathhub.info/alignments/Public
- [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
- [RK13]Rabe, F., Kohlhase, M.: A scalable module system. Inf. Comput. 230(1), 1–54 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
- [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
- [Wie06]Wiedijk, F. (ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)Google Scholar
- [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