Parallel Theorem Proving

Chapter

Abstract

This chapter surveys the research in parallel or distributed strategies for mechanical theorem proving in first-order logic, and explores some of its connections with the research in the parallelization of decision procedures for satisfiability in propositional logic (SAT). We clarify the key role played by the Clause-Diffusion methodology for distributed deduction in moving parallel reasoning from the parallelization of inferences to the parallelization of search, which is the dominating paradigm today. Since the quest for parallel first-order proof procedures has not been pursued recently, we endeavour to relate lessons learned from investigations of parallel theorem proving and parallel SAT-solving with novel advances in theorem proving, such as SGGS (Semantically-Guided Goal-Sensitive reasoning), a method that lifts the CDCL (Conflict-Driven Clause Learning) procedure for SAT to first-order logic.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Martin Aigner, Armin Biere, Christoph M. Kirsch, Aina Niemetz, and Mathias Preiner. Analysis of portfolio-style parallel SAT solving on current multi-core architectures. In Daniel Le Berre and Allen Van Gelder, editors, Notes of the Fourth Workshop on Pragmatics of SAT (POS), Sixteenth International Conference on Theory and Applications of Satisfiability Testing (SAT), pages 28–40, 2013.Google Scholar
  2. [2]
    Gábor Alagi and Christoph Weidenbach. NRCL – a model building approach to the Bernays-Schönfinkel fragment. In Carsten Lutz and Silvio Ranise, editors, Proceedings of the Tenth International Symposium on Frontiers of Combining Systems (FroCoS), volume 9322 of Lecture Notes in Artificial Intelligence, pages 69–84. Springer, 2015.Google Scholar
  3. [3]
    Iliès Alouini. Concurrent garbage collector for concurrent rewriting. In Jieh Hsiang, editor, Proceedings of the Sixth International Conference on Rewriting Techniques and Applications (RTA), volume 914 of Lecture Notes in Computer Science, pages 132–146. Springer, 1995.Google Scholar
  4. [4]
    Iliès Alouini. Étude et mise en oeuvre de la réecriture conditionnelle concurrente sur des machines parallèles à mémoire distribuée. PhD thesis, Université Henri Poincaré Nancy 1, May 1997.Google Scholar
  5. [5]
    Siva Anantharaman and Nirina Andrianarivelo. Heuristical criteria in refutational theorem proving. In Alfonso Miola, editor, Proceedings of the First International Symposium on Design and Implementation of Symbolic Computation Systems (DISCO), volume 429 of Lecture Notes in Computer Science, pages 184–193. Springer, 1990.Google Scholar
  6. [6]
    Siva Anantharaman and Jieh Hsiang. Automated proofs of the Moufang identities in alternative rings. Journal of Automated Reasoning, 6(1):76–109, 1990.Google Scholar
  7. [7]
    Owen L. Astrachan and Donald W. Loveland. METEORs: high performance theorem provers using model elimination. In Robert S. Boyer, editor, Automated Reasoning: Essays in Honor of Woody Bledsoe, pages 31–60. Kluwer Academic Publishers, The Netherlands, 1991.Google Scholar
  8. [8]
    Owen L. Astrachan and Mark E. Stickel. Caching and lemmaizing in model elimination theorem provers. In Deepak Kapur, editor, Proceedings of the Eleventh International Conference on Automated Deduction (CADE), volume 607 of Lecture Notes in Artificial Intelligence, pages 224–238. Springer, 1992.Google Scholar
  9. [9]
    Jürgen Avenhaus and Jörg Denzinger. Distributing equational theorem proving. In Claude Kirchner, editor, Proceedings of the Fifth International Conference on Rewriting Techniques and Applications (RTA), volume 690 of Lecture Notes in Computer Science, pages 62–76. Springer, 1993.Google Scholar
  10. [10]
    Jürgen Avenhaus, Jörg Denzinger, and Matthias Fuchs. DISCOUNT: a system for distributed equational deduction. In Jieh Hsiang, editor, Proceedings of the Sixth International Conference on Rewriting Techniques and Applications (RTA), volume 914 of Lecture Notes in Computer Science, pages 397–402. Springer, 1995.Google Scholar
  11. [11]
    Leo Bachmair and Nachum Dershowitz. Critical pair criteria for completion. Journal of Symbolic Computation, 6(1):1–18, 1988.Google Scholar
  12. [12]
    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, Cambridge, England, 1989.Google Scholar
  13. [13]
    Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217–247, 1994.Google Scholar
  14. [14]
    Leo Bachmair, Harald Ganzinger, Christopher Lynch, and Wayne Snyder. Basic paramodulation. Information and Computation, 121(2):172–192, 1995.Google Scholar
  15. [15]
    Leo Bachmair, Harald Ganzinger, David McAllester, and Christopher A. Lynch. Resolution theorem proving. In John Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, volume 1, chapter 2, pages 535–610. Elsevier, Amsterdam, The Netherlands, 2001.Google Scholar
  16. [16]
    Peter Baumgartner. Hyper tableaux – the next generation. In Harrie de Swart, editor, Proceedings of the Seventh International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX), volume 1397 of Lecture Notes in Artificial Intelligence, pages 60–76. Springer, 1998.Google Scholar
  17. [17]
    Peter Baumgartner, Alexander Fuchs, and Cesare Tinelli. Implementing the model evolution calculus. International Journal on Artificial Intelligence Tools, 15(1):21–52, 2006.Google Scholar
  18. [18]
    Peter Baumgartner, Alexander Fuchs, and Cesare Tinelli. Lemma learning in the model evolution calculus. In Miki Hermann and Andrei Voronkov, editors, Proceedings of the Thirteenth Conference on Logic, Programming and Automated Reasoning (LPAR), volume 4246 of Lecture Notes in Artificial Intelligence, pages 572–586. Springer, 2006.Google Scholar
  19. [19]
    Peter Baumgartner and Ulrich Furbach. Consolution as a framework for comparing calculi. Journal of Symbolic Computation, 16(5):445–477, 1993.Google Scholar
  20. [20]
    Peter Baumgartner and Ulrich Furbach. Variants of clausal tableaux. In Wolfgang Bibel and Peter H. Schmitt, editors, Automated Deduction - A Basis for Applications, volume I: Foundations - Calculi and Methods, chapter 3, pages 73–102. Kluwer Academic Publishers, The Netherlands, 1998.Google Scholar
  21. [21]
    Peter Baumgartner, Björn Pelzer, and Cesare Tinelli. Model evolution calculus with equality - revised and implemented. Journal of Symbolic Computation, 47(9):1011–1045, 2012.Google Scholar
  22. [22]
    Peter Baumgartner and Cesare Tinelli. The model evolution calculus as a first-order DPLL method. Artificial Intelligence, 172(4/5):591–632, 2008.Google Scholar
  23. [23]
    Peter Baumgartner and Uwe Waldmann. Superposition and model evolution combined. In Renate Schmidt, editor, Proceedings of the Twenty-Second International Conference on Automated Deduction (CADE), volume 5663 of Lecture Notes in Artificial Intelligence, pages 17–34. Springer, 2009.Google Scholar
  24. [24]
    Markus Bender, Björn Pelzer, and Claudia Schon. E-KRHyper 1.4: extensions for unique names and description logic. In Maria Paola Bonacina, editor, Proceedings of the Twenty-Fourth International Conference on Automated Deduction (CADE), volume 7898 of Lecture Notes in Artificial Intelligence, pages 126–134. Springer, 2013.Google Scholar
  25. [25]
    Wolfgang Bibel and Elmer Eder. Methods and calculi for deduction. In Dov M. Gabbay, Christopher J. Hogger, and John Alan Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume I: Logical Foundations, pages 68–183. Oxford University Press, Oxford, England, 1993.Google Scholar
  26. [26]
    Jean-Paul Billon. The disconnection method. In Pierangelo Miglioli, Ugo Moscato, Daniele Mundici, and Mario Ornaghi, editors, Proceedings of the Fifth International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX), volume 1071 of Lecture Notes in Artificial Intelligence, pages 110–126. Springer, 1996.Google Scholar
  27. [27]
    Maria Paola Bonacina. Distributed automated deduction. PhD thesis, Department of Computer Science, State University of New York at Stony Brook, December 1992.Google Scholar
  28. [28]
