# MPTP – Motivation, Implementation, First Experiments

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## Abstract

We describe a number of new possibilities for current theorem provers that arise with the existence of large integral bodies of formalized mathematics. Then we describe the implementation of the MPTP system, which makes the largest existing corpus of formalized mathematics available to theorem provers. MPTP (Mizar Problems for Theorem Proving) is a system for translating the Mizar Mathematical Library (MML) into untyped first-order format suitable for automated theorem provers and for generating theorem-proving problems corresponding to MML. The first version generates about 30,000 problems from complete proofs of Mizar theorems and about 630,000 problems from the simple (one-step) justifications done by the Mizar checker. We describe the design and structure of the system, the main problems encountered in this kind of system, their solutions, current limitations, and planned extensions. We present results of first experiments with re-proving the MPTP problems with theorem provers. We also describe first implementation of the Mizar Proof Advisor (MPA) used for selecting suitable axioms from the large library for an arbitrary problem and, again, present first results of this combined MPA/ATP architecture on MPTP.

## Keywords

ATP Mizar MPA MPTP## Preview

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## References

- Bancerek, G. (2000) Development of the theory of continuous lattices in Mizar, in M. Kerber and M. Kohlhase (eds.),
*Symbolic Computation and Automated Reasoning, The CALCULEMUS-2000 Symposium*, A. K. Peters, Natick, MA, pp. 65–80. Google Scholar - Carlson, A. J., Cumby, C. M., Rosen, J. L. and Roth, D. (1999) SNoW user’s guide, UIUC Tech Report UIUC-DCS-R-99-210. Google Scholar
- Dahn, I. and Wernhard, C. (1997) First order proof problems extracted from an article in the MIZAR mathematical library, in
*Proceedings of the International Workshop on First Order Theorem Proving*, RISC-Linz Report Series, No. 97-50, Johannes Kepler Universität Linz. Google Scholar - Dahn, I. (1998) Interpretation of a Mizar-like logic in first order logic, in
*Proceedings of FTP 1998*, pp. 137–151. Google Scholar - Dahn, I. (2001) Slicing book technology – Providing online support for textbooks, in
*Proc. ICDE 2001, International Conference on Distant Education*, Düsseldorf. Google Scholar - Ganzinger, H., Meyer, C. and Weidenbach, C. (1997) Soft typing for ordered resolution, in
*Proc. CADE-14*, Springer, pp. 321–335. Google Scholar - Ganzinger, H. and Stuber, J. (2003) Superposition with equivalence reasoning and delayed clause normal form transformation, in
*Proc. 19th Int. Conf. on Automated Deduction (CADE-19)*, Miami, Florida, LNAI 2741, Springer, pp. 335–349. Google Scholar - Hähnle, R., Kerber, M. and Weidenbach, C. (1996) Common syntax of the DFGSchwerpunktprogramm deduction, Technical Report TR 10/96, Fakultät für Informatik, Universität Karlsruhe, Karlsruhe, Germany. Google Scholar
- Lenat, D. B. (1979) On automated scientific theory formation: A case study using the AM program, in J. Hayes, D. Michie and L. I. Mikulich (eds.),
*Machine Intelligence 9*, Halstead, New York, 1979, pp. 251–283. Google Scholar - McCune, W. W. (1997) Solution of the Robbins problem,
*J. Automated Reasoning***19**(3), 263–276. Google Scholar - MoMM (2004) The MoMM interreduction system by Josef Urban, available online at http://alioth.uwb.edu.pl/twiki/bin/view/Mizar/MoMM.
- MPTPResults.sql – SQL structure of the MPTP result database, published online at http://alioth.uwb.edu.pl/twiki/bin/view/Mizar/MpTP.
- Naumowicz, A. and Byliński, C. (2002) Basic elements of computer algebra in MIZAR,
*Mechanized Mathematics and Its Applications***2**(1), 9–16. Google Scholar - Nonnengart, A. and Weidenbach, C. (2001) Computing small clause normal forms, in A. Robinson and A. Voronkov (eds.),
*Handbook of Automated Reasoning*, Vol. 1, Elsevier, Chapter 6, pp. 335–367. Google Scholar - Pollock, J. L. (1996) OSCAR: A general purpose defeasible reasoner,
*J. Appl. Non-Classical Logics***6**, 89–113. Google Scholar - Rudnicki, P. (1992) An overview of the Mizar project, in
*Proceedings of the 1992 Workshop on Types for Proofs and Programs*, Chalmers University of Technology, Bastad. Google Scholar - Schulz, S. (2001) Learning search control knowledge for equational theorem proving, in F. Baader, G. Brewka and T. Eiter (eds.),
*Proceedings of the Joint German/Austrian Conference on Artificial Intelligence (KI-2001)*, LNAI 2174, Springer, pp. 320–334. Google Scholar - Schumann, J. M. (2001)
*Automated Theorem-Proving in Software Engineering*, Springer-Verlag. Google Scholar - Sutcliffe, G. and Suttner, C. B. (1998) The TPTP problem library: CNF release v1.2.1,
*J. Automated Reasoning***21**(2), 177–203. Google Scholar - Sutcliffe, G., Zimmer, J. and Schulz, S. (2003) Communication formalisms for automated theorem proving tools, in V. Sorge, S. Colton, M. Fisher and J. Gow (eds.),
*Proceedings of the IJCAI-18 Workshop on Agents and Automated Reasoning*, pp. 53–58. Google Scholar - Urban, J. (2003) Translating Mizar for first order theorem provers, in A. Asperti, B. Buchberger and J. Davenport (eds.),
*Mathematical Knowledge Management, Proceedings of MKM 2003*, LNCS 2594. Google Scholar - Urban, J. (2004) MoMM – Fast interreduction and retrieval in large libraries of formalized mathematics, in G. Sutcliffe, S. Schultz and T. Tammit (eds.),
*Proceedings of the IJCAR 2004 Workshop on Empirically Successful First Order Reasoning*, ENTCS, accepted. Available online at http://ktiml.mff.cuni.cz/~urban/MoMM/momm.ps - Weidenbach, C. (2001) SPASS: Combining superposition, sorts and splitting, in A. Robinson and A. Voronkov (eds.),
*Handbook of Automated Reasoning*, Vol. II, Elsevier Science and MIT Press, Chapter 27, pp. 1965–2013. Google Scholar - Wiedijk, F. (2000) CHECKER – Notes on the basic inference step in Mizar, available at http://www.cs.kun.nl/~freek/mizar/by.dvi
- Wiedijk, F. (2002) Estimating the cost of a standard library for a mathematical proof checker, http://www.cs.kun.nl/~freek/notes
- Wiedijk, F. (2003) Comparing mathematical provers, in A. Asperti, B. Buchberger and J. Davenport (eds.),
*Mathematical Knowledge Management, Proceedings of MKM 2003*, LNCS 2594, pp. 188–202. Google Scholar