Skip to main content
SpringerLink
Log in

Proof-Search in Intuitionistic Logic with Equality, or Back to Simultaneous Rigid E-Unification

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Buss, S.: Bounded Arithmetic, Vol. 3 of Studies in Proof Theory, Bibliopolis, Napoly, 1986.

    Google Scholar 

  2. Degtyarev, A., Matiyasevich, T., and Voronkov, A.: Simultaneous rigid E-unification and related algorithmic problems, in Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS’96), New Brunswick, NJ, 1996, pp. 494-502.

  3. Degtyarev, A. and Voronkov, A.: Equality elimination for the inverse method and extension procedures, in C. Mellish (ed.), Proc. International Joint Conference on Artificial Intelligence (IJCAI), Vol. 1, Montreal, 1995, pp. 342-347.

  4. Degtyarev, A. and Voronkov, A.: General connections via equality elimination, in M. De Glas and Z. Pawlak (eds.), Second World Conference on the Fundamentals of Artificial Intelligence (WOCFAI-95), Paris, 1995, pp. 109-120.

  5. Degtyarev, A. and Voronkov, A.: A new procedural interpretation of Horn clauses with equality, in L. Sterling (ed.), Proceedings of the Twelfth International Conference on Logic Programming, 1995, pp. 565-579.

  6. Degtyarev, A. and Voronkov, A.: Decidability problems for the prenex fragment of intuitionistic logic, in Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS’96), New Brunswick, NJ, 1996, pp. 503-512.

  7. Degtyarev, A. and Voronkov, A.: Equality elimination for the tableau method, in J. Calmet and C. Limongelli (eds.), Design and Implementation of Symbolic Computation Systems. International Symposium, DISCO’96, Lecture Notes in Comput. Sci. 1128, Karlsruhe, Germany, 1996, pp. 46-60.

  8. Degtyarev, A. and Voronkov, A.: Simultaneous rigid E-unification is undecidable, in H. Kleine Büning (ed.), Computer Science Logic. 9th International Workshop, CSL’95, Lecture Notes in Comput. Sci. 1092, Paderborn, Germany, 1996, pp. 178-190.

  9. Degtyarev, A. and Voronkov, A.: The undecidability of simultaneous rigid E-unification, Theoret. Comput. Sci. 166(1-2) (1996), 291-300.

    Article  Google Scholar 

  10. Degtyarev, A. and Voronkov, A.: What you always wanted to know about rigid E-unification, in J. Alferes, L. Pereira, and E. Orlowska (eds.), Logics in Artificial Intelligence. European Workshop, JELIA’96, Lecture Notes in Artif. Intell. 1126, Évora, Portugal, 1996, pp. 50-69.

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

    Google Scholar 

  12. Gallier, J., Raatz, S., and Snyder, W.: Theorem proving using rigid E-unification: Equational matings, in Proc. IEEE Conference on Logic in Computer Science (LICS), 1987, pp. 338-346.

  13. Garson, J.: Quantification in modal logic, in D. Gabbay and F. Guenther (eds.), Handbook in Philosophical Logic, Vol. II, Chapt. II.5, D. Reidel Publishing Company, 1984, pp. 249-307.

  14. Girard, J.-Y.: Proof Theory and Logical Complexity, Studies in Proof Theory, Bibliopolis, Napoli, 1987.

    Google Scholar 

  15. Gurevich, Y. and Veanes, M.: Some undecidable problems related to the Herbrand theorem, UPMAIL Technical Report 138, Uppsala University, Computing Science Department, 1997.

  16. Gurevich, Y. and Voronkov, A.: Monadic simultaneous rigid E-unification and related problems, in P. Degano, R. Corrieri, and A. Marchetti-Spaccamella (eds.), Automata, Languages and Programming, 24th International Colloquium, ICALP’97, Lecture Notes in Comput. Sci. 1256, Bologna, Italy, 1997, pp. 154-165.

  17. Kanger, S.: Provability in Logic, Vol. 1 of Studies in Philosophy, Almqvist and Wicksell, Stockholm, 1957.

    Google Scholar 

  18. Kanger, S.: A simplified proof method for elementary logic, in J. Siekmann and G. Wrightson (eds.), Automation of Reasoning. Classical Papers on Computational Logic, Vol. 1, Springer-Verlag, 1983, pp. 364-371. Originally appeared in 1963.

  19. Lee, R. and Chang, C.: Symbolic Logic and Mechanical Theorem Proving, Academic Press, 1973.

  20. Lifschitz, V.: Problem of decidability for some constructive theories of equalities, Zapiski Nauchnyh Seminarov LOMI 4 (1967), 78-85 (in Russian). English Translation in Seminars in Mathematics: Steklov Math. Inst. 4, Consultants Bureau, NY, 1969, pp. 29-31.

    Google Scholar 

  21. Lifschitz, V.: Specialized forms of derivation in predicate calculus with equality and functional symbols, in Trudy MIAN, Vol. 98, 1968, pp. 5-25 (in Russian). English translation in Proc. Steklov Institute of Math., AMS, Providence, RI, 1971.

    Google Scholar 

  22. Lincoln, P. and Shankar, N.: Proof search in first-order linear logic and other cut-free sequent calculi, in S. Abramski (ed.), Ninth Annual Symposium on Logic in Computer Science, Paris, France, 1994, pp. 282-291.