    Maria Paola Bonacina. On the reconstruction of proofs in distributed theorem proving with contraction: a modified Clause-Diffusion method. In Hoon Hong, editor, Proceedings of the First International Symposium on Parallel Symbolic Computation (PASCO), volume 5 of Lecture Notes Series in Computing, pages 22–33. World Scientific, 1994.Google Scholar
  29. [29]
    Maria Paola Bonacina. On the reconstruction of proofs in distributed theorem proving: a modified Clause-Diffusion method. Journal of Symbolic Computation, 21(4–6):507–522, 1996.Google Scholar
  30. [30]
    Maria Paola Bonacina. The Clause-Diffusion theorem prover Peers-mcd. In William W. McCune, editor, Proceedings of the Fourteenth International Conference on Automated Deduction (CADE), volume 1249 of Lecture Notes in Artificial Intelligence, pages 53–56. Springer, 1997.Google Scholar
  31. [31]
    Maria Paola Bonacina. Experiments with subdivision of search in distributed theorem proving. In Markus Hitz and Erich Kaltofen, editors, Proceedings of the Second International Symposium on Parallel Symbolic Computation (PASCO), pages 88–100. ACM Press, 1997.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 Sixth European Workshop on Logics in Artificial Intelligence (JELIA), volume 1489 of Lecture Notes in Artificial Intelligence, pages 107–121. Springer, 1998.Google Scholar
  33. [33]
    Maria Paola Bonacina. Mechanical proofs of the Levi commutator problem. In Peter Baumgartner et al., editor, Notes of the Workshop on Problem Solving Methodologies with Automated Deduction, Fifteenth International Conference on Automated Deduction (CADE), pages 1–10, 1998.Google Scholar
  34. [34]
    Maria Paola Bonacina. A model and a first analysis of distributed-search contraction-based strategies. Annals of Mathematics and Artificial Intelligence, 27(1–4):149–199, 1999.Google Scholar
  35. [35]
    Maria Paola Bonacina. A taxonomy of theorem-proving strategies. In Michael J. Wooldridge and Manuela Veloso, editors, Artificial Intelligence Today - Recent Trends and Developments, volume 1600 of Lecture Notes in Artificial Intelligence, pages 43–84. Springer, Berlin, Germany, 1999.Google Scholar
  36. [36]
    Maria Paola Bonacina. Ten years of parallel theorem proving: a perspective. In Bernhard Gramlich, Hélène Kirchner, and Frank Pfenning, editors, Notes of the Third Workshop on Strategies in Automated Deduction (STRATEGIES), Second Federated Logic Conference (FLoC), pages 3–15, 1999.Google Scholar
  37. [37]
    Maria Paola Bonacina. A taxonomy of parallel strategies for deduction. Annals of Mathematics and Artificial Intelligence, 29(1–4):223–257, 2000.Google Scholar
  38. [38]
    Maria Paola Bonacina. Combination of distributed search and multi-search in Peers-mcd.d. In Rajeev P. Gore, Alexander Leitsch, and Tobias Nipkow, editors, Proceedings of the First International Joint Conference on Automated Reasoning (IJCAR), volume 2083 of Lecture Notes in Artificial Intelligence, pages 448–452. Springer, 2001.Google Scholar
  39. [39]
    Maria Paola Bonacina. Towards a unified model of search in theorem proving: subgoal-reduction strategies. Journal of Symbolic Computation, 39(2):209–255, 2005.Google Scholar
  40. [40]
    Maria Paola Bonacina. On theorem proving for program checking – Historical perspective and recent developments. In Maribel Fernandez, editor, Proceedings of the Twelfth International Symposium on Principles and Practice of Declarative Programming (PPDP), pages 1–11. ACM Press, 2010.Google Scholar
  41. [41]
    Maria Paola Bonacina and Nachum Dershowitz. Abstract canonical inference. ACM Transactions on Computational Logic, 8(1):180–208, 2007.Google Scholar
  42. [42]
    Maria Paola Bonacina and Nachum Dershowitz. Canonical ground Horn theories. In Andrei Voronkov and Christoph Weidenbach, editors, Programming Logics: Essays in Memory of Harald Ganzinger, volume 7797 of Lecture Notes in Artificial Intelligence, pages 35–71. Springer, 2013.Google Scholar
  43. [43]
    Maria Paola Bonacina and Mnacho Echenim. Theory decision by decomposition. Journal of Symbolic Computation, 45(2):229–260, 2010.Google Scholar
  44. [44]
    Maria Paola Bonacina, Ulrich Furbach, and Viorica Sofronie-Stokkermans. On first-order model-based reasoning. In Narciso Martí-Oliet, Peter Olveczky, and Carolyn Talcott, editors, Logic, Rewriting, and Concurrency: Essays Dedicated to José Meseguer, volume 9200 of Lecture Notes in Computer Science, pages 181–204. Springer, Berlin, Germany, 2015.Google Scholar
  45. [45]
    Maria Paola Bonacina and Jieh Hsiang. High performance simplificationbased automated deduction. In Transactions of the Ninth U.S. Army Conference on Applied Mathematics and Computing, number 92-1, pages 321–335. Army Research Office, 1991.Google Scholar
  46. [46]
    Maria Paola Bonacina and Jieh Hsiang. A system for distributed simplificationbased theorem proving. In Bertrand Fronhöfer and Graham Wrightson, editors, Proceedings of the First International Workshop on Parallelization in Inference Systems (December 1990), volume 590 of Lecture Notes in Artificial Intelligence, pages 370–370. Springer, Berlin, Germany, 1992.Google Scholar
  47. [47]
    Maria Paola Bonacina and Jieh Hsiang. Distributed deduction by Clause-Diffusion: the Aquarius prover. In Alfonso Miola, editor, Proceedings of the Third International Symposium on Design and Implementation of Symbolic Computation Systems (DISCO), volume 722 of Lecture Notes in Computer Science, pages 272–287. Springer, 1993.Google Scholar
  48. [48]
    Maria Paola Bonacina and Jieh Hsiang. On fairness in distributed deduction. In Patrice Enjalbert, Alain Finkel, and Klaus W. Wagner, editors, Proceedings of the Tenth Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 665 of Lecture Notes in Computer Science, pages 141–152. Springer, 1993.Google Scholar
  49. [49]
    Maria Paola Bonacina and Jieh Hsiang. On subsumption in distributed derivations. Journal of Automated Reasoning, 12:225–240, 1994.Google Scholar
  50. [50]
    Maria Paola Bonacina and Jieh Hsiang. Parallelization of deduction strategies: an analytical study. Journal of Automated Reasoning, 13:1–33, 1994.Google Scholar
  51. [51]
    Maria Paola Bonacina and Jieh Hsiang. The Clause-Diffusion methodology for distributed deduction. Fundamenta Informaticae, 24(1–2):177–207, 1995.Google Scholar
  52. [52]
    Maria Paola Bonacina and Jieh Hsiang. Distributed deduction by Clause-Diffusion: distributed contraction and the Aquarius prover. Journal of Symbolic Computation, 19:245–267, 1995.Google Scholar
  53. [53]
    Maria Paola Bonacina and Jieh Hsiang. Towards a foundation of completion procedures as semidecision procedures. Theoretical Computer Science, 146:199–242, 1995.Google Scholar
  54. [54]
    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.Google Scholar
  55. [55]
    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.Google Scholar
  56. [56]
    Maria Paola Bonacina, Christopher A. Lynch, and Leonardo de Moura. On deciding satisfiability by DPLL (\(\varGamma + {\mathscr{T}} \)) and unsound theorem proving. In Renate Schmidt, editor, Proceedings of the Twenty-second International Conference on Automated Deduction (CADE), volume 5663 of Lecture Notes in Artificial Intelligence, pages 35–50. Springer, 2009.Google Scholar
  57. [57]
    Maria Paola Bonacina, Christopher A. Lynch, and Leonardo de Moura. On deciding satisfiability by theorem proving with speculative inferences. Journal of Automated Reasoning, 47(2):161–189, 2011.Google Scholar
  58. [58]
    Maria Paola Bonacina and William W. McCune. Distributed theorem proving by Peers. In Alan Bundy, editor, Proceedings of the Twelfth International Conference on Automated Deduction (CADE), volume 814 of Lecture Notes in Artificial Intelligence, pages 841–845. Springer, 1994.Google Scholar
  59. [59]
    Maria Paola Bonacina and David A. Plaisted. Constraint manipulation in SGGS. In Temur Kutsia and Christophe Ringeissen, editors, Proceedings of the Twenty-Eighth Workshop on Unification (UNIF), Sixth Federated Logic Conference (FLoC), Technical Reports of the Research Institute for Symbolic Computation, pages 47–54. Johannes Kepler Universität, 2014. Available at http://vsl2014.at/meetings/UNIF-index.html.
