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Classification of Alignments Between Concepts of Formal Mathematical Systems

  • Dennis MüllerEmail author
  • Thibault GauthierEmail author
  • Cezary KaliszykEmail author
  • Michael KohlhaseEmail author
  • Florian RabeEmail author
Conference paper
First Online:
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.

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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.

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© Springer International Publishing AG 2017

Authors and Affiliations

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

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