A Taxonomy of Theorem-Proving Strategies

  • Maria Paola Bonacina
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1600)

Abstract

This article presents a taxonomy of strategies for fully-automated general-purpose first-order theorem proving. It covers forward-reasoning ordering-based strategies and backward-reasoning subgoal-reduction strategies, which do not appear together often. Unlike traditional presentations that emphasize logical inferences, this classification strives to give equal weight to the inference and search components of theorem proving, which are equally important in practice. For this purpose, a formal notion of search plan is given and shown to apply to all classes of strategies. For each class, the form of derivation is specified, and it is shown how inference system and search plan cooperate to generate it.

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References

  1. 1.
    Siva Anantharaman and Nirina Andrianarivelo. Heuristical criteria in refutational theorem proving. In Alfonso Miola, editor, Proceedings of the 1st DISCO, volume 429 of LNCS, pages 184–193. Springer Verlag, 1990.Google Scholar
  2. 2.
    Siva Anantharaman and Maria Paola Bonacina. An application of automated equational reasoning to many-valued logic. In Stéphane Kaplan and Mitsuhiro Okada, editors, Proceedings of CTRS-90, volume 516 of LNCS, pages 156–161. Springer Verlag, 1990.Google Scholar
  3. 3.
    Siva Anantharaman and Jieh Hsiang. Automated proofs of the Moufang identities in alternative rings. Journal of Automated Reasoning, 6(1):76–109, 1990.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Owen L. Astrachan and Don W. Loveland. METEORs: High performance theorem provers using model elimination. Pages 31–60 in [40].Google Scholar
  5. 5.
    Owen L. Astrachan and Don W. Loveland. The use of lemmas in the model elimination procedure. Journal of Automated Reasoning, 19(1):117–141, 1997.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Owen L. Astrachan and Mark E. Stickel. Caching and lemmaizing in model elimination theorem provers. Pages 224–238 in [77].Google Scholar
  7. 7.
    Franz Baader and Jörg H. Siekmann. Unification theory. In [62].Google Scholar
  8. 8.
    Leo Bachmair and Nachum Dershowitz. Inference rules for rewrite-based first-order theorem proving. In Proceedings of LICS-87, pages 331–337. IEEE Computer Society Press, 1987.Google Scholar
  9. 9.
    Leo Bachmair and Nachum Dershowitz. Critical pair criteria for completion. Journal of Symbolic Computation, 6(1):1–18, 1988.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Leo Bachmair, Nachum Dershowitz, and David A. Plaisted. Completion without failure. In Hassam Aït-Kaci and Maurice Nivat, editors, Resolution of Equations in Algebraic Structures, volume II: Rewriting Techniques, pages 1–30. Academic Press, 1989.Google Scholar
  11. 11.
    Leo Bachmair and Harald Ganzinger. Non-clausal resolution and superposition with selection and redundancy criteria. In Andrei Voronkov, editor, Proceedings of LPAR-92, volume 624 of LNAI, pages 273–284. Springer Verlag, 1992.Google Scholar
  12. 12.
    Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217–247, 1994.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Leo Bachmair and Harald Ganzinger. A theory of resolution. Technical Report MPI-I-97-2-005, Max Planck Institut für Informatik. 1997. To appear in J. Alan Robinson and Andrei Voronkov, eds., Handbook of Automated Reasoning.Google Scholar
  14. 14.
    Leo Bachmair, Harald Ganzinger, Christopher Lynch, and Wayne Snyder. Basic paramodulation. Information and Computation, 121(2):172–192, 1995.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Peter Baumgartner. Theory Reasoning in Connection Calculi, volume 1527 of LNAI. Springer, 1998.Google Scholar
  16. 16.
    Peter Baumgartner and Stefan Brüning. A disjunctive positive refinement of model elimination and its application to subsumption deletion. Journal of Auto mated Reasoning, 19:205–262, 1997.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Peter Baumgartner and Ulrich Furbach. PROTEIN: a PROver with a Theory Extension INterface. Pages 769–773 in [41].Google Scholar
  18. 18.
    Peter Baumgartner and Ulrich Furbach. Consolution as a framework for comparing calculi. Journal of Symbolic Computation, 16(5), 1993.Google Scholar
  19. 19.
