Automated Proof Compression by Invention of New Definitions

  • Jiří Vyskočil
  • David Stanovský
  • Josef Urban
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6355)


State-of-the-art automated theorem provers (ATPs) are today able to solve relatively complicated mathematical problems. But as ATPs become stronger and more used by mathematicians, the length and human unreadability of the automatically found proofs become a serious problem for the ATP users. One remedy is automated proof compression by invention of new definitions.

We propose a new algorithm for automated compression of arbitrary sets of terms (like mathematical proofs) by invention of new definitions, using a heuristics based on substitution trees. The algorithm has been implemented and tested on a number of automatically found proofs. The results of the tests are included.


Compression Ratio Formal Proof Automate Reasoning Weight Assignment Substitution Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Dahn, I.: Robbins algebras are boolean: A revision of mccune’s computer-generated solution of robbins problem. Journal of Algebra 208(2), 526–532 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Gonthier, G.: The four colour theorem: Engineering of a formal proof. In: Kapur, D. (ed.) ASCM 2007. LNCS (LNAI), vol. 5081, p. 333. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. Graf, P.: Term Indexing. LNCS, vol. 1053. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  4. Grabowski, A.: Robbins algebras vs. boolean algebras. Formalized Mathematics 9(4), 681–690 (2001)zbMATHGoogle Scholar
  5. Huang, X.: Translating machine-generated resolution proofs into nd-proofs at the assertion level. In: Foo, N.Y., Göbel, R. (eds.) PRICAI 1996. LNCS, vol. 1114, pp. 399–410. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  6. McCune, W.: Solution of the Robbins problem. J. Autom. Reasoning 19(3), 263–276 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Meng, J., Paulson, L.C.: Translating higher-order clauses to first-order clauses. J. Autom. Reasoning 40(1), 35–60 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Nonnengart, A., Weidenbach, C.: Computing small clause normal forms. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, pp. 335–367. Elsevier and MIT Press (2001)Google Scholar
  9. Puzis, Y., Gao, Y., Sutcliffe, G.: Automated generation of interesting theorems. In: FLAIRS Conference, pp. 49–54 (2006)Google Scholar
  10. Plotkin, G.D.: A note on inductive generalization. Machine Intelligence 5, 153–163 (1969)zbMATHGoogle Scholar
  11. Proietti, M., Pettorossi, A.: Unfolding–definition–folding, in this order, for avoiding unnecessary variables in logic programs. Theoretical Computer Science 142(1), 89–124 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Phillips, J.D., Stanovský, D.: Automated theorem proving in loop theory. In: Sutcliffe, G., Colton, S., Schulz, S. (eds.) ESARM: Empirically Successful Automated Reasoning in Mathematics. CEUR Workshop Proceedings, vol. 378, pp. 54–54. CEUR (2008)Google Scholar
  13. Schulz, S.: E – a brainiac theorem prover. Journal of AI Communications 15(2-3), 111–126 (2002)zbMATHGoogle Scholar
  14. Stanovský, D.: Distributive groupoids are symmetric-by-medial: An elementary proof. Commentationes Mathematicae Universitatis Carolinae 49(4), 541–546 (2008)MathSciNetzbMATHGoogle Scholar
  15. Sutcliffe, G.: The tptp problem library and associated infrastructure. J. Autom. Reasoning 43(4), 337–362 (2009)CrossRefzbMATHGoogle Scholar
  16. Trac, S., Puzis, Y., Sutcliffe, G.: An interactive derivation viewer. In: UITP 2006. Electronic Notes in Theoretical Computer Science, vol. 174, pp. 109–123. Elsevier, Amsterdam (2007)Google Scholar
  17. Tamaki, H., Sato, T.: Unfold/fold transformations of logic programs. In: Tärnlund, S.-Å. (ed.) Proceedings of The Second International Conference on Logic Programming, pp. 127–139 (1984)Google Scholar
  18. Urban, J.: Automated reasoning for mizar: Artificial intelligence through knowledge exchange. In: Rudnicki, P., Sutcliffe, G., Konev, B., Schmidt, R.A., Schulz, S. (eds.) LPAR Workshops. CEUR Workshop Proceedings, vol. 418. (2008)Google Scholar
  19. Urban, J., Sutcliffe, G., Pudlák, P., Vyskočil, J.: Malarea sg1- machine learner for automated reasoning with semantic guidance. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 441–456. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. Veroff, R.: Solving open questions and other challenge problems using proof sketches. J. Autom. Reasoning 27(2), 157–174 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Vyskočil, J., Štěpánek, P.: Improving efficiency of prolog programs by fully automated unfold/fold transformation. In: Gelbukh, A., Kuri Morales, Á.F. (eds.) MICAI 2007. LNCS (LNAI), vol. 4827, pp. 305–315. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jiří Vyskočil
    • 1
  • David Stanovský
    • 2
  • Josef Urban
    • 3
  1. 1.Czech Technical UniversityPragueCzech Republic
  2. 2.Charles UniversityPragueCzech Republic
  3. 3.Radboud UniversityNijmegenNetherlands

Personalised recommendations