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Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry

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Abstract

The power of state-of-the-art automated and interactive theorem provers has reached the level at which a significant portion of non-trivial mathematical contents can be formalized almost fully automatically. In this paper we present our framework for the formalization of mathematical knowledge that can produce machine verifiable proofs (for different proof assistants) but also human-readable (nearly textbook-like) proofs. As a case study, we focus on one of the twentieth century classics – a book on Tarski’s geometry. We tried to automatically generate such proofs for the theorems from this book using resolution theorem provers and a coherent logic theorem prover. In the first experiment, we used only theorems from the book, in the second we used additional lemmas from the existing Coq formalization of the book, and in the third we used specific dependency lists from the Coq formalization for each theorem. The results show that 37 % of the theorems from the book can be automatically proven (with readable and machine verifiable proofs generated) without any guidance, and with additional lemmas this percentage rises to 42 %. These results give hope that the described framework and other forms of automation can significantly aid mathematicians in developing formal and informal mathematical knowledge.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000, Belgrade, Serbia

    Sana Stojanović Ðurđević & Predrag Janičić

  2. ICube, UMR 7357 CNRS, University of Strasbourg Pôle API, Bd Sébastien Brant, BP 10413, 67412, Illkirch, France

    Julien Narboux

Authors
  1. Sana Stojanović Ðurđević
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  2. Julien Narboux
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  3. Predrag Janičić
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Correspondence to Sana Stojanović Ðurđević.

Additional information

The research presented in this paper was partly funded by the Serbian-French Technology Co-Operation grant EGIDE/“Pavle Savi´c” 680-00-132/2012-09/12. The first and the third author are partly supported by the grant ON174021 of the Ministry of Science of Serbia.

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Ðurđević, S.S., Narboux, J. & Janičić, P. Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry. Ann Math Artif Intell 74, 249–269 (2015). https://doi.org/10.1007/s10472-014-9443-5

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  • DOI: https://doi.org/10.1007/s10472-014-9443-5

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