    Peter Baumgartner and Ulrich Furbach. Model elimination without contrapositives and its application to PTTP. Journal of Automated Reasoning, 13:339–359, 1994.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Peter Baumgartner, Ulrich Furbach, and Frieder Stolzenburg. Model elimination, logic programming and computing answers. Artificial Intelligence, 90(1–2):135–176, 1997.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Bernhard Beckert. Semantic tableaux with equality. Journal of Logic and Com putation, 7(1):39–58, 1997.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Bernhard Beckert, Reiner Hähnle, P. Oel, and M. Sulzmann. The tableau-based theorem prover 3TAP, version 4.0. Pages 303–307 in [101].Google Scholar
  23. 23.
    Bernhard Beckert and Joachim Posegga. leanTAP: lean tableau-based theorem proving. Pages 793–797 in [41].Google Scholar
  24. 24.
    Wolfgang Bibel. Deduction: Automated Logic. Academic Press, 1993.Google Scholar
  25. 25.
    Wolfgang Bibel, Stefan Brüning, Uwe Egly, and T. Rath. KoMeT. Pages 783–788 in [41].Google Scholar
  26. 26.
    Wolfgang Bibel and E. Eder. Methods and calculi for deduction. Pages 68–183 in [62].Google Scholar
  27. 27.
    Wolfgang Bibel and P. H. Schmitt, Eds. Automated Deduction — A Basis for Applications. Kluwer, 1998.Google Scholar
  28. 28.
    Wolgang Bibel. Automated Theorem Proving. Friedr. Vieweg & Sohn, 2nd edition, 1987.Google Scholar
  29. 29.
    Carsten Bierwald and Thomas Käufl. Tableau prover Tatzelwurm: hyper-links and UR-resolution. In Maria Paola Bonacina and Ulrich Furbach, editors, Proceedings of the 1st FTP, number 97-50 in Technical Reports of RISC, pages 22–28. Johannes Kepler Universität, 1997.Google Scholar
  30. 30.
    Roland N. Bol, Krzysztof R. Apt, and J. W. Klop. An analysis of loop checking mechanisms in logic programming. Theoretical Computer Science, 86:35–79, 1991.MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Maria Paola Bonacina. A note on the analysis of theorem-proving strategies. AAR Newsletter, No. 36, pages 2–8, April 1997. Full version available as Technical Report, Department of Computer Science, University of Iowa, May 1996.Google Scholar
  32. 32.
    Maria Paola Bonacina. Analysis of distributed-search contraction-based strategies. In Jürgen Dix, Luis Fariñas del Cerro, and Ulrich Furbach, editors, Proceedings of the 6th JELIA, volume 1489 of LNAI, pages 107–121. Springer, 1998. Full version available as Tech. Rep., Dept. of Comp. Sci., Univ. of Iowa, April 1998.Google Scholar
  33. 33.
    Maria Paola Bonacina and Jieh Hsiang. On rewrite programs: semantics and relationship with Prolog. Journal of Logic Programming, 14(1 & 2):155–180, 1992.MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Maria Paola Bonacina and Jieh Hsiang. On subsumption in distributed derivations. Journal of Automated Reasoning, 12:225–240, 1994.MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Maria Paola Bonacina and Jieh Hsiang. Parallelization of deduction strategies: an analytical study. Journal of Automated Reasoning, 13:1–33, 1994.MathSciNetCrossRefGoogle Scholar
  36. 36.
    Maria Paola Bonacina and Jieh Hsiang. Towards a foundation of completion procedures as semidecision procedures. Theoretical Computer Science, 146:199–242, 1995.MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Maria Paola Bonacina and Jieh Hsiang. A category-theoretic treatment of automated theorem proving. Journal of Information Science and Engineering, 12(1):101–125, 1996.Google Scholar
  38. 38.
    Maria Paola Bonacina and Jieh Hsiang. On semantic resolution with lemmaizing and contraction and a formal treatment of caching. New Generation Computing, 16(2):163–200, 1998.CrossRefGoogle Scholar
  39. 39.
