Programming Logics pp 345-391

Part of the Lecture Notes in Computer Science book series (LNCS, volume 7797) | Cite as

First-Order Resolution Methods for Modal Logics

  • Renate A. Schmidt
  • Ullrich Hustadt

Abstract

In this paper we give an overview of results for modal logic which can be shown using techniques and methods from first-order logic and resolution. Because of the breadth of the area and the many applications we focus on the use of first-order resolution methods for modal logics. In addition to traditional propositional modal logics we consider more expressive PDL-like dynamic modal logics closely related to description logics. Without going into too much detail, we survey different ways of translating modal logics into first-order logic, we explore different ways of using first-order resolution theorem provers to solve a range of reasoning problems for modal logics, and we discuss a variety of results which have been obtained in the setting of first-order resolution.

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References

  1. 1.
    AlBarakati, R.G.: Development of a tableaux resolution prover. Master’s thesis, The University of Manchester, UK (2009)Google Scholar
  2. 2.
    AlBarakati, R.G.: spass-tab (2009), http://www.cs.man.ac.uk/~schmidt/spass-tab/
  3. 3.
    Andréka, H., Németi, I., van Benthem, J.: Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic 27(3), 217–274 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Andréka, H., van Benthem, J., Németi, I.: Back and forth between modal logic and classical logic. Bulletin of the IGPL 3(5), 685–720 (1995)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Areces, C., Gennari, R., Heguiabehere, J., de Rijke, M.: Tree-based heuristics in modal theorem proving. In: Proc. ECAI 2000, pp. 199–203. IOS Press (2000)Google Scholar
  6. 6.
    Auffray, Y., Enjalbert, P.: Modal theorem proving: An equational viewpoint. Journal of Logic and Computation 2(3), 247–297 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation 4(3), 217–247 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bachmair, L., Ganzinger, H.: Equational reasoning in saturation-based theorem proving. In: Bibel, W., Schmitt, P.H. (eds.) Automated Deduction—A Basis for Applications, vol. I, pp. 353–397. Kluwer (1998)Google Scholar
  9. 9.
    Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 19–99. Elsevier (2001)Google Scholar
  10. 10.
    Bachmair, L., Ganzinger, H., Waldmann, U.: Superposition with Simplification as a Decision Procedure for the Monadic Class with Equality. In: Mundici, D., Gottlob, G., Leitsch, A. (eds.) KGC 1993. LNCS, vol. 713, pp. 83–96. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  11. 11.
    Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational theorem proving for hierarchic first-order theories. Applicable Algebra in Engineering, Communication and Computing 5(3/4), 193–212 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Baumgartner, P.: A First-order Davis-Putnam-Logeman-Loveland Procedure. In: McAllester, D. (ed.) CADE-17. LNCS (LNAI), vol. 1831, pp. 200–219. Springer, Heidelberg (2000)Google Scholar
  13. 13.
    Baumgartner, P., Horton, J.D., Spencer, B.: Merge Path Improvements for Minimal Model Hyper Tableaux. In: Murray, N.V. (ed.) TABLEAUX 1999. LNCS (LNAI), vol. 1617, pp. 51–66. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. 14.
    Baumgartner, P., Schmidt, R.A.: Blocking and Other Enhancements for Bottom-Up Model Generation Methods. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 125–139. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Baumgartner, P., Schmidt, R.A.: Blocking and other enhancements for bottom-up model generation methods. Manuscript (2008)Google Scholar
  16. 16.
    Baumgartner, P., Tinelli, C.: The Model Evolution Calculus. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 350–364. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Blackburn, P., de Rijke, M., Venema, V.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press (2001)Google Scholar
  18. 18.
    Bledsoe, W.W.: Splitting and reduction heuristics in automatic theorem proving. Artificial Intelligence 2, 55–77 (1971)CrossRefMATHGoogle Scholar
  19. 19.
    Brink, C., Britz, K., Schmidt, R.A.: Peirce algebras. Formal Aspects of Computing 6(3), 339–358 (1994)CrossRefMATHGoogle Scholar
  20. 20.
    Bry, F., Yahya, A.: Positive unit hyperresolution tableaux for minimal model generation. Journal of Automated Reasoning 25(1), 35–82 (2000)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Castilho, M.A., Fariñas del Cerro, L., Gasquet, O., Herzig, A.: Modal tableaux with propagation rules and structural rules. Fundamenta Informaticae 32(3-4), 281–297 (1997)MathSciNetMATHGoogle Scholar
  22. 22.
