Skip to main content
Log in

Automatic theorem proving. II

  • Published:
Cybernetics Aims and scope

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

Access this article

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

Instant access to the full article PDF.

Literature Cited

  1. W. Bibel, “Mating in matrices,” Comm. ACM,26, No. 11, 844–867 (1983).

    Google Scholar 

  2. J. Corbin and M. Bidoit, “A rehabilitation of Robinson's unification algorithm,” in: R. E. A. Mason (ed.), Information Processing 83, North-Holland, Amsterdam (1983), pp. 909–914.

    Google Scholar 

  3. A. V. Sochilina, “An algorithm and a program to establish deducibility, which decides wide classes of formulas,” Kibernetika, No. 3, 136–138 (1978).

    Google Scholar 

  4. M. S. Paterson and M. Wegman, “Linear unification,” J. Comput. Syst. Sci.,16, No. 20, 158–167 (1978).

    Google Scholar 

  5. D. de Champeaux, “About the Paterson-Wegman linear unification algorithm,” J. Comput. Syst. Sci.,32, No. 1, 79–90 (1986).

    Google Scholar 

  6. A. Martelli and U. Montanari, “An efficient unification algorithm,” ACM Trans. Progr. Lang. and Syst.,4, No. 2, 258–282 (1982).

    Google Scholar 

  7. A. Martelli and G. Rossi, “Efficient unification with infinite terms in logic programming,” Proc. Int. Conf. on Fifth-Generation Computer Systems 1984, ICOT, Tokyo (1984), pp. 202–209.

  8. A. I. Degtyarev, “Some special tools of the obviousness algorithm for handling equality,” Automated Processing of Mathematical Texts [in Russian], IK AN UkrSSR, Kiev (1981), pp. 30–36.

    Google Scholar 

  9. H. Yasuura, “On parallel computational complexity of unification,” Proc. Int. conf. on Fifth-Generation Computer Systems 1984, ICOT, Tokyo (1984), pp. 235–243.

  10. C. Dwork, P. C. Kanellakis, and J. C. Mitchell, “On the sequential nature of unification,” J. Logic Progr., No. 1, 35–50 (1984).

    Google Scholar 

  11. J. Siekmann, “Universal unification,” Lecture Notes Comput. Sci.,170, 1–42 (1984).

    Google Scholar 

  12. J. H. Siekmann (ed.), Proc. 8th Int. Conf. on Automated Deduction (Oxford, England, July 27–Aug. 1, 1986), Lecture Notes Comput. Sci., Vol. 232, Springer, Berlin (1986).

    Google Scholar 

  13. A. V. Lyaletskii, “A variant of the Herbrand theorem for prenex formulas,” Kibernetika, No. 1, 112–116 (1981).

    Google Scholar 

  14. A. V. Lyaletskii, “Tools for generating sufficient assertions and consequences in the SAD system,” Mathematical Software in Computerized Logical Deduction and Deductive Construction Systems [in Russian], IK AN UkrSSR, Kiev (1983), pp. 55–63

    Google Scholar 

  15. D. A. Miller, E. L. Cohen, and P. B. Andrews, “A look at TPS,” Lecture Notes Comput. Sci.,138, 50–69 (1982).

    Google Scholar 

  16. Z. Manna and R. Waldinger, “A deductive approach to program synthesis,” ACM Trans. Progr. Lang. and Syst.,2, No. 1, 90–121 (1980).

    Google Scholar 

  17. M. E. Stickel, “A nonclausal connection graph resolution theorem proving program,” Proc. Nat. Conf. on Artif. Intel., Pittsburgh (1982), pp. 229–233.

  18. N. V. Murray, “Completely nonclausal theorem proving,” Artif. Intel.,18, No. 1, 67–85 (1982).

    Google Scholar 

  19. J. A. Robinson, “Mechanizing higher order logic,” Machine Intel.,4, 151–170 (1969).

    Google Scholar 

  20. G. P. Huet, “The indecidability of unification in third order logic,” Inform. and Control,22, 257–267 (1973).

    Google Scholar 

  21. D. Goldfarb, “The undecidability of the second order unification problem,” Theor. Comput Sci.,13, 47–61 (1981).

    Google Scholar 

  22. G. P. Huet, “A unification algorithm for typed λ-calculus,” Theor. Comput. Sci.,1, 27–57 (1975).

    Google Scholar 

  23. P. B. Andrews, “Resolution in type theory,” J. Symbol. Logic,36, No. 3, 414–432 (1971).

    Google Scholar 

  24. A. Church, “A formulation of the simple theory of types,” J. Symbol. Logic,5, 56–68 (1940).

    Google Scholar 

  25. G. P. Huet, “A mechanization of type theory,” Proc. 3rd Int. Conf. on Artif. Intel. (Stanford, 1973), Stanford Univ. (1973), pp. 139–146.