  60. [60]
    Maria Paola Bonacina and David A. Plaisted. SGGS theorem proving: an exposition. In Stephan Schulz, Leonardo De Moura, and Boris Konev, editors, Proceedings of the Fourth Workshop on Practical Aspects in Automated Reasoning (PAAR), Sixth Federated Logic Conference (FLoC), July 2014, volume 31 of EasyChair Proceedings in Computing (EPiC), pages 25–38, 2015.Google Scholar
  61. [61]
    Maria Paola Bonacina and David A. Plaisted. Semantically-guided goalsensitive reasoning: model representation. Journal of Automated Reasoning, 56(2):113–141, 2016.Google Scholar
  62. [62]
    Maria Paola Bonacina and David A. Plaisted. Semantically-guided goalsensitive reasoning: inference system and completeness. Journal of Automated Reasoning, 56(2):165–218, 2016.Google Scholar
  63. [63]
    Soumitra Bose, Edmund M. Clarke, David E. Long, and Spiro Michaylov. Parthenon: A parallel theorem prover for non-Horn clauses. Journal of Automated Reasoning, 8(2):153–182, 1992.Google Scholar
  64. [64]
    Bruno Buchberger. An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal (in German). PhD thesis, Department of Mathematics, Universität Innsbruck, 1965.Google Scholar
  65. [65]
    Bruno Buchberger. History and basic features of the critical-pair/completion procedure. Journal of Symbolic Computation, 3:3–38, 1987.Google Scholar
  66. [66]
    Reinhard Bündgen, Manfred Göbel, and Wolfgang Küchlin. Strategycompliant multi-threaded term completion. Journal of Symbolic Computation, 21(4–6):475–506, 1996.Google Scholar
  67. [67]
    Ralph M. Butler and Ewing L. Lusk. User’s guide to the p4 programming system. Technical Report 92/17, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, October 1992.Google Scholar
  68. [68]
    Soumen Chakrabarti and Katherine A. Yelick. Implementing an irregular application on a distributed memory multiprocessor. In Proceedings of the Fourth ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, pages 169–178, 1993.Google Scholar
  69. [69]
    Soumen Chakrabarti and Katherine A. Yelick. On the correctness of a distributed memory Gröbner basis algorithm. In Claude Kirchner, editor, Proceedings of the Fifth International Conference on Rewriting Techniques and Applications (RTA), volume 690 of Lecture Notes in Computer Science, pages 77–91. Springer, 1993.Google Scholar
  70. [70]
    K. Many Chandy and Stephen Taylor. An Introduction to Parallel Programming. Jones and Bartlett, Burlington, Massachusetts, 1991.Google Scholar
  71. [71]
    Chin-Liang Chang and Richard Char-Tung Lee. Symbolic Logic and Mechanical Theorem Proving. Academic Press, Cambridge, England, 1973.Google Scholar
  72. [72]
    P. Daniel Cheng and J. Y. Juang. A parallel resolution procedure based on connection graph. In Proceedings of the Sixth Annual Conference of the American Association for Artificial Intelligence (AAAI), pages 13–17, 1987.Google Scholar
  73. [73]
    Heng Chu and David A. Plaisted. Model finding in semantically guided instance-based theorem proving. Fundamenta Informaticae, 21(3):221–235, 1994.Google Scholar
  74. [74]
    Heng Chu and David A. Plaisted. CLINS-S: a semantically guided first-order theorem prover. Journal of Automated Reasoning, 18(2):183–188, 1997.Google Scholar
  75. [75]
    Edmund M. Clarke, David E. Long, Spiro Michaylov, Stephen A. Schwab, Jean-Philippe Vidal, and Shinji Kimura. Parallel symbolic computation algorithms. Technical Report CMU-CS-90-182, School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania, October 1990.Google Scholar
  76. [76]
    Susan E. Conry, Douglas J. MacIntosh, and Robert A. Meyer. DARES: a Distributed Automated REasoning System. In Proceedings of the Eleventh Annual Conference of the American Association for Artificial Intelligence (AAAI), pages 78–85, 1990.Google Scholar
  77. [77]
    Simon Cruanes. Extending superposition with integer arithmetic, structural induction, and beyond. PhD thesis, École Polytechnique, Université Paris-Saclay, September 2015.Google Scholar
  78. [78]
    Bernd I. Dahn. Robbins algebras are Boolean: a revision of McCune’s computer-generated solution of Robbins problem. Journal of Algebra, 208:526–532, 1998.Google Scholar
  79. [79]
    Martin Davis, George Logemann, and Donald Loveland. A machine program for theorem-proving. Communications of the ACM, 5(7):394–397, 1962.Google Scholar
  80. [80]
    Martin Davis and Hilary Putnam. A computing procedure for quantification theory. Journal of the ACM, 7:201–215, 1960.Google Scholar
  81. [81]
    Leonardo de Moura and Nikolaj Bjørner. Engineering DPLL(T) + saturation. In Alessandro Armando, Peter Baumgartner, and Gilles Dowek, editors, Proceedings of the Fourth International Conference on Automated Reasoning (IJCAR), volume 5195 of Lecture Notes in Artificial Intelligence, pages 475–490. Springer, 2008.Google Scholar
  82. [82]
    Leonardo de Moura and Nikolaj Bjørner. Bugs, moles and skeletons: Symbolic reasoning for software development. In Jürgen Giesl and Reiner Hähnle, editors, Proceedings of the Fifth International Conference on Automated Reasoning (IJCAR), volume 6173 of Lecture Notes in Artificial Intelligence, pages 400–411. Springer, 2010.Google Scholar
  83. [83]
    Leonardo de Moura and Nikolaj Bjørner. Satisfiability modulo theories: introduction and applications. Communications of the ACM, 54(9):69–77, 2011.Google Scholar
  84. [84]
    Jörg Denzinger. Team-Work: a method to design distributed knowledge based theorem provers. PhD thesis, Department of Computer Science, Universität Kaiserslautern, 1993.Google Scholar
  85. [85]
    Jörg Denzinger and Bernd Ingo Dahn. Cooperating theorem provers. In Wolfgang Bibel and Peter H. Schmitt, editors, Automated Deduction – A Basis for Applications, volume II: Systems and Implementation, chapter 14, pages 383–416. Kluwer Academic Publishers, Amsterdam, The Netherlands, 1998.Google Scholar
  86. [86]
    Jörg Denzinger and Dirk Fuchs. Cooperation of heterogeneous provers. In Thomas Dean, editor, Proceedings of the Sixeenth International Joint Conference on Artificial Intelligence (IJCAI), pages 10–15. Morgan Kaufmann Publishers, 1999.Google Scholar
  87. [87]
    Jörg Denzinger, Marc Fuchs, and Matthias Fuchs. High performance ATP systems by combining several AI methods. In Martha E. Pollack, editor, Proceedings of the Fifteenth International Joint Conference on Artificial Intelligence (IJCAI), pages 102–107. Morgan Kaufmann Publishers, 1997.Google Scholar
  88. [88]