    Maria Paola Bonacina and Jieh Hsiang. On the modelling of search in theorem proving — towards a theory of strategy analysis. Information and Computation, 147:171–208, 1998.MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Robert S. Boyer, Ed. Automated Reasoning — Essays in Honor of Woody Bledsoe. Kluwer, 1991.Google Scholar
  41. 41.
    Alan Bundy, Ed. Proceedings of the 12th CADE, volume 814 of LNAI. Springer, 1994.Google Scholar
  42. 42.
    H.-J. Bürckert. A Resolution Principle for a Logic with Restricted Quantifiers, volume 568 of LNAI. Springer Verlag, 1991.Google Scholar
  43. 43.
    Ricardo Caferra and Nicolas Peltier. Model building in the cross-roads of con sequence and non-consequence relations. In Maria Paola Bonacina and Ulrich Furbach, editors, Proceedings of the 1st FTP, number 97-50 in Technical Reports of RISC, pages 40–44. Johannes Kepler Universität, 1997.Google Scholar
  44. 44.
    Ricardo Caferra and N. Zabel. A method for simultaneous search for refutations and models by equational constraint solving. Journal of Symbolic Computation, 13:613–641, 1992.MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Chin-Liang Chang and Richard Char-Tung Lee. Symbolic Logic and Mechanical Theorem Proving. Academic Press, 1973.Google Scholar
  46. 46.
    Jim Christian. Flatterms, discrimination nets and fast term rewriting. Journal of Automated Reasoning, 10:95–113, 1993.MathSciNetCrossRefGoogle Scholar
  47. 47.
    Heng Chu and David A. Plaisted. CLINS-S: a semantically guided first-order theorem prover. In [136].Google Scholar
  48. 48.
    Heng Chu and David A. Plaisted. Model finding in semantically guided instance-based theorem proving. Fundamenta Informaticae, 21(3):221–235, 1994.MathSciNetMATHGoogle Scholar
  49. 49.
    M. D’Agostino, Dov M. Gabbay, Reiner Hähnle, and Joachim Posegga, Eds. Handbook of Tableau Methods. Kluwer, 1998.Google Scholar
  50. 50.
    Martin Davis and Hilary Putnam. A computing procedure for quantification theory. Journal of the ACM, 7:201–215, 1960.MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Jörg Denzinger and M. Fuchs. Goal-oriented equational theorem proving using team-work. In Proceedings of the 18th KI, volume 861 of LNAI, pages 343–354. Springer, 1994.Google Scholar
  52. 52.
    Nachum Dershowitz. Orderings for term-rewriting systems. Theoretical Computer Science, 17:279–301, 1982.MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Nachum Dershowitz. Computing with rewrite systems. Information and Control, 65:122–157, 1985.MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Nachum Dershowitz. Canonical sets of Horn clauses. In J. Leach Albert, B. Monien, and Mario Rodriguez Artalejo, editors, Proceedings of the 18th ICALP, volume 510 of LNCS, pages 267–278. Springer Verlag, 1991.Google Scholar
  55. 55.
    Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 243–320. Elsevier, 1990.Google Scholar
  56. 56.
    Roland Dietrich. Relating resolution and algebraic completion for Horn logic. Pages 62–78 in [123].Google Scholar
  57. 57.
    Norbert Eisinger and Hans Jürgen Ohlbach. Deduction systems based on resolution. Pages 184–273 in [62].Google Scholar
  58. 58.
    Christian Fermüller and Alexander Leitsch. Hyperresolution and automated model-building. Journal of Logic and Computation, 6(2):173–203, 1996.MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Christian Fermüller and Alexander Leitsch. Decision procedures and model-building in equational clause logic. Journal of the IGPL, 6(1):17–41, 1998.MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Melvin Fitting. First-order Logic and Automated Theorem Proving. Springer, 1990.Google Scholar
  61. 61.
    S. Fleisig, Don W. Loveland, A. Smiley, and D. Yarmush. An implementation of the model elimination proof procedure. Journal of the ACM, 21:124–139, 1974.MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Dov M. Gabbay, Christopher J. Hogger, and J. Alan Robinson, Eds. Handbook of Logic in Artificial Intelligence and Logic Programming (Vol. 1 & 2). Oxford University Press, 1993.Google Scholar
  63. 63.
