Nothing Special   »   [go: up one dir, main page]

Skip to main content

Proof-search in intuitionistic logic with equality, or back to simultaneous rigid E-unification

  • Session 1A
  • Conference paper
  • First Online:
Automated Deduction — Cade-13 (CADE 1996)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1104))

Included in the following conference series:

Abstract

We characterize provability in intuitionistic logic with equality in terms of a constraint calculus. This characterization uncovers close connections between provability in intuitionistic logic with equality and solutions to simultaneous rigid E-unification. We show that the problem of existence of a sequent proof with a given skeleton is polynomial-time equivalent to simultaneous rigid E-unifiability. This gives us a proof procedure for intuitionistic logic with equality modulo simultaneous rigid E-unification. We also show that simultaneous rigid E-unifiability is polynomial-time reducible to intuitionistic logic with equality. Thus, any proof procedure for intuitionistic logic with equality can be considered as a procedure for simultaneous rigid E-unifiability. In turn, any procedure for simultaneous rigid E-unifiability gives a procedure for establishing provability in intuitionistic logic with equality.

Supported by a TFR grant.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. A. Beltyukov. Decidability of the universal theory of natural numbers with addition and divisibility (in Russian). Zapiski Nauchnyh Seminarov LOMI, 60:15–28, 1976. English translation in: Journal of Soviet Mathematics.

    Google Scholar 

  2. A. Degtyarev, Yu. Matiyasevich, and A. Voronkov. Simultaneous rigid E-unification and related algorithmic problems. In LICS'96, page 11, 1996.

    Google Scholar 

  3. A. Degtyarev and A. Voronkov. Equality elimination for semantic tableaux. UP-MAIL Technical Report 90, Uppsala University, Computing Science Department, December 1994.

    Google Scholar 

  4. A. Degtyarev and A. Voronkov. Simultaneous rigid E-unification is undecidable. UPMAIL Technical Report 105, Uppsala University, Computing Science Department, May 1995. To appear in Theoretical Computer Science.

    Google Scholar 

  5. A. Degtyarev and A. Voronkov. Reduction of second-order unification to simultaneous rigid E-unification. UPMAIL Technical Report 109, Uppsala University, Computing Science Department, June 1995.

    Google Scholar 

  6. A. Degtyarev and A. Voronkov. A new procedural interpretation of Horn clauses with equality. In Leon Sterling, editor, Proceedings of the Twelfth International Conference on Logic Programming, pages 565–579. The MIT Press, 1995.

    Google Scholar 

  7. A. Degtyarev and A. Voronkov. General connections via equality elimination. In M. De Glas and Z. Pawlak, editors, Second World Conference on the Fundamentals of Artificial Intelligence (WOCFAI-95), pages 109–120, Paris, July 1995. Angkor.

    Google Scholar 

  8. A. Degtyarev and A. Voronkov. Equality elimination for the inverse method and extension procedures. In C.S. Mellish, editor, Proc. International Joint Conference on Artificial Intelligence (IJCAI), volume 1, pages 342–347, Montréal, August 1995.

    Google Scholar 

  9. A. Degtyarev and A. Voronkov. Skolemization and decidability problems for fragments of intuitionistic logic. Submitted, 10 pages, 1996.

    Google Scholar 

  10. M. Fitting. First Order Logic and Automated Theorem Proving. Springer Verlag, New York, 1990.

    Google Scholar 

  11. J.H. Gallier, S. Raatz, and W. Snyder. Theorem proving using rigid E-unification: Equational matings. In Proc. IEEE Conference on Logic in Computer Science (LICS), pages 338–346. IEEE Computer Society Press, 1987.

    Google Scholar 

  12. S. Kanger. A simplified proof method for elementary logic. In J. Siekmann and G. Wrightson, editors, Automation of Reasoning. Classical Papers on Computational Logic, volume 1, pages 364–371. Springer Verlag, 1983. Originally appeared in 1963.

    Google Scholar 

  13. R.C.T. Lee and C.L. Chang. Symbolic Logic and Mechanical Theorem Proving. Academic Press, 1973.

    Google Scholar 

  14. V. Lifschitz. The decidability problem for some constructive theories of equality (in Russian). Zapiski Nauchnyh Seminarov LOMI, 4:78–85, 1967.