    Chortaras, A., Trivela, D., Stamou, G.: Optimized Query Rewriting for OWL 2 QL. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 192–206. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  23. 23.
    de Nivelle, H.: Splitting through New Proposition Symbols. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 172–185. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  24. 24.
    de Nivelle, H., Schmidt, R.A., Hustadt, U.: Resolution-based methods for modal logics. Logic Journal of the IGPL 8(3), 265–292 (2000)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    de Rijke, M.: Extending Modal Logic. PhD thesis, University of Amsterdam, The Netherlands (1993)Google Scholar
  26. 26.
    Degtyarev, A., Fisher, M., Konev, B.: Monodic temporal resolution. ACM Transactions in Computational Logic 7(1), 108–150 (2006)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Demri, S., de Nivelle, H.: Deciding regular grammar logics with converse through first-order logic. Journal of Logic, Language and Information 14(3), 289–329 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Demri, S., Gabbay, D.: On modal logics characterized by models with relative accessibility relations: Part II. Studia Logica 66(3), 349–384 (2000)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Doherty, P., Lukaszewicz, W., Szalas, A.: Computing circumscription revisited: A reduction algorithm. Journal of Automated Reasoning 18(3), 297–336 (1997)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Doherty, P., Lukaszewicz, W., Szalas, A., Gustafsson, J.: dls (1996), http://www.ida.liu.se/labs/kplab/projects/dls/
  31. 31.
    Engel, T.: Quantifier elimination in second-order predicate logic. Diplomarbeit, Fachbereich Informatik, Universität des Saarlandes, Saarbrücken, Germany (1996)Google Scholar
  32. 32.
    Fariñas del Cerro, L., Herzig, A.: Modal deduction with applications in epistemic and temporal logics. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming: Epistemic and Temporal Reasoning, pp. 499–594. Clarendon Press (1995)Google Scholar
  33. 33.
    Fermüller, C., Leitsch, A., Hustadt, U., Tammet, T.: Resolution decision procedures. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 1791–1849. Elsevier (2001)Google Scholar
  34. 34.
    Fisher, M., Dixon, C., Peim, M.: Clausal temporal resolution. ACM Transactions on Computational Logic 2(1), 12–56 (2001)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Gabbay, D.M., Ohlbach, H.J.: Quantifier elimination in second-order predicate logic. South African Computer Journal 7, 35–43 (1992)Google Scholar
  36. 36.
    Gabbay, D.M., Schmidt, R.A., Szałas, A.: Second-Order Quantifier Elimination: Foundations, Computational Aspects and Applications. Studies in Logic: Mathematical Logic and Foundations, vol. 12. College Publications (2008)Google Scholar
  37. 37.
    Ganzinger, H., de Nivelle, H.: A superposition decision procedure for the guarded fragment with equality. In: Proc. LICS-14, pp. 295–303. IEEE (1999)Google Scholar
  38. 38.
    Ganzinger, H., Hustadt, U., Meyer, C., Schmidt, R.A.: A resolution-based decision procedure for extensions of K4. In: Advances in Modal Logic. Lecture Notes, vol. 2, 119, pp. 225–246. CSLI Publications (2001)Google Scholar
  39. 39.
    Ganzinger, H., Korovin, K.: New directions in instantiation-based theorem proving. In: Proc. LICS-18, pp. 55–64. IEEE (2003)Google Scholar
  40. 40.
    Ganzinger, H., Meyer, C., Veanes, M.: The two-variable guarded fragment with transitive relations. In: Proc. LICS-14, pp. 24–34. IEEE (1999)Google Scholar
  41. 41.
    Ganzinger, H., Sofronie-Stokkermans, V.: Chaining techniques for automated theorem proving in finitely-valued logics. In: Proc. ISMVL 2000, pp. 337–344. IEEE (2000)Google Scholar
  42. 42.
    Gargov, G., Passy, S.: A note on Boolean modal logic. In: Mathematical Logic: Proceedings of the 1988 Heyting Summerschool, pp. 299–309. Plenum Press (1990)Google Scholar
  43. 43.