    Jean Gallier. Logic for Computer Science — Foundations of Automatic Theorem Proving. Harper & Row, 1986.Google Scholar
  64. 64.
    Chr. Goller, Reinhold Letz, K. Mayr, and Johann Schumann. SETHEO v3.2: recent developments. Pages 778–782 in [41].Google Scholar
  65. 65.
    John Harrison. Optimizing proof search in model elimination. Pages 313–327 in [101].Google Scholar
  66. 66.
    Ryuzo Hasegawa, Miyuki Koshimura, and Hiroshi Fujita. MGTP: a parallel the orem prover based on lazy model generation. Pages 776–780 in [77].Google Scholar
  67. 67.
    Jieh Hsiang. Refutational theorem proving using term rewriting systems. Artificial Intelligence, 25:255–300, 1985.MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Jieh Hsiang. Rewrite method for theorem proving in first order theories with equality. Journal of Symbolic Computation, 3:133–151, 1987.MathSciNetCrossRefMATHGoogle Scholar
  69. 69.
    Jieh Hsiang and Michaël Rusinowitch. On word problems in equational theories. In Th. Ottman, editor, Proceedings of the 14th ICALP, volume 267 of LNCS, pages 54–71. Springer Verlag, 1987.Google Scholar
  70. 70.
    Jieh Hsiang and Michaël Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559–587, 1991.MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Jieh Hsiang, Michaël Rusinowitch, and Ko Sakai. Complete inference rules for the cancellation laws. In Proceedings of the 10th IJCAI, pages 990–992, 1987.Google Scholar
  72. 72.
    Jean-Pierre Jouannaud and Claude Kirchner. Solving equations in abstract algebras: a rule-based survey of unification. Pages 257–321 in [83].Google Scholar
  73. 73.
    Deepak Kapur, Dave Musser, and Paliath Narendran. Only prime superposition need be considered in the Knuth-Bendix completion procedure. Journal of Symbolic Computation, 6:19–36, 1988.MathSciNetCrossRefMATHGoogle Scholar
  74. 74.
    Deepak Kapur and Paliath Narendran. An equational approach to theorem proving in first order predicate calculus. In Proceedings of the 9th IJCAI, pages 1146–1153, 1985.Google Scholar
  75. 75.
    Deepak Kapur and Hantao Zhang. A case study of the completion procedure: proving ring commutativity problems. Pages 360–394 in [83].Google Scholar
  76. 76.
    Deepak Kapur and Hantao Zhang. An overview of RRL: rewrite rule laboratory. In Nachum Dershowitz, editor, Proceedings of the 3rd RTA, volume 355 of LNCS, pages 513–529. Springer Verlag, 1989.Google Scholar
  77. 77.
    Deepak Kapur, Ed. Proceedings of the 11th CADE, volume 607 of LNAI. Springer, 1992.Google Scholar
  78. 78.
    Claude Kirchner, Hélène Kirchner, and Michaël Rusinowitch. Deduction with symbolic constraints. Revue Française d’Intelligence Artificielle, 4(3):9–52, 1990.Google Scholar
  79. 79.
    Donald E. Knuth and Peter B. Bendix. Simple word problems in universal algebras. In J. Leech, editor, Proceedings of the Conf. on Computational Problems in Abstract Algebras, pages 263–298. Pergamon Press, 1970.Google Scholar
  80. 80.
    R. E. Korf. Depth-first iterative deepening: an optimal admissible tree search. Artificial Intelligence, 27(1):97–109, 1985.MathSciNetCrossRefMATHGoogle Scholar
  81. 81.
    Robert Kowalski. Search strategies for theorem proving. In B. Meltzer and D. Michie, editors, Machine Intelligence, volume 5, pages 181–201. Edinburgh University Press, 1969.Google Scholar
  82. 82.
    Robert Kowalski and D. Kuehner. Linear resolution with selection function. Artificial Intelligence, 2:227–260, 1971.MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    Jean-Louis Lassez and Gordon Plotkin, Eds. Computational Logic — Essays in Honor of Alan Robinson. The MIT Press, 1991.Google Scholar
  84. 84.
