Abstract
Even though only a tiny fraction of mathematical knowledge is available digitally (e.g in theorem prover or computer algebra libraries, in documents with content-markup), our current retrieval methods are already inadequate. With further increase in digitalization of mathematics the situation will get worse without significant advances.
When searching a formula, we often want to find not only structurally identical occurrences, but also all (logically) equivalent ones. Furthermore, we want to retrieve whole mathematical theories (i.e. objects with prescribed properties), and we want to find them irrespective of the nomenclature chosen in the respective formalization.
In this paper, we propose a normalization-based approach to mathematical formula and theory retrieval modulo an equivalence theory and concept renaming, and apply the proposed algorithm to end-user querying and knowledge sharing. We test the implementation by applying it to a first-order translation of the Mizar library.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bancerek, G.: On the structure of Mizar types. Electronic Notes in Theoretical Computer Science, 85(7) (2003)
Bancerek, G., Kohlhase, M.: The mizar mathematical library in omdoc (submitted, 2007)
Robert, L., Constable, S., Allen, H., Bromly, W., Cleaveland, J., Cremer, R., Harper, D., Howe, T., Knoblock, N., Mendler, P., Panangaden, J., Sasaki, J., Smith, S.: Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, Englewood Cliffs (1986)
Dalmas, S., Gaëtano, M., Huchet, C.: A deductive database for mathematical formulas. In: Limongelli, C., Calmet, J. (eds.) DISCO 1996. LNCS, vol. 1128, pp. 287–296. Springer, Heidelberg (1996)
Hähnle, R., Kerber, M., Weidenbach, C.: Common syntax of dfg-schwerpunktprogramm “deduktion”. Interner Bericht 10/96, Universität Karlsruhe, Fakultät für Informatik (1996)
Kohlhase, M., Şucan, I.: A search engine for mathematical formulae. In: Calmet, J., Ida, T., Wang, D. (eds.) AISC 2006. LNCS (LNAI), vol. 4120, pp. 241–253. Springer, Heidelberg (2006)
Kohlhase, M., Şucan, I.: System description: MathWebSearch 0.3, a semantic search engine. submitted to CADE 21 (2007)
Kohlhase, M.: OMDoc – An Open Markup Format for Mathematical Documents [version 1.2]. LNCS (LNAI), vol. 4180. Springer, Heidelberg (2006)
Naylor, W., Padget, J.A.: Semantic matching for mathematical services. In: Kohlhase, M. (ed.) MKM 2005. LNCS (LNAI), vol. 3863, pp. 174–189. Springer, Heidelberg (2006)
The NuPrl online theory library. Internet interface at http://simon.cs.cornell.edu/Info/Projects/NuPrl/Nuprl4.2/Libraries/Welc.html
Owre, S., Rushby, J.M., Shankar, N.: PVS: a prototype verification system. In: Kapur, D. (ed.) Automated Deduction - CADE-11. LNCS, vol. 607, pp. 748–752. Springer, Heidelberg (1992)
Pvs libraries, http://pvs.csl.sri.com/libraries.html
Rudnicki, P., Schwarzweller, C., Trybulec, A.: Commutative algebra in the Mizar system. Journal of Symbolic Computation 32, 143–169 (2001)
Rudnicki, P.: An overview of the mizar project. In: Proceedings of the 1992 Workshop on Types and Proofs as Programs, pp. 311–332 (1992)
Coq Development Team. The Coq Proof Assistant Reference Manual. INRIA, see http://coq.inria.fr/doc/main.html
Trybulec, A.: Tarski Grothendieck set theory. Formalized Mathematics 1(1), 9–11 (1990)
Urban, J.: Translating mizar for first-order theorem provers. In: Asperti, A., Buchberger, B., Davenport, J.H. (eds.) MKM 2003. LNCS, vol. 2594, pp. 203–215. Springer, Heidelberg (2003)
Weidenbach, C., Gaede, B., Rock, G.: Spass & flotter, version 0.42. In: McRobbie, M.A., Slaney, J.K. (eds.) Automated Deduction - Cade-13. LNCS, vol. 1104, Springer, Heidelberg (1996)
Wiedijk, F.: Mizar: An impression (1999), http://www.cs.kun.nl/~freek/notes
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Normann, I., Kohlhase, M. (2007). Extended Formula Normalization for ε-Retrieval and Sharing of Mathematical Knowledge. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds) Towards Mechanized Mathematical Assistants. MKM Calculemus 2007 2007. Lecture Notes in Computer Science(), vol 4573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73086-6_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-73086-6_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73083-5
Online ISBN: 978-3-540-73086-6
eBook Packages: Computer ScienceComputer Science (R0)