    Gargov, G., Passy, S., Tinchev, T.: Modal environment for Boolean speculations. In: Mathematical Logic and its Applications: Proceedings of the 1986 Gödel Conference, pp. 253–263. Plenum Press (1987)Google Scholar
  44. 44.
    Georgieva, L., Hustadt, U., Schmidt, R.A.: Computational Space Efficiency and Minimal Model Generation for Guarded Formulae. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 85–99. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  45. 45.
    Georgieva, L., Hustadt, U., Schmidt, R.A.: A New Clausal Class Decidable by Hyperresolution. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 260–274. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  46. 46.
    Georgieva, L., Hustadt, U., Schmidt, R.A.: Hyperresolution for guarded formulae. Journal of Symbolic Computation 36(1–2), 163–192 (2003)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Goranko, V., Hustadt, U., Schmidt, R.A., Vakarelov, D.: SCAN is Complete for all Sahlqvist Formulae. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, pp. 149–162. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  48. 48.
    Goré, R.: Tableau methods for modal and temporal logics. In: D’Agostino, M., Gabbay, D., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, pp. 297–396. Kluwer (1999)Google Scholar
  49. 49.
    Grädel, E.: Decision Procedures for Guarded Logics. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 31–51. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  50. 50.
    Grädel, E.: On the restraining power of guards. Journal of Symbolic Logic 64, 1719–1742 (1999)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Hasegawa, R., Fujita, H., Koshimura, M.: Efficient Minimal Model Generation using Branching Lemmas. In: McAllester, D. (ed.) CADE-17. LNCS (LNAI), vol. 1831, pp. 184–199. Springer, Heidelberg (2000)Google Scholar
  52. 52.
    Herzig, A.: Raisonnement automatique en logique modale et algorithmes d’unification. PhD thesis, University Paul-Sabatier, Toulouse, France (1989)Google Scholar
  53. 53.
    Herzig, A.: A new decidable fragment of first order logic. In: Abstracts of 3rd Logical Biennial, Summer School & Conf. in honour of S. C. Kleene, Bulgaria (1990)Google Scholar
  54. 54.
    Horrocks, I., Hustadt, U., Sattler, U., Schmidt, R.A.: Computational modal logic. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic. Studies in Logic and Practical Reasoning, pp. 181–245. Elsevier (2007)Google Scholar
  55. 55.
    Humberstone, I.L.: Inaccessible worlds. Notre Dame Journal of Formal Logic 24(3), 346–352 (1983)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Humberstone, I.L.: The modal logic of ‘all and only’. Notre Dame Journal of Formal Logic 28(2), 177–188 (1987)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Hustadt, U.: Resolution-Based Decision Procedures for Subclasses of First-Order Logic. PhD thesis, Universität des Saarlandes, Saarbrücken, Germany (1999)Google Scholar
  58. 58.
    Hustadt, U., Konev, B.: TRP++: A temporal resolution prover. In: Collegium Logicum, pp. 65–79. Kurt Gödel Society (2004)Google Scholar
  59. 59.
    Hustadt, U., Motik, B., Sattler, U.: Deciding expressive description logics in the framework of resolution. Information and Computation 206(5) (2008)Google Scholar
  60. 60.
    Hustadt, U., Schmidt, R.A.: An empirical analysis of modal theorem provers. Journal of Applied Non-Classical Logics 9(4), 479–522 (1999)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Hustadt, U., Schmidt, R.A.: Maslov’s Class K Revisited. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 172–186. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  62. 62.
    Hustadt, U., Schmidt, R.A.: On the relation of resolution and tableaux proof systems for description logics. In: IJCAI 1999, pp. 110–115. Morgan Kaufmann (1999)Google Scholar
  63. 63.
    Hustadt, U., Schmidt, R.A.: Issues of Decidability for Description Logics in the Framework of Resolution. In: Caferra, R., Salzer, G. (eds.) FTP 1998. LNCS (LNAI), vol. 1761, pp. 191–205. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  64. 64.
    Hustadt, U., Schmidt, R.A.: MSPASS: Modal Reasoning by Translation and First-Order Resolution. In: Dyckhoff, R. (ed.) TABLEAUX 2000. LNCS (LNAI), vol. 1847, pp. 67–71. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  65. 65.