    Shie-Jue Lee and David A. Plaisted. Eliminating duplication with the hyperlinking strategy. Journal of Automated Reasoning, 9:25–42, 1992.MathSciNetCrossRefMATHGoogle Scholar
  85. 85.
    Alexander Leitsch. The Resolution Calculus. Springer, 1997.Google Scholar
  86. 86.
    Reinhold Letz, K. Mayr, and Chr. Goller. Controlled integration of the cut rule into connection tableau calculi. Journal of Automated Reasoning, 13(3):297–338, 1994.MathSciNetCrossRefMATHGoogle Scholar
  87. 87.
    Reinhold Letz, Johann Schumann, S. Bayerl, and Wolfgang Bibel. SETHEO: a high performance theorem prover. Journal of Automated Reasoning, 8(2):183–212, 1992.MathSciNetCrossRefMATHGoogle Scholar
  88. 88.
    John W. Lloyd. Foundations of Logic Programming. Springer Verlag, 2nd edition, 1987.Google Scholar
  89. 89.
    Don W. Loveland. A simplified format for the model elimination procedure. Journal of the ACM, 16(3):349–363, 1969.MathSciNetCrossRefMATHGoogle Scholar
  90. 90.
    Don W. Loveland. A unifying view of some linear Herbrand procedures. Journal of the ACM, 19(2):366–384, 1972.MathSciNetCrossRefMATHGoogle Scholar
  91. 91.
    Don W. Loveland. Automated Theorem Proving: A Logical Basis. North-Holland, 1978.Google Scholar
  92. 92.
    Ewing Lusk and Ross Overbeek, Eds. Proceedings of the 9th CADE, volume 310 of LNCS. Springer Verlag, 1988.Google Scholar
  93. 93.
    Rainer Manthey and François Bry. SATCHMO: a theorem prover implemented in Prolog. Pages 415–434 in [92].Google Scholar
  94. 94.
    J. McCharen, Ross Overbeek, and Larry Wos. Complexity and related enhancements for automated theorem proving programs. Computers and Mathematics with Applications, 2(1): 1–16, 1976.CrossRefMATHGoogle Scholar
  95. 95.
    William W. McCune. Experiments with discrimination tree indexing and path indexing for term retrieval. Journal of Automated Reasoning, 9(2):147–167, 1992.MathSciNetCrossRefMATHGoogle Scholar
  96. 96.
    William W. McCune. A Davis-Putnam program and its application to finite first-order model search: quasigroup existence problems. Unpublished manuscript, May 1994.Google Scholar
  97. 97.
    William W. McCune. Otter 3.0 reference manual and guide. Technical Report 94/6, MCS Division, Argonne National Laboratory, 1994.Google Scholar
  98. 98.
    William W. McCune. 33 Basic test problems: a practical evaluation of some paramodulation strategies. In Robert Veroff, editor, Automated Reasoning and its Applications: Essays in Honor of Larry Wos, pages 71–114. MIT Press, 1997.Google Scholar
  99. 99.
    William W. McCune. Solution of the Robbins problem. Journal of Automated Reasoning, 19(3):263–276, 1997.MathSciNetCrossRefMATHGoogle Scholar
  100. 100.
    William W. McCune, Ed. Proceedings of the 14th CADE, volume 1249 of LNAI. Springer, 1997.Google Scholar
  101. 101.
    Michael McRobbie and John Slaney, Eds. Proceedings of the 13th CADE, volume 1104 of LNAI. Springer, 1996.Google Scholar
  102. 102.
    Jürgen Müller and Rolf Socher-Ambrosius. Topics in completion theorem proving. Technical Report SEKI SR-88-13, Fachbereich Informatik, Univ. Kaiserslautern, 1988.Google Scholar
  103. 103.
    Xumin Nie and David A. Plaisted. A complete semantic back chaining proof system. Pages 16–27 in [135].Google Scholar
  104. 104.
    Robert Niewenhuis, José Miguel Rivero, and Miguel Angel Vallejo. The Barcelona prover. In [136].Google Scholar
  105. 105.
    Robert Niewenhuis and A. Rubio. Theorem proving with ordering and equality constrained clauses. Journal of Symbolic Computation, 19(4):321–351, 1995.MathSciNetCrossRefMATHGoogle Scholar
  106. 106.