  23. Loveland, D.: Automated Theorem Proving: A Logical Basis, North-Holland, Amsterdam, 1978.

    Google Scholar 

  24. Maslov, S.: Invertible sequential variant of constructive predicate calculus, Zapiski Nauchnyh Seminarov LOMI 4 (1967), 96-111 (in Russian). English Translation in Seminars in Mathematics: Steklov Math. Inst. 4, Consultants Bureau, NY, 1969, pp. 36-42.

    Google Scholar 

  25. Maslov, S.: An inverse method for establishing deducibility of nonprenex formulas of the predicate calculus, in J. Siekmann and G. Wrightson (eds.), Automation of Reasoning (Classical papers on Computational Logic), Vol. 2, Springer-Verlag, 1983, pp. 48-54.

  26. Matulis, V.: On variants of classical predicate calculus with the unique deduction tree, Soviet Math. Dokl. 148 (1963), 768-770 (in Russian).

    Google Scholar 

  27. Mints, G.: Choice of terms in quantifier rules of constructive predicate calculus, Zapiski Nauchnyh Seminarov LOMI 4 (1967), 78-85 (in Russian). English translation in Seminars in Mathematics: Steklov Math. Inst. 4, Consultants Bureau, NY, 1969, pp. 43-46.

    Google Scholar 

  28. Orevkov, V.: A specialized form of derivation in Gentzen calculi and its applications, Zapiski Nauchnyh Seminarov LOMI 32 (1974), 110-118 (in Russian).

    Google Scholar 

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

    Google Scholar 

  30. Petermann, U.: A complete connection calculus with rigid E-unification, in JELIA’94, Lecture Notes in Comput. Sci. 838, 1994, pp. 152-166.

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

  32. Plotkin, G.: Building-in equational theories, in B. Meltzer and D. Michie (eds.), Machine Intelligence, Vol. 7, Edinburgh University Press, 1972, pp. 73-90.

  33. Robinson, G. and Wos, L.: Paramodulation and theorem-proving in first order theories with equality, in B. Meltzer and D. Michie (eds.),Machine Intelligence, Vol. 4, Edinburgh University Press, 1969, pp. 135-150.

  34. Robinson, J.: A machine-oriented logic based on the resolution principle, J. Assoc. Comput. Mach. 12(1) (1965), 23-41.

    Google Scholar 

  35. Sahlin, D., Franzén, T., and Haridi, S.: An intuitionistic predicate logic theorem prover, J. Logic and Comput. 2(5) (1992), 619-656.

    Google Scholar 

  36. Sazonov, V.: A logical aproach to the problem “P=NP?”, in Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 88, 1980, pp. 562-575.

  37. Shankar, N.: Proof search in the intuitionistic sequent calculus, in D. Kapur (ed.), 11th International Conference on Automated Deduction, Lecture Notes in Artif. Intell. 607, Saratoga Springs, NY, USA, 1992, pp. 522-536.

  38. Tammet, T.: A resolution theorem prover for intuitionistic logic, in M. McRobbie and J. Slaney (eds.), Automated Deduction - CADE-13, Lecture Notes in Comput. Sci. 1104, New Brunswick, NJ, USA, 1996, pp. 2-16.

  39. Veanes, M.: Uniform representation of recursively enumerable sets with simultaneous rigid E-unification, UPMAIL Technical Report 126, Uppsala University, Computing Science Department, 1996.

  40. Veanes, M.: On simultaneous rigid E-unification, Ph.D. Thesis, Uppsala University, 1997.

  41. Voda, P. and Komara, J.: On Herbrand skeletons, Technical Report, Institute of Informatics, Comenius University Bratislava, 1995.

  42. Voronkov, A.: Theorem proving in non-standard logics based on the inverse method, in D. Kapur (ed.), 11th International Conference on Automated Deduction, Lecture Notes in Artif. Intell. 607, Saratoga Springs, NY, USA, 1992, pp. 648-662.

  43. Voronkov, A.: Proof search in intuitionistic logic based on constraint satisfaction, in P. Miglioli, U. Moscato, D. Mundici, and M. Ornaghi (eds.), Theorem Proving with Analytic Tableaux and Related Methods. 5th International Workshop, TABLEAUX’ 96, Lecture Notes in Artif. Intell. 1071, Terrasini, Palermo, Italy, 1996, pp. 312-329.

  44. Voronkov, A.: Proof search in intuitionistic logic with equality, or back to simultaneous rigid E-unification, in M. McRobbie and J. Slaney (eds.), Automated Deduction - CADE-13, Lecture Notes in Comput. Sci. 1104, New Brunswick, NJ, USA, 1996, pp. 32-46.

Download references

Author information

Authors and Affiliations

  1. Computing Science Department, Uppsala University, Box 311, S 751 05, Uppsala, Sweden

    Andrei Voronkov

Authors
  1. Andrei Voronkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Voronkov, A. Proof-Search in Intuitionistic Logic with Equality, or Back to Simultaneous Rigid E-Unification. Journal of Automated Reasoning 30, 121–151 (2003). https://doi.org/10.1023/A:1023260415982

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023260415982

Search

Navigation