    Hustadt, U., Schmidt, R.A.: Using resolution for testing modal satisfiability and building models. Journal of Automated Reasoning 28(2), 205–232 (2002)MathSciNetCrossRefMATHGoogle Scholar
  66. 66.
    Hustadt, U., Schmidt, R.A., Georgieva, L.: A survey of decidable first-order fragments and description logics. Journal of Relational Methods in Computer Science 1, 251–276 (2004)Google Scholar
  67. 67.
    Kazakov, Y.: Consequence-driven reasoning for horn \(\mathcal{SHIQ}\) ontologies. In: Proc. IJCAI 2009, pp. 2040–2045 (2009)Google Scholar
  68. 68.
    Kazakov, Y., Motik, B.: A resolution-based decision procedure for \(\mathcal{SHOIQ}\). Journal of Automated Reasoning 40(2-3), 89–116 (2008); Erratum in Journal of Automated Reasoning 40(4), 357 (2008)Google Scholar
  69. 69.
  70. 70.
    Konev, B., Degtyarev, A., Dixon, C., Fisher, M., Hustadt, U.: Mechanising first-order temporal resolution. Information and Computation 199(1–2), 55–86 (2005)MathSciNetCrossRefMATHGoogle Scholar
  71. 71.
    Kracht, M.: Tools and Techniques in Modal Logic. Studies in Logic, vol. 142. Elsevier (1999)Google Scholar
  72. 72.
    Kurucz, Á., Németi, I., Sain, I., Simon, A.: Undecidable varieties of semilattice-ordered semigroups, of Boolean algebras with operators and logics extending lambek calculus. Bulletin of the IGPL 1(1), 91–98 (1993)MathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Ludwig, M.: Advancing Formal Verification: Resolution-Based Methods for Linear-Time Temporal Logics. PhD thesis, University of Liverpool, UK (2010)Google Scholar
  74. 74.
  75. 75.
    Ludwig, M., Hustadt, U.: Fair Derivations in Monodic Temporal Reasoning. In: Schmidt, R.A. (ed.) CADE-22. LNCS (LNAI), vol. 5663, pp. 261–276. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  76. 76.
    Ludwig, M., Hustadt, U.: Implementing a fair monodic temporal logic prover. AI Communication 23(2-3), 69–96 (2010)MathSciNetMATHGoogle Scholar
  77. 77.
    Maslov, S.J.: The inverse method for establishing deducibility for logical calculi. In: Orevkov, V.P., Petrovskiǐ, I.G., Nikol’skiǐ, S.M. (eds.) Proc. of the Steklov Institute of Mathematics, vol. 98, pp. 25–96. Amer. Math. Soc., Providence (1968)Google Scholar
  78. 78.
    Massacci, F.: Single step tableaux for modal logics: Computational properties, complexity and methodology. Journal of Automated Reasoning 24(3), 319–364 (2000)MathSciNetCrossRefMATHGoogle Scholar
  79. 79.
    Motik, B., Shearer, R., Horrocks, I.: Hypertableau reasoning for description logics. Journal of Artifical Intelligence Research 36, 165–228 (2009)MathSciNetMATHGoogle Scholar
  80. 80.
    Nellas, K.: Reasoning about sets and relations: A tableaux-based automated theorem prover for Peirce logic. Master’s thesis, The University of Manchester, UK (2001)Google Scholar
  81. 81.
    Niemelä, I.: A Tableau Calculus for Minimal Model Reasoning. In: Miglioli, P., Moscato, U., Ornaghi, M., Mundici, D. (eds.) TABLEAUX 1996. LNCS (LNAI), vol. 1071, pp. 278–294. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  82. 82.
    Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 371–443. Elsevier (2001)Google Scholar
  83. 83.
    Nonnengart, A.: First-order modal logic theorem proving and functional simulation. In: Proc. IJCAI 1993, pp. 80–85. Morgan Kaufmann (1993)Google Scholar
  84. 84.
    Nonnengart, A.: A Resolution-Based Calculus For Temporal Logics. PhD thesis, Universität des Saarlandes, Saarbrücken, Germany (1995)Google Scholar
  85. 85.
    Nonnengart, A.: Resolution-Based Calculi for Modal and Temporal Logics. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS (LNAI), vol. 1104, pp. 598–612. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  86. 86.