    Judea Pearl. Heuristics — Intelligent Search Strategies for Computer Problem Solving. Addison Wesley, 1984.Google Scholar
  107. 107.
    Gerald E. Peterson. A technique for establishing completeness results in theorem proving with equality. SIAM Journal of Computing, 12(1):82–100, 1983.MathSciNetCrossRefMATHGoogle Scholar
  108. 108.
    David A. Plaisted. Equational reasoning and term rewriting systems. Pages 273–364 in [62].Google Scholar
  109. 109.
    David A. Plaisted. A simplified problem reduction format. Artificial Intelligence, 18:227–261, 1982.MathSciNetCrossRefMATHGoogle Scholar
  110. 110.
    David A. Plaisted. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning, 4(3):287–325, 1988.MathSciNetCrossRefMATHGoogle Scholar
  111. 111.
    David A. Plaisted. Mechanical theorem proving. In Ranan B. Banerji, editor, Formal Techniques in Artificial Intelligence. Elsevier, 1990.Google Scholar
  112. 112.
    David A. Plaisted. A sequent-style model elimination strategy and a positive refinement. Journal of Automated Reasoning, 6(4):389–402, 1990.MathSciNetCrossRefMATHGoogle Scholar
  113. 113.
    David A. Plaisted and Yunshan Zhu. The Efficiency of Theorem Proving Strate gies. Friedr. Vieweg & Sohns, 1997.Google Scholar
  114. 114.
    David A. Plaisted and Yunshan Zhu. Ordered semantic hyper linking. In Proceedings of AAAI-97, 1997.Google Scholar
  115. 115.
    Allan Ramsay. Formal Methods in Artificial Intelligence. Cambridge University Press, 1988.Google Scholar
  116. 116.
    G. Robinson and Larry Wos. Paramodulation and theorem-proving in first-order theories with equality. In D. Michie and R. Meltzer, editors, Machine Intelligence, volume IV, pages 135–150. Edinburgh Univ. Press, 1969.Google Scholar
  117. 117.
    J. Alan Robinson. Automatic deduction with hyper-resolution. International Journal of Computer Mathematics, 1:227–234, 1965.MathSciNetMATHGoogle Scholar
  118. 118.
    J. Alan Robinson. A machine oriented logic based on the resolution principle. Journal of the ACM, 12(1):23–41, 1965.MathSciNetCrossRefMATHGoogle Scholar
  119. 119.
    Michaël Rusinowitch. Theorem-proving with resolution and superposition. Journal of Symbolic Computation, 11(1 & 2):21–50, 1991.MathSciNetCrossRefMATHGoogle Scholar
  120. 120.
    Johann Schumann. Delta: a bottom-up pre-processor for top-down theorem provers. Pages 774–777 in [41].Google Scholar
  121. 121.
    Robert E. Shostak. Refutation graphs. Artificial Intelligence, 7:51–64, 1976.MathSciNetCrossRefMATHGoogle Scholar
  122. 122.
    Jörg H. Siekmann and Graham Wrightson, Eds. Automation of reasoning-Classical Papers on Computational Logic. Springer Verlag, 1983.Google Scholar
  123. 123.
    Jörg H. Siekmann, Ed. Proceedings of the 8th CADE, volume 230 of LNCS. Springer, 1986.Google Scholar
  124. 124.
    James R. Slagle. Automatic theorem proving with renamable and semantic resolution. Journal of the ACM, 14(4):687–697, 1967.MathSciNetCrossRefMATHGoogle Scholar
  125. 125.
    John Slaney. FINDER: finite domain enumerator. Pages 798–801 in [41].Google Scholar
  126. 126.
    Raymond M. Smullyan. First-Order Logic. Dover, 1995. (Republication of the work first published as “Band 43” Series Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag, 1968).Google Scholar
  127. 127.
    Rolf Socher-Ambrosius. How to avoid the derivation of redundant clauses in reasoning systems. Journal of Automated Reasoning, 9(1):77–98, 1992.MathSciNetCrossRefMATHGoogle Scholar
  128. 128.