    Jörg Denzinger and Matthias Fuchs. Goal-oriented equational theorem proving using Team-Work. In Bernhard Nebel and Leonie Dreschler-Fischer, editors, Proceedings of the Eighteenth German Conference on Artificial Intelligence(KI), volume 861 of Lecture Notes in Artificial Intelligence, pages 343–354. Springer, 1994.Google Scholar
  89. [89]
    Jörg Denzinger and Martin Kronenburg. Planning for distributed theorem proving: the Team-Work approach. In Steffen Hölldobler, editor, Proceedings of the Twentieth German Conference on Artificial Intelligence (KI), volume 1137 of Lecture Notes in Artificial Intelligence, pages 43–56. Springer, 1996.Google Scholar
  90. [90]
    Jörg Denzinger, Martin Kronenburg, and Stephan Schulz. DISCOUNT: a distributed and learning equational prover. Journal of Automated Reasoning, 18(2):189–198, 1997.Google Scholar
  91. [91]
    Jörg Denzinger and Jürgen Lind. TWlib: a library for distributed search applications. In Chu-Sing Yang, editor, Proceedings of the International Conference on Artificial Intelligence, International Computer Symposium (ICS), pages 101–108. National Sun-Yat Sen University, 1996.Google Scholar
  92. [92]
    Jörg Denzinger and Stephan Schulz. Recording and analyzing knowledgebased distributed deduction processes. Journal of Symbolic Computation, 21(4–6):523–541, 1996.Google Scholar
  93. [93]
    Nachum Dershowitz. Orderings for term-rewriting systems. Theoretical Computer Science, 17(3):279–301, 1982.Google Scholar
  94. [94]
    Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, pages 243–320. Elsevier, Amsterdam, The Netherlands, 1990.Google Scholar
  95. [95]
    Nachum Dershowitz and Naomi Lindenstrauss. An abstract concurrent machine for rewriting. In Hélène Kirchner and W. Wechler, editors, Proceedings of the Second International Conference on Algebraic and Logic Programming (ALP), volume 463 of Lecture Notes in Computer Science, pages 318–331. Springer, 1990.Google Scholar
  96. [96]
    Nachum Dershowitz and Zohar Manna. Proving termination with multiset orderings. Communications of the ACM, 22(8):465–476, 1979.Google Scholar
  97. [97]
    Nachum Dershowitz and David A. Plaisted. Rewriting. In John Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, volume 1, chapter 9, pages 535–610. Elsevier, Amsterdam, The Netherlands, 2001.Google Scholar
  98. [98]
    Norbert Eisinger and Hans Jürgen Ohlbach. Deduction systems based on resolution. In Dov M. Gabbay, Christopher J. Hogger, and John Alan Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume I: Logical Foundations, pages 184–273. Oxford University Press, Oxford, England, 1993.Google Scholar
  99. [99]
    Zachary Ernst and Seth Kurtenbach. Toward a procedure for data mining proofs. In Maria Paola Bonacina and Mark E. Stickel, editors, Automated Reasoning and Mathematics: Essays in Memory of William W. McCune, volume 7788 of Lecture Notes in Artificial Intelligence, pages 229–239. Springer, 2013.Google Scholar
  100. [100]
    Michael Fisher. An alternative approach to concurrent theorem proving. In James Geller, Hiroaki Kitano, and Christian B. Suttner, editors, Parallel Processing for Artificial Intelligence 3, pages 209–230. Elsevier, Amsterdam, The Netherlands, 1997.Google Scholar
  101. [101]
    Ian Foster and Steve Tuecke. Parallel programming with PCN. Technical Report 91/32, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, December 1991.Google Scholar
  102. [102]
    Dirk Fuchs. Requirement-based cooperative theorem proving. In Jürgen Dix, Luis Fariñas del Cerro, and Ulrich Furbach, editors, Proceedings of the Sixth Joint European Workshop on Logic in Artificial Intelligence (JELIA), volume 1489 of Lecture Notes in Artificial Intelligence, pages 139–153. Springer, 1998.Google Scholar
  103. [103]
    Marc Fuchs. Controlled use of clausal lemmas in connection tableau calculi. Journal of Symbolic Computation, 29(2):299–341, 2000.Google Scholar
  104. [104]
    Marc Fuchs and Andreas Wolf. Cooperation in model elimination: CPTHEO. In Claude Kirchner and Hélène Kirchner, editors, Proceedings of the Fifteenth International Conference on Automated Deduction (CADE), volume 1421 of Lecture Notes in Artificial Intelligence, pages 42–46. Springer, 1998.Google Scholar
  105. [105]
    Harald Ganzinger and Konstantin Korovin. New directions in instantiationbased theorem proving. In Proceedings of the Eighteenth IEEE Symposium on Logic in Computer Science (LICS), pages 55–64. IEEE Computer Society Press, 2003.Google Scholar
  106. [106]
    Harald Ganzinger and Konstantin Korovin. Theory instantiation. In Miki Hermann and Andrei Voronkov, editors, Proceedings of the Thirteenth Conference on Logic, Programming and Automated Reasoning (LPAR), volume 4246 of Lecture Notes in Artificial Intelligence, pages 497–511. Springer, 2006.Google Scholar
  107. [107]
    Luís Gil, Paulo F. Flores, and Luis Miguel Silveira. PMSat: a parallel version of Minisat. Journal on Satisfiability, Boolean Modeling and Computation, 6:71–98, 2008.Google Scholar
  108. [108]
    Joseph A. Goguen, Sany Leinwand, José Meseguer, and Timothy Winkler. The rewrite rule machine 1988. Technical Report PRG-76, Oxford University Computing Laboratory, Oxford, England, August 1989.Google Scholar
  109. [109]
    Joseph A. Goguen, José Meseguer, Sany Leinwand, Timothy Winkler, and Hitoshi Aida. The rewrite rule machine. Technical Report SRI-CSL-89-6, Computer Science Laboratory, SRI International, Menlo Park, California, March 1989.Google Scholar
  110. [110]
    William Gropp, Ewing Lusk, and Anthony Skjellum. Using MPI: Portable Parallel Programming with the Message Passing Interface. MIT Press, Cambridge, Massachusetts, 1994.Google Scholar
  111. [111]
    Long Guo, Youssef Hamadi, Said Jabbour, and Lakhdar Sais. Diversification and intensification in parallel SAT solving. In Dave Cohen, editor, Proceedings of the Sixteenth International Conference on Principles and Practice of Constraint Programming (CP), volume 6308 of Lecture Notes in Computer Science, pages 252–265. Springer, 2010.Google Scholar
  112. [112]
    Youssef Hamadi, Said Jabbour, and Lakhdar Sais. Control-based clause sharing in parallel SAT solving. In Craig Boutilier, editor, Proceedings of the Twenty-First International Joint Conference on Artificial Intelligence (IJCAI), pages 409–504. AAAI Press, 2009.Google Scholar
  113. [113]
    Youssef Hamadi, Said Jabbour, and Lakhdar Sais. ManySAT: a parallel SAT solver. Journal on Satisfiability, Boolean Modeling and Computation, 6:245–262, 2009.Google Scholar
  114. [114]
    Youssef Hamadi and Christoph M. Wintersteiger. Seven challenges in parallel SAT solving. AI Magazine, 34(2):99–106, 2013.Google Scholar
  115. [115]