  26. D. C. Jensen and T. Pietrzykowski, “Mechanizing ω-order type theory through unification,” Theor. Comput. Sci.,3, 123–171 (1976).

    Google Scholar 

  27. L. Henshen, “N-sorted logic for automatic theorem-proving in higher-order logic,” Proc. ACM Conf., Boston (1972), pp. 71–81.

  28. Hao Wang, “Logic of many-sorted theories,” J. Symbol. Logic,17, 105–116 (1952).

    Google Scholar 

  29. W. V. O. Quine, “Unification of universes in set theory,” J. Symbol. Logic,21, 267–279.

  30. C. Walter, “A many-sorted calculus based on resolution and paramodulation,” Proc. 8th Int. Joint Conf. on Artif. Intel., Karlsruhe (1983), pp. 882–891.

  31. M. Schmidt-Schauss, “A many-sorted calculus with polymorphic functions based on resolution and paramodulation,” Proc. 9th Int. Joint Conf. on Artif. Intel., Los Angeles (1985), pp. 1162–1168.

  32. A. C. Cohn, “On the solution of Schubert's steamroller in many-sorted logic,” Proc. 9th Int. Joint Conf. on Artif. Intel., Los Angeles (1985), pp. 1169–1174.

  33. K. B. Irani and D. G. Shin, “A many-sorted resolution based on an extension of a firstorder language,” Proc. 9th Int. Joint Conf. on Artif. Intel., Los Angeles (1985), pp. 1175–1177.

  34. M. A. Frisch, “An investigation into inference with restricted quantification and a taxonomic representation,” Logic Progr. Newsletter, No. 6, 5–8 (1985).

    Google Scholar 

  35. K. P. Vershinin and M. K. Morokhovets, “A deduction strategy for assertions with bounded quantifiers,” Kibernetika, No. 3, 9–15 (1983).

    Google Scholar 

  36. C. L. Chag, “Resolution plans in theorem proving,” Proc. 6th Int. Joint Conf. on Artif. Intel., Tokyo (1979), pp. 143–148.

  37. R. Caferra, “Proof by matrix reduction as plan validation,” Lecture Notes Comput. Sci.,138, 309–325 (1982).

    Google Scholar 

  38. P. T. Cox and T. Pietrzykowski, “Deduction plans: a basis for intelligent backtracking,” IEEE Trans., Pattern Analysis and Machine Intel.,3, No. 1, 52–65 (1981).

    Google Scholar 

  39. W. Dilger and H.-A. Schneider, “Assip-T: A theorem proving machine,” Proc. 9th Int. Joint Conf. on Artif. Intel., Los Angeles (1985), pp. 1194–1200.

  40. A. I. Degtyarev, “Monotone paramodulation in planning and backtracking of automatic proofs,” All-Union Conf. on Application of Methods of Math. Logic, abstracts of papers and communications [in Russian], Inst. Kibernet. AN ESSR, Tallin (1986), pp. 61–63.

    Google Scholar 

  41. A. I. Degtyarev, A. P. Zhezherun, and A. V. Lyaletskii, “On some deductive tools in a mathematical text processing system,” Kibernetika, No. 5, 105–108 (1978).

    Google Scholar 

  42. D. A. Plaisted, “Theorem proving by abstractions,” Kibernet. Sb., No. 21, 139–212 (1984).

    Google Scholar 

  43. S. Yu. Maslov, “Application of the reverse method of establishing deducibility to the theory of decidable fragments of classical predicate calculus,” Dokl. AN SSSR,6, 1282–1285 (1966).

    Google Scholar 

  44. S. Yu. Maslov, “The inverse method for establishing deducibility for logical calculi,” Tr. Mat. Inst. im. V. A. Steklova AN SSSR,98, 26087 (1968).

    Google Scholar 

  45. W. H. Joyner, “Resolution strategies in decision procedures,” J. ACM,23, No. 3, 327–364 (1976).

    Google Scholar 

  46. S. A. Tarnlung, “Horn clause computability,” BIT, No. 17, 215–226 (1975).

    Google Scholar 

  47. L. Henshen, “Semantic resolution for Horn sets,” IEEE Trans., Computers,25, No. 8, 816–822 (1976).

    Google Scholar 

  48. W. W. McCune and L. J. Henshen, “Semantic paramodulation for Horn sets,” Proc. 8th Int. Joint Conf. on Artif. Intel., Karlsruhe (1983), pp. 902–908.