    Rolf Socher-Ambrosius and Patricia Johann. Deduction systems. Springer, 1997.Google Scholar
  129. 129.
    Mark E. Stickel. PTTP and linked inference. Pages 283–296 in [40].Google Scholar
  130. 130.
    Mark E. Stickel. A Prolog technology theorem prover. New Generation Computing, 2(4):371–383, 1984.CrossRefGoogle Scholar
  131. 131.
    Mark E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, 1:333–355, 1985.MathSciNetCrossRefMATHGoogle Scholar
  132. 132.
    Mark E. Stickel. A Prolog technology theorem prover: implementation by an extended Prolog compiler. Journal of Automated Reasoning, 4:353–380, 1988.MathSciNetCrossRefMATHGoogle Scholar
  133. 133.
    Mark E. Stickel. The path-indexing method for indexing terms. Technical Report 473, SRI International, 1989.Google Scholar
  134. 134.
    Mark E. Stickel, Richard Waldinger, Michael Lowry, Thomas Pressburger, and Ian Underwood. Deductive composition of astronomical software from subroutine libraries. Pages 341–355 in [41].Google Scholar
  135. 135.
    Mark E. Stickel, Ed. Proceedings of the 10th CADE, volume 449 of LNAI. Springer, 1990.Google Scholar
  136. 136.
    Geoff Sutcliffe and Christian Suttner, Eds. The CADE-13 ATP system competition. Journal of Automated Reasoning, 18(2), 1997.Google Scholar
  137. 137.
    Tanel Tammet. Gandalf. Pages 199–204 in [136].Google Scholar
  138. 138.
    Laurent Vigneron. Automated deduction techniques for studying rough algebras. Fundamenta Informaticae, 33:85–103, 1998.MathSciNetMATHGoogle Scholar
  139. 139.
    Kevin Wallace and Graham Wrightson. Regressive merging in model elimination tableau-based theorem provers. Journal of the IGPL, 3(6):921–937, 1995.MathSciNetCrossRefMATHGoogle Scholar
  140. 140.
    David H. D. Warren. An abstract Prolog instruction set. Technical Report 309, SRI International, 1983.Google Scholar
  141. 141.
    David S. Warren. Memoing for logic programs. Communications of the ACM, 35(3):94–111, 1992.CrossRefGoogle Scholar
  142. 142.
    Christoph Weidenbach, B. Gaede, and G. Rock. SPASS & FLOTTER, version 0.42. Pages 141–145 in [101].Google Scholar
  143. 143.
    Larry Wos, D. Carson, and G. Robinson. Efficiency and completeness of the set of support strategy in theorem proving. Journal of the ACM, 12:536–541, 1965.MathSciNetCrossRefMATHGoogle Scholar
  144. 144.
    Larry Wos, G. Robinson, D. Carson, and L. Shalla. The concept of demodulation in theorem proving. Journal of the ACM, 14(4):698–709, 1967.CrossRefMATHGoogle Scholar
  145. 145.
    Lary Wos, Ross Overbeek, Ewing Lusk, and J. Boyle. Automated Reasoning: Introduction and Applications. McGraw-Hill, 2nd edition, 1992.Google Scholar
  146. 146.
    Hantao Zhang. SATO: an efficient propositional prover. Pages 272–275 in [100].Google Scholar
  147. 147.
    Hantao Zhang. A new method for the boolean ring based theorem proving. Journal of Symbolic Computation, 17(2):189–211, 1994.MathSciNetCrossRefMATHGoogle Scholar
  148. 148.
    Hantao Zhang, Maria Paola Bonacina, and Jieh Hsiang. PSATO: a distributed propositional prover and its application to quasigroup problems. Journal of Symbolic Computation, 21:543–560, 1996.MathSciNetCrossRefMATHGoogle Scholar
  149. 149.
    Hantao Zhang and Mark E. Stickel. Implementing the Davis-Putnam algorithm by tries. Technical Report 94-12, Department of Computer Science, University of Iowa, 1994.Google Scholar
  150. 150.
    Jian Zhang and Hantao Zhang. Generating models by SEM. Pages 308–312 in [101].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Maria Paola Bonacina
    • 1
  1. 1.Department of Computer ScienceThe University of IowaIowa CityUSA

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