    D. J. Hawley. A Buchberger algorithm for distributed memory multiprocessors. In Hans P. Zima, editor, Proceedings of the First International Conference of the Austrian Center for Parallel Computation (ACPC), volume 591 of Lecture Notes in Computer Science. Springer, 1991.Google Scholar
  116. [116]
    Marijn Heule, Oliver Kullmann, Siert Wieringa, and Armin Biere. Cube and conquer: guiding CDCL SAT solvers by lookaheads. In Kerstin Eder, João Lourenço, and Onn M. Shehory, editors, Proceedings of the Seventh International Haifa Verification Conference (HVC), volume 7261 of Lecture Notes in Computer Science, pages 50–65. Springer, 2012.Google Scholar
  117. [117]
    Thomas Hillenbrand. Citius, altius, fortius: lessons learned from the theorem prover WALDMEISTER. In Ingo Dahn and Laurent Vigneron, editors, Proceedings of the Fourth International Workshop On First-Order Theorem Proving (FTP), volume 86 of Electronic Notes in Theoretical Computer Science. Elsevier, 2003.Google Scholar
  118. [118]
    Christoph M. Hoffmann and Michael J. O’Donnell. Programming with equations. ACM Transactions on Programming Languages and Systems, 4(1):83–112, 1982.Google Scholar
  119. [119]
    Alfred Horn. On sentences which are true in direct unions of algebras. Journal of Symbolic Logic, 16:14–21, 1951.Google Scholar
  120. [120]
    Jieh Hsiang and Michaël Rusinowitch. On word problems in equational theories. In Thomas Ottman, editor, Proceedings of the Fourteenth International Colloquium on Automta, Languages, and Programming (ICALP), volume 267 of Lecture Notes in Computer Science, pages 54–71. Springer, 1987.Google Scholar
  121. [121]
    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.Google Scholar
  122. [122]
    Jieh Hsiang, Michaël Rusinowitch, and Ko Sakai. Complete inference rules for the cancellation laws. In John McDermott, editor, Proceedings of the Tenth International Joint Conference on Artificial Intelligence (IJCAI), pages 990–992. Morgan Kaufmann Publishers, 1987.Google Scholar
  123. [123]
    Antti E. J. Hyvärinen, Tommi Junttila, and Ilka Niemelä. Incorporating clause learning in grid-based randomized SAT solving. Journal on Satisfiability, Boolean Modeling and Computation, 6:223–244, 2009.Google Scholar
  124. [124]
    Daniyar Itegulov, John Slaney, and Bruno Woltzenlogel Paleo. Scavenger 0.1: a theorem prover based on conflict resolution. In Leonardo de Moura, editor, Proceedings of the Twenty-Sixth Conference on Automated Deduction (CADE), volume 10395 of Lecture Notes in Artificial Intelligence, pp. 344–356, Springer, 2017.Google Scholar
  125. [125]
    Swen Jacobs and Uwe Waldmann. Comparing instance generation methods for automated reasoning. Journal of Automated Reasoning, 38:57–78, 2007.Google Scholar
  126. [126]
    Himanshu Jain. Verification using satisfiability checking, predicate abstraction and Craig interpolation. PhD thesis, School of Computer Science, Carnegie Mellon University, September 2008.Google Scholar
  127. [127]
    Anita Jindal, Ross Overbeek, and Waldo C. Kabat. Exploitation of parallel processing for implementing high-performance deduction systems. Journal of Automated Reasoning, 8:23–38, 1992.Google Scholar
  128. [128]
    Deepak Kapur, David Musser, and Paliath Narendran. Only prime superposition need be considered in the Knuth-Bendix completion procedure. Journal of Symbolic Computation, 6:19–36, 1988.Google Scholar
  129. [129]
    Owen Kaser, Shaunak Pawagi, C. R. Ramakrishnan, I. V. Ramakrishnan, and R. C. Sekar. Fast parallel implementations of lazy languages – the EQUALS experience. In John L. White, editor, Proceedings of the ACM Conference on LISP and Functional Programming, pages 335–344. ACM Press, 1992.Google Scholar
  130. [130]
    Claude Kirchner, Christopher Lynch, and Christelle Scharff. Fine-grained concurrent completion. In Harald Ganzinger, editor, Proceedings of the Seventh International Conference on Rewriting Techniques and Applications (RTA), volume 1103 of Lecture Notes in Computer Science, pages 3–17. Springer, 1996.Google Scholar
  131. [131]
    Claude Kirchner and Patrick Viry. Implementing parallel rewriting. In Bertrand Fronhöfer and Graham Wrightson, editors, Proceedings of the First International Workshop on Parallelization in Inference Systems (December 1990), volume 590 of Lecture Notes in Artificial Intelligence, pages 123–138. Springer, Berlin, Germany, 1992.Google Scholar
  132. [132]
    Donald E. Knuth and Peter B. Bendix. Simple word problems in universal algebras. In John Leech, editor, Proceedings of the Conference on Computational Problems in Abstract Algebras, pages 263–298. Pergamon Press, Oxford, England, 1970.Google Scholar
  133. [133]
    Richard E. Korf. Depth-first iterative deepening: an optimal admissible tree search. Artificial Intelligence, 27(1):97–109, 1985.Google Scholar
  134. [134]
    Konstantin Korovin. An invitation to instantiation-based reasoning: from theory to practice. In Renate Schmidt, editor, Proceedings of the Twenty-Second International Conference on Automated Deduction (CADE), volume 5663 of Lecture Notes in Artificial Intelligence, pages 163–166. Springer, 2009.Google Scholar
  135. [135]
    Konstantin Korovin. Inst-Gen: a modular approach to instantiation-based automated reasoning. In Andrei Voronkov and Christoph Weidenbach, editors, Programming Logics: Essays in Memory of Harald Ganzinger, volume 7797 of Lecture Notes in Artificial Intelligence, pages 239–270. Springer, 2013.Google Scholar
  136. [136]
    Konstantin Korovin and Christoph Sticksel. iProver-Eq: An instantiationbased theorem prover with equality. In Jürgen Giesl and Reiner Hähnle, editors, Proceedings of the Fifth International Conference on Automated Reasoning (IJCAR), volume 6173 of Lecture Notes in Artificial Intelligence, pages 196–202. Springer, 2010.Google Scholar
  137. [137]
    Laura Kovàcs and Andrei Voronkov. First order theorem proving and Vampire. In Natasha Sharygina and Helmut Veith, editors, Proceedings of the Twenty Fifth International Conference on Computer-Aided Verification (CAV), volume 8044 of Lecture Notes in Computer Science, pages 1–35. Springer, 2013.Google Scholar
  138. [138]
    Robert Kowalski and Donald Kuehner. Linear resolution with selection function. Artificial Intelligence, 2:227–260, 1971.Google Scholar
  139. [139]
    Dallas S. Lankford and A. M. Ballantyne. The refutation completeness of blocked permutative narrowing and resolution. InWilliam H. Joyner Jr., editor, Proceedings of the Fourth Conference on Automated Deduction (CADE), pages 168–174, 1979. Available at http://www.cadeinc.org/.