    Google Scholar 

  15. D.W. Loveland. Automated Theorem Proving: a Logical Basis. North Holland, 1978.

    Google Scholar 

  16. S.Yu. Maslov. An inverse method for establishing deducibility of nonprenex formulas of the predicate calculus. In J.Siekmann and G.Wrightson, editors, Automation of Reasoning (Classical papers on Computational Logic), volume 2, pages 48–54. Springer Verlag, 1983.

    Google Scholar 

  17. S.Yu. Maslov. An invertible sequential variant of constructive predicate calculus (in Russian). Zapiski Nauchnyh Seminarov LOMI, 4, 1967. English Translation in: Seminars in Mathematics: Steklov Math. Inst. 4, Consultants Bureau, NY-London, 1969, p.36–42.

    Google Scholar 

  18. V. A. Matulis. On variants of classical predicate calculus with the unique deduction tree (in Russian). Soviet Mathematical Doklady, 148:768–770, 1963.

    Google Scholar 

  19. G.E. Mints. Collecting terms in the quantifier rules of the constructive predicate calculus (in Russian). Zapiski Nauchnyh Seminarov LOMI, 4:78–85, 1967.

    Google Scholar 

  20. V.P. Orevkov. Solvable classes of pseudo-prenex formulas (in Russian). Zapiski Nauchnyh Seminarov LOMI, 60:109–170, 1976. English translation in: Journal of Soviet Mathematics.

    Google Scholar 

  21. D.A. Plaisted. Special cases and substitutes for rigid E-unification. Technical Report MPI-I-95-2-010, Max-Planck-Institut für Informatik, November 1995.

    Google Scholar 

  22. G. Plotkin. Building-in equational theories. In Meltzer and Michie, editors, Machine Intelligence, volume 7, pages 73–90. Edinburgh University Press, Edinburgh, 1972.

    Google Scholar 

  23. G. Robinson and L.T. Wos. Paramodulation and theorem-proving in first order theories with equality. In Meltzer and Michie, editors, Machine Intelligence, volume 4, pages 135–150. Edinburgh University Press, Edinburgh, 1969.

    Google Scholar 

  24. D. Sahlin, T. Franzén, and S. Haridi. An intuitionistic predicate logic theorem prover. Journal of Logic and Computation, 2(5):619–656, 1992.

    Google Scholar 

  25. N. Shankar. Proof search in the intuitionistic sequent calculus. In D. Kapur, editor, 11th International Conference on Automated Deduction, volume 607 of Lecture Notes in Artificial Intelligence, pages 522–536, Saratoga Springs, NY, USA, June 1992. Springer Verlag.

    Google Scholar 

  26. T. Tammet. A resolution theorem prover for intuitionistic logic. This volume.

    Google Scholar 

  27. M. Veanes. Undecidability proofs of simultaneous rigid E-unification. Upmail technical report, Uppsala University, Computing Science Department, 1996. To appear.

    Google Scholar 

  28. P.J. Voda and J. Komara. On Herbrand skeletons. Technical report, Institute of Informatics, Comenius University Bratislava, July 1995.

    Google Scholar 

  29. A. Voronkov. Theorem proving in non-standard logics based on the inverse method. In D. Kapur, editor, 11th International Conference on Automated Deduction, volume 607 of Lecture Notes in Artificial Intelligence, pages 648–662, Saratoga Springs, NY, USA, June 1992. Springer Verlag.

    Google Scholar 

  30. A. Voronkov. Proof-search in intuitionistic logic based on the constraint satisfaction. UPMAIL Technical Report 120, Uppsala University, Computing Science Department, January 1996.

    Google Scholar 

  31. A. Voronkov. On proof-search in intuitionistic logic with equality, or back to simultaneous rigid E-Unification. UPMAIL Technical Report 121, Uppsala University, Computing Science Department, January 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

M. A. McRobbie J. K. Slaney

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Voronkov, A. (1996). Proof-search in intuitionistic logic with equality, or back to simultaneous rigid E-unification. In: McRobbie, M.A., Slaney, J.K. (eds) Automated Deduction — Cade-13. CADE 1996. Lecture Notes in Computer Science, vol 1104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61511-3_67

Download citation

  • DOI: https://doi.org/10.1007/3-540-61511-3_67

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61511-8

  • Online ISBN: 978-3-540-68687-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics