Journal of Automated Reasoning

, Volume 37, Issue 1–2, pp 21–43 | Cite as

MPTP 0.2: Design, Implementation, and Initial Experiments

  • Josef UrbanEmail author


This paper describes the second version of the Mizar Problems for Theorem Proving (MPTP) system and first experimental results obtained with it. The goal of the MPTP project is to make the large formal Mizar Mathematical Library (MML) available to current first-order automated theorem provers (ATPs) (and vice versa) and to boost the development of domain-based, knowledge-based, and generally AI-based ATP methods. This version of MPTP switches to a generic extended TPTP syntax that adds term-dependent sorts and abstract (Fraenkel) terms to the TPTP syntax. We describe these extensions and explain how they are transformed by MPTP to standard TPTP syntax using relativization of sorts and deanonymization of abstract terms. Full Mizar proofs are now exported and also encoded in the extended TPTP syntax, allowing a number of ATP experiments. This covers, for example, consistent handling of proof-local constants and proof-local lemmas and translating of a number of Mizar proof constructs into the TPTP formalism. The proofs using second-order Mizar schemes are now handled by the system, too, by remembering (and, if necessary, abstracting from the proof context) the first-order instances that were actually used. These features necessitated changes in Mizar, in the Mizar-to-TPTP exporter, and in the problem-creating tools. Mizar has been reimplemented to produce and use natively a detailed XML format, suitable for communication with other tools. The Mizar-to-TPTP exporter is now just a XSLT stylesheet translating the XML tree to the TPTP syntax. The problem creation and other MPTP processing tasks are now implemented in about 1,300 lines of Prolog. All these changes have made MPTP more generic, more complete, and more correct. The largest remaining issue is the handling of the Mizar arithmetical evaluations. We describe several initial ATP experiments, both on the easy and on the hard MML problems, sometimes assisted by machine learning. It is shown that on the nonarithmetical problems, countersatisfiability (completions) is no longer detected by the ATP systems, suggesting that the ‘Mizar deconstruction’ done by MPTP is in this case already complete. About every fifth nonarithmetical theorem is proved in a fully autonomous mode, in which the premises are selected by a machine-learning system trained on previous proofs. In 329 of these cases, the newly discovered proofs are shorter than the MML originals and therefore are likely to be used for MML refactoring. This situation suggests that even a simple inductive or deductive system trained on formal mathematics can be sometimes smarter than MML authors and usable for general discovery in mathematics.

Key words

MPTP Mizar MML ATP MPA re-proving proof discovery 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asperti, A., Bancerek, G., Trybulec, A. (eds.): Mathematical Knowledge Management, Third International Conference, MKM 2004, Bialowieza, Poland, September 19–21, 2004, Proceedings, Vol. 3119 of Lecture Notes in Computer Science. Springer, Berlin Heidelberg New York (2004)Google Scholar
  2. 2.
    Avron, A.: Formalizing set theory as it is actually used. In: Mathematical Knowledge Management, pp. 32–43 (2004)Google Scholar
  3. 3.
    Bancerek, G., Rudnicki, P.: Information retrieval in MML. In: MKM, Vol. 2594 of Lecture Notes in Computer Science, pp. 119–132 (2003)Google Scholar
  4. 4.
    Bancerek, G., Urban, J.: Integrated semantic browsing of the Mizar mathematical library for authoring Mizar articles. In: [1], pp. 44–57 (2004)Google Scholar
  5. 5.
    Bylinski, C.: The complex numbers. Formaliz. Math. 2(2), (1990).
  6. 6.
    Carlson, A.J., Cumby, C.M., Rosen, J.L., Roth, D.: SNoW user’s guide. Technical Report UIUC-DCS-R-99-210, UIUC (1999)Google Scholar
  7. 7.
    Dahn, I.: Interpretation of a Mizar-like logic in first-order logic. In: FTP (LNCS Selection), pp. 137–151 (1998)Google Scholar
  8. 8.
    Dahn, I., Wernhard, C.: First order proof problems extracted from an article in the MIZAR mathematical library. In: Bonacina, M.P., Furbach, U. (eds.), Int. Workshop on First-Order Theorem Proving (FTP’97), pp. 58–62 (1997)Google Scholar
  9. 9.
    Ganzinger, H., Stuber, J., Superposition with equivalence reasoning and delayed clause normal form transformation. In: Baader, F. (ed.), CADE, Vol. 2741 of Lecture Notes in Computer Science, pp. 335–349 (2003)Google Scholar
  10. 10.
    Goguen, J.A., Meseguer, J.: Order-sorted algebra. I. Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci. 105(2), 217–273 (1992)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Hähnle, R., Kerber, M., Weidenbach, C.: Common syntax of the DFGSchwerpunktprogramm deduction. Technical Report TR 10/96, Fakultät für Informatik, Universität Karlsruhe, Karlsruhe, Germany (1996)Google Scholar
  12. 12.
    Jaskowski, S.: On the rules of suppositions. Stud. Log. 1 (1934). Reprinted in S. McCall (1967). Polish Logic 1920–1939. Oxford, Oxford University Press, pp. 232–258Google Scholar
  13. 13.
    Matuszewski, R., Rudnicki, P.: Mizar: The first 30 years. In: Bancerek, G. (ed.), MKM Workshop on 30 Years of Mizar (2004)Google Scholar
  14. 14.
    Naumowicz, A., Bylinski, C.: Improving Mizar texts with properties and requirements. In: Mathematical Knowledge Management, pp. 290–301 (2004)Google Scholar
  15. 15.
    Nonnengart, A., Weidenbach, C.: Handbook of Automated Reasoning, Vol. I, Chapt. Computing small clause normal forms, pp. 335–367. Elsevier, Amsterdam, The Netherlands (2001)Google Scholar
  16. 16.
    Pelletier, F.J.: A brief history of natural deduction. Hist. Philos. Logic 20, 1–31 (1999)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Riazanov, A., Voronkov, A.: The design and implementation of VAMPIRE. Journal of AI Communications 15(2–3), 91–110 (2002)Google Scholar
  18. 18.
    Rudnicki, P.: An overview of the MIZAR project. In: 1992 Workshop on Types for Proofs and Programs, pp. 311–332 (1992)Google Scholar
  19. 19.
    Rudnicki, P., Trybulec, A.: On equivalents of well-foundedness. J. Autom. Reason. 23(3–4), 197–234 (1999)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Schulz, S.: E – A brainiac theorem prover. Journal of AI Communications 15(2–3), 111–126 (2002)Google Scholar
  21. 21.
    Sutcliffe, G.: The design and implementation of a compositional competition–cooperation parallel ATP system. In: de Nivelle, H., Schulz, S. (eds.), Proceedings of the 2nd International Workshop on the Implementation of Logics, pp. 92–102 (2001)Google Scholar
  22. 22.
    Sutcliffe, G., Suttner, C.: The TPTP problem library: CNF release v1.2.1. J. Autom. Reasoning 21(2), 177–203 (1998)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Urban, J.: Translating Mizar for first order theorem provers. In: MKM, Vol. 2594 of Lecture Notes in Computer Science, pp. 203–215 (2003)Google Scholar
  24. 24.
    Urban, J.: MPTP – Motivation, implementation, first experiments. J. Autom. Reasoning 33(3–4), 319–339 (2004)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Urban, J.: MizarMode – An integrated proof assistance tool for the Mizar way of formalizing mathematics. Journal of Applied Logic, 2005 (Article In Press). doi:10.1016/j.jal.2005.10.004, available online at
  26. 26.
    Urban, J.: MoMM – Fast interreduction and retrieval in large libraries of formalized mathematics. Int. J. Artif. Intell. Tools 15(1), 109–130 (2006a)CrossRefGoogle Scholar
  27. 27.
    Urban, J.: XML-izing Mizar: Making semantic processing and presentation of MML easy’. In: Kohlhase, M. (ed.), MKM 2005, Vol. 3863 of Lecture Notes in Artificial Intelligence, pp. 346–360 (2006b)Google Scholar
  28. 28.
    Weidenbach, C.: Handbook of Automated Reasoning, Vol. II, Chapt. SPASS: Combining Superposition, Sorts and Splitting, pp. 1965–2013. Elsevier, Amsterdam, The Netherlands (2001)Google Scholar
  29. 29.
    Wiedijk, F.: CHECKER – Notes on the basic inference step in Mizar’. available at, 2000
  30. 30.
    Wiedijk, F.: Comparing mathematical provers. In: MKM, Vol. 2594 of Lecture Notes in Computer Science, pp. 188–202 (2003)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Theoretical Computer ScienceCharles UniversityPragueCzech Republic

Personalised recommendations