  140. [140]
    Shie-Jue Lee and David A. Plaisted. Eliminating duplication with the hyperlinking strategy. Journal of Automated Reasoning, 9:25–42, 1992.Google Scholar
  141. [141]
    K. Rustan M. Leino and Aleksandar Milicevic. Program extrapolation with Jennisys. In Proceedings of the Twenty-Seventh Conference on Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA), pages 411–430. ACM, 2012.Google Scholar
  142. [142]
    Reinhold Letz. Clausal tableaux. In Wolfgang Bibel and Peter H. Schmitt, editors, Automated Deduction - A Basis for Applications, volume I: Foundations - Calculi and Methods, chapter 2, pages 43–72. Kluwer Academic Publishers, Amsterdam, The Netherlands, 1998.Google Scholar
  143. [143]
    Reinhold Letz, Klaus Mayr, and Christian Goller. Controlled integration of the cut rule into connection tableau calculi. Journal of Automated Reasoning, 13(3):297–338, 1994.Google Scholar
  144. [144]
    Reinhold Letz, Johann Schumann, Stephan Bayerl, and Wolfgang Bibel. SETHEO: a high performance theorem prover. Journal of Automated Reasoning, 8(2):183–212, 1992.Google Scholar
  145. [145]
    Reinhold Letz and Gernot Stenz. DCTP - a disconnection calculus theorem prover. In Rajeev P. Goré, Alexander Leitsch, and Tobias Nipkow, editors, Proceedings of the First International Joint Conference on Automated Reasoning (IJCAR), volume 2083 of Lecture Notes in Artificial Intelligence, pages 381–385. Springer, 2001.Google Scholar
  146. [146]
    Reinhold Letz and Gernot Stenz. Model elimination and connection tableau procedures. In John Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, chapter 28, pages 2015–2114. Elsevier, Amsterdam, The Netherlands, 2001.Google Scholar
  147. [147]
    Reinhold Letz and Gernot Stenz. Proof and model generation with disconnection tableaux. In Robert Nieuwenhuis and Andrei Voronkov, editors, Proceedings of the Eighth International Conference on Logic, Programming and Automated Reasoning (LPAR), volume 2250 of Lecture Notes in Artificial Intelligence, pages 142–156. Springer, 2001.Google Scholar
  148. [148]
    Reinhold Letz and Gernot Stenz. Integration of equality reasoning into the disconnection calculus. In Uwe Egly and Christian G. Fermüller, editors, Proceedings of the Fifteenth International Conference on Analytic Tableaux and Related Methods (TABLEAUX), volume 2381 of Lecture Notes in Artificial Intelligence, pages 176–190. Springer, 2002.Google Scholar
  149. [149]
    Vladimir Lifschitz, Leora Morgenstern, and David A. Plaisted. Knowledge representation and classical logic. In Frank van Harmelen, Vladimir Lifschitz, and Bruce Porter, editors, Handbook of Knowledge Representation, volume 1, pages 3–88. Elsevier, Amsterdam, The Netherlands, 2008.Google Scholar
  150. [150]
    Rasiah Loganantharaj. Theoretical and implementational aspects of parallel link resolution in connection graphs. PhD thesis, Department of Computer Science, Colorado State University, 1985.Google Scholar
  151. [151]
    Rasiah Loganantharaj and Robert A. Müller. Parallel theorem proving with connection graphs. In Jörg Siekmann, editor, Proceedings of the Eighth International Conference on Automated Deduction (CADE), volume 230 of Lecture Notes in Computer Science, pages 337–352. Springer, 1986.Google Scholar
  152. [152]
    DonaldW. Loveland. A simplified format for the model elimination procedure. Journal of the ACM, 16(3):349–363, 1969.Google Scholar
  153. [153]
    Donald W. Loveland. A unifying view of some linear Herbrand procedures. Journal of the ACM, 19(2):366–384, 1972.Google Scholar
  154. [154]
    Ewing L. Lusk and William W. McCune. Experiments with ROO: a parallel automated deduction system. In Bertrand Fronhöfer and Graham Wrightson, editors, Proceedings of the First International Workshop on Parallelization in Inference Systems (December 1990), volume 590 of Lecture Notes in Artificial Intelligence, pages 139–162. Springer, Berlin, Germany, 1992.Google Scholar
  155. [155]
    Ewing L. Lusk, William W. McCune, and John K. Slaney. Parallel closurebased automated reasoning. In Bertrand Fronhöfer and Graham Wrightson, editors, Proceedings of the First International Workshop on Parallelization in Inference Systems (December 1990), volume 590 of Lecture Notes in Artificial Intelligence, pages 347–347. Springer, Berlin, Germany, 1992.Google Scholar
  156. [156]
    Ewing L. Lusk, William W. McCune, and John K. Slaney. ROO: a parallel theorem prover. In Deepak Kapur, editor, Proceedings of the Eleventh International Conference on Automated Deduction (CADE), volume 607 of Lecture Notes in Artificial Intelligence, pages 731–734. Springer, 1992.Google Scholar
  157. [157]
    Sharad Malik and Lintao Zhang. Boolean satisfiability: from theoretical hardness to practical success. Communications of the ACM, 52(8):76–82, 2009.Google Scholar
  158. [158]
    Norbert Manthey. Towards next generation sequential and parallel SAT solvers. Constraints, 20(4):504–505, 2015.Google Scholar
  159. [159]
    Rainer Manthey and François Bry. SATCHMO: a theorem prover implemented in Prolog. In Ewing Lusk and Ross Overbeek, editors, Proceedings of the Ninth International Conference on Automated Deduction (CADE), volume 310 of Lecture Notes in Computer Science, pages 415–434. Springer, 1988.Google Scholar
  160. [160]
    João P. Marques Silva, Inês Lynce, and Sharad Malik. Conflict-driven clause learning SAT solvers. In Armin Biere, Marjin Heule, Hans Van Maaren, and Toby Walsh, editors, Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications, chapter 4, pages 131–153. IOS Press, Amsterdam, The Netherlands, 2009.Google Scholar
  161. [161]
    João P. Marques-Silva and Karem A. Sakallah. GRASP: A new search algorithm for satisfiability. In Proceedings of the International Conference on Computer-Aided Design (ICCAD), pages 220–227, 1997.Google Scholar
  162. [162]
    João P. Marques Silva and Karem A. Sakallah. GRASP: A search algorithm for propositional satisfiability. IEEE Transactions on Computers, 48(5):506–521, 1999.Google Scholar
  163. [163]
    Ruben Martins, Vasco M. Manquinho, and Inês Lynce. An overview of parallel SAT solving. Constraints, 17(3):304–347, 2012.Google Scholar
  164. [164]
    William W. McCune. OTTER 2.0 users guide. Technical Report 90/9, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, March 1990.Google Scholar
  165. [165]
    William W. McCune. What’s new in OTTER 2.2. Technical Report TM-153, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, July 1991.Google Scholar
  166. [166]
    William W. McCune. OTTER 3.0 reference manual and guide. Technical Report 94/6, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, January 1994. Revised August 1995.Google Scholar
  167. [167]
    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, Cambridge, Massachusetts, 1997.Google Scholar
  168. [168]
    William W. McCune. Solution of the Robbins problem. Journal of Automated Reasoning, 19(3):263–276, 1997.Google Scholar
  169. [169]
    WilliamW. McCune. OTTER 3.3 reference manual. Technical Report TM-263, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, August 2003.Google Scholar
  170. [170]
    Max Moser, Ortrun Ibens, Reinhold Letz, Joachim Steinbach, Christoph Goller, Johann Schumann, and Klaus Mayr. The model elimination provers SETHEO and E-SETHEO. Journal of Automated Reasoning, 18(2):237–246, 1997.Google Scholar
  171. [171]
    Matthew W. Moskewicz, Conor F. Madigan, Ying Zhao, Lintao Zhang, and Sharad Malik. Chaff: Engineering an efficient SAT solver. In David Blaauw and Luciano Lavagno, editors, Proceedings of the Thirty-Ninth Design Automation Conference (DAC), pages 530–535, 2001.Google Scholar
  172. [172]
    Robert Nieuwenhuis and Albert Rubio. Paramodulation-based theorem proving. In John Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, volume 1, chapter 7, pages 371–443. Elsevier, Amsterdam, The Netherlands, 2001.Google Scholar
  173. [173]
    Robert Niewenhuis and A. Rubio. Theorem proving with ordering and equality constrained clauses. Journal of Symbolic Computation, 19(4):321–351, 1995.Google Scholar
  174. [174]
    Gerald E. Peterson. A technique for establishing completeness results in theorem proving with equality. SIAM Journal of Computing, 12(1):82–100, 1983.Google Scholar
  175. [175]
    Gerald E. Peterson and Mark E. Stickel. Complete sets of reductions for some equational theories. Journal of the ACM, 28(2):233–264, 1981.Google Scholar
  176. [176]
    Ruzica Piskac, Leonardo de Moura, and Nikolaj Bjørner. Deciding effectively propositional logic using DPLL and substitution sets. Journal of Automated Reasoning, 44(4):401–424, 2010.Google Scholar
  177. [177]
    David A. Plaisted. Mechanical theorem proving. In Ranan B. Banerji, editor, Formal Techniques in Artificial Intelligence, pages 269–320. Elsevier, Amsterdam, The Netherlands, 1990.Google Scholar
  178. [178]
    David A. Plaisted. Equational reasoning and term rewriting systems. In Dov M. Gabbay, Christopher J. Hogger, and John Alan Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, volume I: Logical Foundations, pages 273–364. Oxford University Press, Oxford, England, 1993.Google Scholar
  179. [179]
    David A. Plaisted. Automated theorem proving. Wiley Interdisciplinary Reviews: Cognitive Science, 5(2):115–128, 2014.Google Scholar
  180. [180]
    David A. Plaisted and Swaha Miller. The relative power of semantics and unification. In Andrei Voronkov and ChristophWeidenbach, editors, Programming Logics: Essays in Memory of Harald Ganzinger, volume 7797 of Lecture Notes in Artificial Intelligence, pages 317–344. Springer, 2013.Google Scholar
  181. [181]
    David A. Plaisted and Yunshan Zhu. Ordered semantic hyper linking. Journal of Automated Reasoning, 25:167–217, 2000.Google Scholar
  182. [182]
    Giles Reger, Martin Suda, and Andrei Voronkov. Playing with AVATAR. In Amy P. Felty and Aart Middeldorp, editors, Proceedings of the Twenty-Fifth International Conference on Automated Deduction (CADE), volume 9195 of Lecture Notes in Artificial Intelligence, pages 399–415. Springer, 2015.Google Scholar
  183. [183]
    George A. Robinson and Larry Wos. Paramodulation and theorem-proving in first-order theories with equality. In Donald Michie and Bernard Meltzer, editors, Machine Intelligence, volume 4, pages 135–150. Edinburgh University Press, Edinburgh, Scotland, 1969.Google Scholar
  184. [184]
    John Alan Robinson. Automatic deduction with hyper-resolution. International Journal of Computer Mathematics, 1:227–234, 1965.Google Scholar
  185. [185]
    John Alan Robinson. A machine oriented logic based on the resolution principle. Journal of the ACM, 12(1):23–41, 1965.Google Scholar
  186. [186]
    Michaël Rusinowitch. Theorem-proving with resolution and superposition. Journal of Symbolic Computation, 11(1 & 2):21–50, 1991.Google Scholar
  187. [187]
    Tobias Schubert, Matthew Lewis, and Bernd Becker. PaMiraXT: parallel SAT solving with threads and message passing. Journal on Satisfiability, Boolean Modeling and Computation, 6:203–222, 2009.Google Scholar
  188. [188]
    Stephan Schulz. E – A brainiac theorem prover. Journal of AI Communications, 15(2–3):111–126, 2002.Google Scholar
  189. [189]
    Stephan Schulz. Simple and efficient clause subsumption with feature vector indexing. In Maria Paola Bonacina and Mark E. Stickel, editors, Automated Reasoning and Mathematics: Essays in Memory of William W. McCune, volume 7788 of Lecture Notes in Artificial Intelligence, pages 45–67. Springer, 2013.Google Scholar
  190. [190]
    Stephan Schulz. System description: E 1.8. In Ken McMillan, Aart Middeldorp, and Andrei Voronkov, editors, Proceedings of the Nineteenth International Conference on Logic, Programming and Automated Reasoning (LPAR), volume 8312 of Lecture Notes in Artificial Intelligence, pages 735–743. Springer, 2013.Google Scholar
  191. [191]
    Stephan Schulz and Martin Möhrmann. Performance of clause selection heuristics for saturation-based theorem proving. In Nicola Olivetti and Ashish Tiwari, editors, Proceedings of the Eighth International Conference on Automated Reasoning (IJCAR), volume 9706 of Lecture Notes in Artificial Intelligence, pages 330–345. Springer, 2016.Google Scholar
  192. [192]
    Johan Schumann. Parallel theorem provers – an overview. In Bertrand Fronhöfer and Graham Wrightson, editors, Proceedings of the First International Workshop on Parallelization in Inference Systems (December 1990), volume 590 of Lecture Notes in Artificial Intelligence, pages 26–50. Springer, Berlin, Germany, 1992.Google Scholar
  193. [193]
    Johann Schumann. Delta: a bottom-up pre-processor for top-down theorem provers. In Alan Bundy, editor, Proceedings of the Twelfth International Conference on Automated Deduction (CADE), volume 814 of Lecture Notes in Artificial Intelligence, pages 774–777. Springer, 1994.Google Scholar
  194. [194]
    Johann Schumann and Reinhold Letz. PARTHEO: a high-performance parallel theorem prover. In Mark E. Stickel, editor, Proceedings of the Tenth International Conference on Automated Deduction (CADE), volume 449 of Lecture Notes in Artificial Intelligence, pages 28–39. Springer, 1990.Google Scholar
  195. [195]
    Robert E. Shostak. Refutation graphs. Artificial Intelligence, 7:51–64, 1976.Google Scholar
  196. [196]
    Kurt Siegl. Gröbner bases computation in STRAND: a case study for concurrent symbolic computation in logic programming languages (Master thesis). Technical Report 90-54.0, Research Institute for Symbolic Computation (RISC), Linz, Austria, November 1990.Google Scholar
  197. [197]
    Carsten Sinz, Jörg Denzinger, Jürgen Avenhaus, and Wolfgang Küchlin. Combining parallel and distributed search in automated equational deduction. In Proceedings of the Fourth International Conference on Parallel Processing and Applied Mathematics (PPAM) – Revised Papers, pages 819–832, 2001.Google Scholar
  198. [198]
    James R. Slagle. Automatic theorem proving with renamable and semantic resolution. Journal of the ACM, 14(4):687–697, 1967.Google Scholar
  199. [199]
    James R. Slagle. Automated theorem proving for theories with simplifiers, commutativity, and associativity. Journal of the ACM, 21:622–642, 1974.Google Scholar
  200. [200]
