Advertisement

Automated Improving of Proof Legibility in the Mizar System

  • Karol Pąk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8543)

Abstract

Both easily readable and obscure proof scripts can be found in the bodies of formalisations around formal proof checking environments such as Mizar. The communities that use this system try to encourage writing legible texts by making available various solutions, e.g., by introduction of phrases and constructs that make formal deductions look closer to the informal ones. Still, many authors do not want to invest additional efforts in enhancing readability of their scripts and assume this can be handled automatically for them. Therefore, it is desirable to create a tool that can automatically improve legibility of proofs. It turns out that this goal is non-trivial since improving features of text that enhance legibility is in general NP-complete.

The successful application of SMT technology to solving computationally difficult problems suggests that available SMT solvers can give progress in legibility enhancement. In this paper we present the first experimental results obtained with automated legibility improving tools for the Mizar system that use Z3 solver in the backend.

Keywords

Operations on languages Legibility of proofs Proof assistants SMT solvers 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Behaghel, O.: Beziehungen zwischen Umfang und Reihenfolge von Satzgliedern. Indogermanische Forschungen 25, 110–142 (1909)Google Scholar
  2. 2.
    Blanchette, J.C.: Redirecting Proofs by Contradiction. In: Third International Workshop on Proof Exchange for Theorem Proving, PxTP 2013. EPiC Series, vol. 14, pp. 11–26. EasyChair (2013)Google Scholar
  3. 3.
    Cowan, N.: The magical number 4 in short-term memory: A reconsideration of mental storage capacity. Behavioral and Brain Sciences 24(1), 87–114 (2001)CrossRefGoogle Scholar
  4. 4.
    de Moura, L., Bjørner, N.: Z3: An efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Ericsson, K.A.: Analysis of memory performance in terms of memory skill. Advances in the psychology of human intelligence, vol. 4. Lawrence Erlbaum Associates Inc., Hillsdale (1988)Google Scholar
  6. 6.
    Fitch, F.B.: Symbolic Logic: an Introduction. The Ronald Press Co. (1952)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. A Series of Books in the Mathematical Science. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  8. 8.
    Gonthier, G.: Formal Proof—The Four-Color Theorem. Notices of the AMS 55(11), 1382–1393 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a Nutshell. Journal of Formalized Reasoning 3(2), 153–245 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Grabowski, A., Schwarzweller, C.: Improving Representation of Knowledge within the Mizar Library. Studies in Logic, Grammar and Rhetoric 18(31), 35–50 (2009)Google Scholar
  11. 11.
    Jaśkowski, S.: On the Rules of Supposition in Formal Logic. Studia Logica (1934), Warszawa Reprinted in Polish Logic, McCall, S., ed. Clarendon Press, Oxford (1967)Google Scholar
  12. 12.
    Kaliszyk, C., Urban, J.: PRocH: Proof Reconstruction for HOL Light. In: Bonacina, M.P. (ed.) CADE-24. LNCS (LNAI), vol. 7898, pp. 267–274. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Ko, A.J., Myers, B.A., Coblenz, M.J., Aung, H.H.: An Exploratory Study of How Developers Seek, Relate, and Collect Relevant Information during Software Maintenance Tasks. IEEE Transactions on Software Engineering 32(12), 971–988 (2006)CrossRefGoogle Scholar
  14. 14.
    Korniłowicz, A.: Tentative Experiments with Ellipsis in Mizar. In: Jeuring, J., Campbell, J.A., Carette, J., Dos Reis, G., Sojka, P., Wenzel, M., Sorge, V. (eds.) CICM 2012. LNCS (LNAI), vol. 7362, pp. 453–457. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Levy, R.: Expectation-based syntactic comprehension. Cognition 106, 1126–1177 (2008)CrossRefGoogle Scholar
  16. 16.
    Matuszewski, R.: On Automatic Translation of Texts from Mizar-QC language into English. Studies in Logic, Grammar and Rhetoric 4 (1984)Google Scholar
  17. 17.
    Miller, G.A.: The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information. Psychological Review 63, 81–97 (1956)CrossRefGoogle Scholar
  18. 18.
    Naumowicz, A., Korniłowicz, A.: A Brief Overview of Mizar. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 67–72. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Naumowicz, A., Korniłowicz, A.: A Brief Overview of Mizar. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 67–72. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Pąk, K.: The Algorithms for Improving and Reorganizing Natural Deduction Proofs. Studies in Logic, Grammar and Rhetoric 22(35), 95–112 (2010)Google Scholar
  21. 21.
    Pąk, K.: Methods of Lemma Extraction in Natural Deduction Proofs. Journal of Automated Reasoning 50(2), 217–228 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pąk, K.: The Algorithms for Improving Legibility of Natural Deduction Proofs. PhD thesis, University of Warsaw (2013)Google Scholar
  23. 23.
    Pąk, K., Trybulec, A.: Laplace Expansion. Formalized Mathematics 15(3), 143–150 (2008)Google Scholar
  24. 24.
    Smolka, S.J., Blanchette, J.C.: Robust, Semi-Intelligible Isabelle Proofs from ATP Proofs. In: Third International Workshop on Proof Exchange for Theorem Proving, PxTP 2013. EPiC Series, vol. 14, pp. 117–132. EasyChair (2013)Google Scholar
  25. 25.
    Urban, J.: XML-izing Mizar: Making Semantic Processing and Presentation of MML Easy. In: Kohlhase, M. (ed.) MKM 2005. LNCS (LNAI), vol. 3863, pp. 346–360. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    Urban, J.: MizarMode - An Integrated Proof Assistance Tool for the Mizar Way of Formalizing Mathematics. Journal of Applied Logic 4(4), 414–427 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wenzel, M.: The Isabelle/Isar Reference Manual. University of Cambridge (2011)Google Scholar
  28. 28.
    Whitehead, A.N., Russell, B.: Principia Mathematica to *56. Cambridge Mathematical Library. Cambridge University Press (1910)Google Scholar
  29. 29.
    Zammit, V.: On the Readability of Machine Checkable Formal Proofs. PhD thesis, The University of Kent at Canterbury (March 1999)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Karol Pąk
    • 1
  1. 1.Institute of Computer ScienceUniversity of BialystokPoland

Personalised recommendations