  49. S. Yamasaki, M. Yoshida, S. Doshita, and M. Hirata, “A new combination of input and unit deductions for Horn sentences,” Inform. Process. Letters,18, No. 4, 209–214 (1984).

    Google Scholar 

  50. A. I. Degtyarev, “Equality handling methods for Horn sets,” Methods of Algorithmization and Realization of Problem Solving Methods in Artificial Intelligence [in Russian], IK AN UkrSSR, Kiev (1986), pp. 19–26.

    Google Scholar 

  51. P. T. Cox and T. Pietrzykowski, “Incorporating equality into logic programming via surface deduction,” Ann. Pure and Appl. Logic,31, 177–189 (1986).

    Google Scholar 

  52. W. W. Bledsoe, “Splitting and reduction heuristics in automated theorem-proving,” Artif. Intel.,2, 55–77 (1971).

    Google Scholar 

  53. L. Henshen and L. Wos, “Unit refutation and Horn sets,” J. ACM,21, No. 4, 590–605 (1974).

    Google Scholar 

  54. G. E. Peterson, “Theorem proving with lemmas,” J. ACM,23, No. 4, 573–581 (1976).

    Google Scholar 

  55. S. Greenbaum, A. Nagasaka, P. O'Rorke, and D. Plaisted, “Comparison of natural deduction and locking resolution implementation,” Lecture Notes Comput. Sci.,138, 159–171 (1982).

    Google Scholar 

  56. D. A. Plaisted, “A simplified problem reduction format,” Artif. Intel.,18, No. 2, 227–261 (1982).

    Google Scholar 

  57. J. Ketonen and A. Weyhrauch, “A decidable fragment of predicate calculus,” Theor. Comput. Sci.,32, 297–307 (1984).

    Google Scholar 

  58. F. V. Anufriev and Z. M. Asel'derov, “An obviousness algorithm,” Kibernetika, No. 5, 29–60 (1972).

    Google Scholar 

  59. S. Yu. Maslov and E. Ya. Dantsin, “A splitting method and other tautology proving systems,” All-Union Conf. on Methods of Math. Logic in Problems of Artificial Intelligence and Systematic Programming, abstracts of papers and communications [in Russian], Part 1, Inst. Matem. i Kibernet. AN LitSSR, Vilnius (1980), pp. 25–30.

    Google Scholar 

  60. G. S. Tseitin, “On complexity of deduction in predicate calculus,” Zapiski Nauchnykh Seminarov LOMI,8, 234–259 (1968).

    Google Scholar 

  61. R. Statman, “Bounds for proof-search and speedup in predicate calculus,” Ann. Math. Logic,15, No. 3, 225–287 (1978).

    Google Scholar 

  62. V. P. Orevkov, “Lower bounds on the increase of complexity after cutting removal,” Zapiski Nauchnykh Seminarov LOMI,88, 137–162 (1979).

    Google Scholar 

  63. S. A. Cook and R. A. Rechnow, “The relative efficiency of propositional proof systems,” J. Symbol. Logic,44, 36–50 (1979).

    Google Scholar 

  64. B. Dreben and W. P. Goldfarb, The Decision Problem: Solvable Classes of Quantificational Formulas, Addison-Wesley, New York (1979).

    Google Scholar 

  65. A. T. Goldberg, On the Complexity of the Satisfiability Problem, Report NSO-16, Courant Inst. of Math. Sci. (1979).

  66. E. Börger, “Decision problems in predicate logic,” Studies in Logic and Found. Math.,112, 236–301 (1984).

    Google Scholar 

  67. V. A. Steklov, Mathematics and Its Role for Humanity [in Russian], Gosizdat, Moscow (1923).

    Google Scholar 

  68. V. M. Glushkov, Artificial Intelligence. Cybernetics. Topics of Theory and Practice [in Russian], Nauka, Moscow (1986).

    Google Scholar 

Download references

Authors

Additional information

Part I of the survey with its bibliography was published in Kibernetika, No. 3 (1986).

Translated from Kibernetika, No. 4, pp. 88–95, 108, July–August, 1987.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Voronkov, A.A., Degtyarev, A.I. Automatic theorem proving. II. Cybern Syst Anal 23, 547–556 (1987). https://doi.org/10.1007/BF01078915

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01078915

Keywords

Navigation