    John Slaney, Ewing Lusk, and William W. McCune. SCOTT: Semantically constrained Otter. In Alan Bundy, editor, Proceedings of the Twelfth International Conference on Automated Deduction (CADE), volume 814 of Lecture Notes in Artificial Intelligence, pages 764–768. Springer, 1994.Google Scholar
  201. [201]
    John Slaney and Bruno Woltzenlogel Paleo. Conflict resolution: a first-order resolution calculus with decision literals and conflict-driven clause learning. Journal of Automated Reasoning, in press:1–27, 2017.Google Scholar
  202. [202]
    Mark E. Stickel. A Prolog technology theorem prover. New Generation Computing, 2(4):371–383, 1984.Google Scholar
  203. [203]
    Mark E. Stickel. A Prolog technology theorem prover: implementation by an extended Prolog compiler. Journal of Automated Reasoning, 4:353–380, 1988.Google Scholar
  204. [204]
    Mark E. Stickel. PTTP and linked inference. In Robert S. Boyer, editor, Automated Reasoning: Essays in Honor of Woody Bledsoe, pages 283–296. Kluwer Academic Publishers, Amsterdam, The Netherlands, 1991.Google Scholar
  205. [205]
    Mark E. Stickel. A Prolog technology theorem prover: new exposition and implementation in Prolog. Theoretical Computer Science, 104:109–128, 1992.Google Scholar
  206. [206]
    Mark E. Stickel and W. Mabry Tyson. An analysis of consecutively bounded depth-first search with applications in automated deduction. In Proceedings of the Ninth International Joint Conference on Artificial Intelligence (IJCAI), pages 1073–1075. Morgan Kaufmann Publishers, 1985.Google Scholar
  207. [207]
    David Sturgill and Alberto Maria Segre. Nagging: a distributed, adversarial search-pruning technique applied to first-order inference. Journal of Automated Reasoning, 19(3):347–376, 1997.Google Scholar
  208. [208]
    Geoff Sutcliffe. A heterogeneous parallel deduction system. In Ryuzo Hasegawa and Mark E. Stickel, editors, Proceedings of the FGCS Workshop on Automated Deduction: Logic Programming and Parallel Computing Approaches, pages 5–13, 1992.Google Scholar
  209. [209]
    Christian B. Suttner. SPTHEO: a parallel theorem prover. Journal of Automated Reasoning, 18(2):253–258, 1997.Google Scholar
  210. [210]
    Christian B. Suttner and Johann Schumann. Parallel automated theorem proving. In Laveen N. Kanal, Vipin Kumar, Hiroaki Kitano, and Christian B. Suttner, editors, Parallel Processing for Artificial Intelligence. Elsevier, Amsterdam, The Netherlands, 1994.Google Scholar
  211. [211]
    Tanel Tammet. Gandalf. Journal of Automated Reasoning, 18(2):199–204, 1997.Google Scholar
  212. [212]
    Stephen Taylor. Parallel Logic Programming Techniques. Prentice Hall, Upper Saddle River, New Jersey, 1989.Google Scholar
  213. [213]
    Josef Urban and Jirí Vyskocil. Theorem proving in large formal mathematics as an emerging AI field. In Maria Paola Bonacina and Mark E. Stickel, editors, Automated Reasoning and Mathematics: Essays in Memory of William W. McCune, volume 7788 of Lecture Notes in Artificial Intelligence, pages 240–257. Springer, 2013.Google Scholar
  214. [214]
    Jean-Philippe Vidal. The computation of Gröbner bases on a shared memory multiprocessor. In Alfonso Miola, editor, Proceedings of the First International Symposium on Design and Implementation of Symbolic Computation Systems (DISCO), volume 429 of Lecture Notes in Computer Science, pages 81–90. Springer, 1990.Google Scholar
  215. [215]
    Kevin Wallace and Graham Wrightson. Regressive merging in model elimination tableau-based theorem provers. Journal of the IGPL, 3(6):921–937, 1995.Google Scholar
  216. [216]
    David H. D. Warren. An abstract Prolog instruction set. Technical Report 309, Artificial Intelligence Center, SRI International, Menlo Park, California, October 1983.Google Scholar
  217. [217]
    David S. Warren. Memoing for logic programs. Communications of the ACM, 35(3):94–111, 1992.Google Scholar
  218. [218]
    ChristophWeidenbach, Dylana Dimova, Arnaud Fietzke, Rohit Kumar, Martin Suda, and PatrickWischnewski. SPASS version 3.5. In Renate Schmidt, editor, Proceedings of the Twenty-Second International Conference on Automated Deduction (CADE), volume 5663 of Lecture Notes in Artificial Intelligence, pages 140–145. Springer, 2009.Google Scholar
  219. [219]
    AndreasWolf. P-SETHEO: strategy parallelism in automated theorem proving. In Harrie de Swart, editor, Proceedings of the Seventh International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX), volume 1397 of Lecture Notes in Artificial Intelligence, pages 320–324. Springer, 1998.Google Scholar
  220. [220]
    Larry Wos. Searching for open questions. Newsletter of the Association for Automated Reasoning, 15, May 1990.Google Scholar
  221. [221]
    Larry Wos, Daniel F. Carson, and George A. Robinson. Efficiency and completeness of the set of support strategy in theorem proving. Journal of the ACM, 12:536–541, 1965.Google Scholar
  222. [222]
    Larry Wos, George A. Robinson, Daniel F. Carson, and Leon Shalla. The concept of demodulation in theorem proving. Journal of the ACM, 14(4):698–709, 1967.Google Scholar
  223. [223]
    Chih-Hung Wu and Shie-Jue Lee. Parallelization of a hyper-linking based theorem prover. Journal of Automated Reasoning, 26(1):67–106, 2001.Google Scholar
  224. [224]
    Katherine A. Yelick. Using abstraction in explicitly parallel programs. PhD thesis, Laboratory for Computer Science, Massachusetts Institute of Technology, July 1991.Google Scholar
  225. [225]
    Katherine A. Yelick and Steven J. Garland. A parallel completion procedure for term rewriting systems. In Deepak Kapur, editor, Proceedings of the Eleventh International Conference on Automated Deduction (CADE), volume 607 of Lecture Notes in Artificial Intelligence, pages 109–123. Springer, 1992.Google Scholar
  226. [226]
    Hantao Zhang. SATO: an efficient propositional prover. In William W. McCune, editor, Proceedings of the Fourteenth International Conference on Automated Deduction (CADE), volume 1249 of Lecture Notes in Artificial Intelligence, pages 272–275. Springer, 1997.Google Scholar
  227. [227]
    Hantao Zhang and Maria Paola Bonacina. Cumulating search in a distributed computing environment: a case study in parallel satisfiability. In Hoon Hong, editor, Proceedings of the First International Symposium on Parallel Symbolic Computation (PASCO), volume 5 of Lecture Notes Series in Computing, pages 422–431. World Scientific, 1994.Google Scholar
  228. [228]
    Hantao Zhang, Maria Paola Bonacina, and Jieh Hsiang. PSATO: a distributed propositional prover and its application to quasigroup problems. Journal of Symbolic Computation, 21(4–6):543–560, 1996.Google Scholar
  229. [229]
    Hantao Zhang and Mark E. Stickel. Implementing the Davis-Putnam method. Journal of Automated Reasoning, 24(1–2):277–296, 2000.Google Scholar
  230. [230]
    Lintao Zhang and Sharad Malik. The quest for efficient Boolean satisfiability solvers. In Andrei Voronkov, editor, Proceedings of the Eighteenth International Conference on Automated Deduction (CADE), volume 2392 of Lecture Notes in Artificial Intelligence, pages 295–313. Springer, 2002.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità degli Studi di VeronaVeronaItaly

Personalised recommendations