    Nonnengart, A., Ohlbach, H.J., Szałas, A.: Elimination of predicate quantifiers. In: Ohlbach, H.J., Reyle, U. (eds.) Logic, Language and Reasoning. Essays in Honor of Dov Gabbay, pp. 159–181. Kluwer (1999)Google Scholar
  87. 87.
    Ohlbach, H.J.: Semantics based translation methods for modal logics. Journal of Logic and Computation 1(5), 691–746 (1991)MathSciNetCrossRefMATHGoogle Scholar
  88. 88.
    Ohlbach, H.J.: Translation methods for non-classical logics: An overview. Bulletin of the IGPL 1(1), 69–89 (1993)MathSciNetCrossRefMATHGoogle Scholar
  89. 89.
    Ohlbach, H.J.: Combining Hilbert Style and Semantic Reasoning in a Resolution Framework. In: Kirchner, C., Kirchner, H. (eds.) CADE 1998. LNCS (LNAI), vol. 1421, pp. 205–219. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  90. 90.
    Ohlbach, H.-J., Engel, T.: scan (1994), http://www.mpi-inf.mpg.de/departments/d2/software/SCAN/
  91. 91.
    Ohlbach, H.J., Nonnengart, A., de Rijke, M., Gabbay, D.: Encoding two-valued nonclassical logics in classical logic. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 1403–1486. Elsevier (2001)Google Scholar
  92. 92.
    Ohlbach, H.J., Schmidt, R.A.: Functional translation and second-order frame properties of modal logics. Journal of Logic and Computation 7(5), 581–603 (1997)MathSciNetCrossRefMATHGoogle Scholar
  93. 93.
    Pérez-Urbina, H., Motik, B., Horrocks, I.: Tractable query answering and rewriting under description logic constraints. Journal of Applied Logic 8(2), 186–209 (2010)MathSciNetCrossRefMATHGoogle Scholar
  94. 94.
    Purdy, W.C.: Decidability of fluted logic with identity. Notre Dame Journal of Formal Logic 37(1), 84–104 (1996)MathSciNetCrossRefMATHGoogle Scholar
  95. 95.
    Purdy, W.C.: Quine’s ‘limits of decision’. Journal of Symbolic Logic 64(4), 1439–1466 (1999)MathSciNetCrossRefMATHGoogle Scholar
  96. 96.
    Quine, W.V.: Variables explained away. In: Proc. American Philosophy Society, vol. 104, pp. 343–347 (1960)Google Scholar
  97. 97.
    Quine, W.V.: Algebraic logic and predicate functors. In: Rudner, R., Scheffler, I. (eds.) Logic and Art: Esssays in Honor of Nelson Goodman. Bobbs-Merrill (1971)Google Scholar
  98. 98.
    Riazanov, A., Voronkov, A.: Vampire. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 292–296. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  99. 99.
    Riazanov, A., Voronkov, A.: Splitting without backtracking. In: Proc. IJCAI 2001, pp. 611–617. Morgan Kaufmann (2001)Google Scholar
  100. 100.
    Robinson, J.A.: A machine-oriented logic based on the resolution principle. Journal of the ACM 12(1), 23–41 (1965)MathSciNetCrossRefMATHGoogle Scholar
  101. 101.
    Sahlqvist, H.: Completeness and correspondence in the first and second order semantics for modal logics. In: Proc. 3rd Scandinavian Logic Symposium, pp. 110–143. North-Holland (1973-1975)Google Scholar
  102. 102.
    Schmidt, R.A.: Optimised Modal Translation and Resolution. PhD thesis, Universität des Saarlandes, Saarbrücken, Germany (1997)Google Scholar
  103. 103.
    Schmidt, R.A.: Decidability by resolution for propositional modal logics. Journal of Automated Reasoning 22(4), 379–396 (1999)MathSciNetCrossRefMATHGoogle Scholar
  104. 104.
    Schmidt, R.A.: MSPASS (1999), http://www.cs.man.ac.uk/~schmidt/mspass/
  105. 105.
    Schmidt, R.A.: Improved Second-Order Quantifier Elimination in Modal Logic. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 375–388. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  106. 106.
    Schmidt, R.A.: A new methodology for developing deduction methods. Annals of Mathematics and Artificial Intelligence 55(1–2), 155–187 (2009)MathSciNetCrossRefMATHGoogle Scholar
  107. 107.
    Schmidt, R.A.: Simulation and synthesis of deduction calculi. Electronic Notes in Theoretical Computer Science 262, 221–229 (2010)MathSciNetCrossRefMATHGoogle Scholar
  108. 108.
    Schmidt, R.A., Hustadt, U.: A Resolution Decision Procedure for Fluted Logic. In: McAllester, D. (ed.) CADE 2000. LNCS (LNAI), vol. 1831, pp. 433–448. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  109. 109.
    Schmidt, R.A., Hustadt, U.: Mechanised Reasoning and Model Generation for Extended Modal Logics. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) Theory and Applications of Relational Structures as Knowledge Instruments. LNCS, vol. 2929, pp. 38–67. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  110. 110.
    Schmidt, R.A., Hustadt, U.: A Principle for Incorporating Axioms into the First-Order Translation of Modal Formulae. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 412–426. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  111. 111.
    Schmidt, R.A., Hustadt, U.: The axiomatic translation principle for modal logic. ACM Transactions on Computational Logic 8(4), 1–55 (2007)MathSciNetCrossRefGoogle Scholar
  112. 112.
    Schmidt, R.A., Orlowska, E., Hustadt, U.: Two Proof Systems for Peirce Algebras. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, pp. 238–251. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  113. 113.
    Schmidt, R.A., Tishkovsky, D.: Using Tableau to Decide Expressive Description Logics with Role Negation. In: Aberer, K., Choi, K.-S., Noy, N., Allemang, D., Lee, K.-I., Nixon, L.J.B., Golbeck, J., Mika, P., Maynard, D., Mizoguchi, R., Schreiber, G., Cudré-Mauroux, P. (eds.) ASWC 2007 and ISWC 2007. LNCS, vol. 4825, pp. 438–451. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  114. 114.
    Schulz, S.: E: A brainiac theorem prover. Journal of AI Communications 15(2–3), 111–126 (2002)MATHGoogle Scholar
  115. 115.
    Smith, K.J.: The axiomatic translation of modal logic into first order logic. Master’s thesis, The University of Manchester, UK (2008)Google Scholar
  116. 116.
    Smith, K.J.: Downloads for project in Axiomatic Translation of Modal Logic 2007/8 Manchester (2008), http://project.kjsmith.net/
  117. 117.
    Stenz, G.: DCTP 1.2 - System Abstract. In: Egly, U., Fermüller, C. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 335–340. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  118. 118.
    Szałas, A.: On the correspondence between modal and classical logic: An automated approach. Journal of Logic and Computation 3(6), 605–620 (1993)MathSciNetCrossRefMATHGoogle Scholar
  119. 119.
    van Benthem, J.: Correspondence theory. In: Gabbay, D., Guenther, F. (eds.) Handbook of Philosophical Logic, pp. 167–247. Reidel, Dordrecht (1984)CrossRefGoogle Scholar
  120. 120.
    Weidenbach, C., Brahm, U., Hillenbrand, T., Keen, E., Theobald, C., Topic, D.: SPASS Version 2.0. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 275–279. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  121. 121.
    Weidenbach, C., Schmidt, R.A., Hillenbrand, T., Rusev, R., Topic, D.: System Description: Spass Version 3.0. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 514–520. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  122. 122.
    Zamov, N.K.: Modal resolutions. Soviet Mathematics 33(9), 22–29 (1989); Translated from Izv. Vyssh. Uchebn. Zaved. Mat. 9(328), 22–29 (1989)Google Scholar
  123. 123.
    Zhang, L.: CTL-RP 00.25 (2010), http://www.csc.liv.ac.uk/~lan/softwares.html
  124. 124.
    Zhang, L.: Clausal Reasoning for Branching-Time Logics. PhD thesis, University of Liverpool, UK (2011)Google Scholar
  125. 125.
    Zhang, L., Hustadt, U., Dixon, C.: A Refined Resolution Calculus for CTL. In: Schmidt, R.A. (ed.) CADE-22. LNCS (LNAI), vol. 5663, pp. 245–260. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  126. 126.
    Zhang, L., Hustadt, U., Dixon, C.: CTL-RP: A computation tree logic resolution prover. AI Communication 23(2-3), 111–136 (2010)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Renate A. Schmidt
    • 1
  • Ullrich Hustadt
    • 2
  1. 1.The University of ManchesterUK
  2. 2.University of LiverpoolUK

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