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Satisfiability Judgement under Incomplete Information

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Transactions on Rough Sets XI

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 5946))

Abstract

In this paper we keep on discussing satisfiability of conditions by objects when information about the situation considered, including objects of some sort and concepts comprised of them, is incomplete. Our approach to satisfiability is that of concept modelling and we have a rough granular view on the problem. Objects considered are known partially, in terms of values of attributes of Pawlak information systems. An additional knowledge (domain knowledge) is assumed to be available. We choose descriptor languages for Pawlak information systems as specification languages in which we will express conditions about objects and concepts.

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References

  1. Hornby, A.S. (ed.): Oxford Advanced Learner’s Dictionary of Current English, 7th edn. with Vocabulary Trainer. Oxford University Press, Oxford (2007)

    Google Scholar 

  2. Klir, G.J., Wierman, M.J.: Uncertainty-based Information: Elements of Generalized Information Theory. Physica, Heidelberg (1998)

    MATH  Google Scholar 

  3. Keefe, R.: Theories of Vagueness. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  4. Demri, S., Orłowska, E. (eds.): Incomplete Information: Structure, Inference, Complexity. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  5. Kephart, J.O.: Research challenges of autonomic computing. In: Proc. 27th Int. Conf. on Software Engineering (ICSE 2005), May 2005, pp. 15–22. ACM Press, New York (2005)

    Google Scholar 

  6. Liu, J.: Autonomy-oriented computing (AOC): The nature and implications of a paradigm for self-organized computing. In: Proc. 4th Int. Conf. on Natural Computation (ICNC 2008), Jinan, China, October 2008, pp. 3–11. IEEE Computer Society Press, Los Alamitos (2008)

    Google Scholar 

  7. Liu, J., Jin, X., Tsui, K.C.: Autonomy Oriented Computing: From Problem Solving to Complex Systems Modeling. Kluwer, Dordrecht (2005)

    MATH  Google Scholar 

  8. Jankowski, A., Skowron, A.: A wistech paradigm for intelligent systems. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 94–132. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  9. Kondratoff, Y., Michalski, R.S. (eds.): Machine Learning: An Artificial Intelligence Approach, vol. 3. Morgan Kaufmann, San Mateo (1990)

    Google Scholar 

  10. Michalski, R.S., Carbonell, T.J., Mitchell, T.M. (eds.): Machine Learning: An Artificial Intelligence Approach. TIOGA Publ., Palo Alto (1983)

    Google Scholar 

  11. Michalski, R.S., Tecuci, G. (eds.): Machine Learning – A Multistrategy Approach, vol. 4. Morgan Kaufmann, San Mateo (1994)

    Google Scholar 

  12. Mitchell, T.M.: Machine Learning. McGraw-Hill, Portland (1998)

    Google Scholar 

  13. Cios, K.J., Pedrycz, W., Swiniarski, R.W., Kurgan, L.A.: Data Mining: A Knowledge Discovery Approach. Springer Science + Business Media, LLC (2007)

    MATH  Google Scholar 

  14. Kloesgen, W., Żytkow, J.: Handbook of Knowledge Discovery and Data Mining. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  15. Maimon, O., Rokach, L. (eds.): The Data Mining and Knowledge Discovery Handbook. Springer, Heidelberg (2005)

    Google Scholar 

  16. Kahneman, D., Slovic, P., Tversky, A. (eds.): Judgment Under Uncertainty: Heuristics and Biases. Cambridge University Press, New York (1982)

    Google Scholar 

  17. Kant, I.: Critique of Judgment. Clarendon, Oxford (1988); Transl. by Meredith, J. C.

    Google Scholar 

  18. Plous, S.: The Psychology of Judgement and Decision Making. McGraw-Hill, New York (1993)

    Google Scholar 

  19. Thiele, L.P.: The Heart of Judgment: Practical Wisdom, Neuroscience, and Narrative. Cambridge University Press, New York (2006)

    Google Scholar 

  20. Tarski, A.: The semantical concept of truth and the foundations of semantics. Philosophy and Phenomenological Research 4, 341–375 (1944)

    Article  MathSciNet  Google Scholar 

  21. Banerjee, M., Chakraborty, M.K.: Rough consequence and rough algebra. In: Ziarko, W. (ed.) Proc. 2nd Int. Workshop on Rough Sets and Knowledge Discovery (RSKD 1993), Banff, Canada, October 1993, pp. 196–207. Springer/British Computer Society, Berlin/London (1994)

    Google Scholar 

  22. Barwise, J., Seligman, J.: Information Flow: The Logic of Distributed Systems. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  23. Belnap, N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-valued Logic, pp. 8–37. Reidel, Dordrecht (1977)

    Google Scholar 

  24. Bolc, L., Borowik, P.: Many-valued Logics, vol. 1. Springer, Berlin (1992)

    MATH  Google Scholar 

  25. Chellas, B.F.: Modal Logic: An Introduction. Cambridge University Press, Cambridge (1980); Reprinted with corrections in 1988

    MATH  Google Scholar 

  26. Emerson, E.A.: Temporal and modal logic. In: Leeuwen, J.v. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 995–1072. Elsevier/The MIT Press (1990)

    Google Scholar 

  27. Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. The MIT Press, Cambridge (1995)

    MATH  Google Scholar 

  28. Kleene, S.C.: Introduction to Metamathematics. North-Holland, Amsterdam (1952)

    MATH  Google Scholar 

  29. Kripke, S.A.: Semantical analysis of modal logic I: Normal propositional calculi. Zeit. Math. Logik. Grund. 9, 67–96 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kripke, S.A.: Semantical analysis of modal logic II: Non-normal propositional calculi. In: Addison, J.W., et al. (eds.) The Theory of Models, pp. 206–220. North-Holland, Amsterdam (1965)

    Google Scholar 

  31. Łukasiewicz, J.: On three-valued logic (in Polish). Ruch Filozoficzny 5, 170–171 (1920); English transl. in [132], pp. 87–88

    Google Scholar 

  32. Łukasiewicz, J.: Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls. C. R. Soc. Sci. Lettr. Varsovie 23, 51–77 (1930); English transl. in [132], pp. 153–178

    Google Scholar 

  33. Pavelka, J.: On fuzzy logic I. Zeit. Math. Logic Grund. Math. 25, 45–52 (1979); See also parts II and III in the same volume, pp. 119–134, 447–464

    Article  MATH  MathSciNet  Google Scholar 

  34. Pawlak, Z.: Rough logic. Bull. Polish Acad. Sci. Tech. 35, 253–258 (1987)

    MATH  MathSciNet  Google Scholar 

  35. Pogorzelski, W.A.: Notions and Theorems of Elementary Formal Logic. Białystok Division of Warsaw University, Białystok (1994)

    Google Scholar 

  36. Rescher, N.: Many-valued Logic. McGraw-Hill, New York (1969)

    MATH  Google Scholar 

  37. Rosser, J.B., Turquette, A.R.: Many-valued Logics. North Holland, Amsterdam (1958)

    Google Scholar 

  38. Segerberg, K.: An Essay in Classical Modal Logic, vol. 1-3. Uppsala Universitet (1971)

    Google Scholar 

  39. Zadeh, L.A.: Fuzzy logic and approximate reasoning. Synthese 30, 407–428 (1975)

    Article  MATH  Google Scholar 

  40. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading (1974)

    MATH  Google Scholar 

  41. Cook, S.A.: The complexity of theorem proving procedure. In: Proc. 3rd Annual ACM Symp. on Theory of Computing, pp. 151–158 (1971)

    Google Scholar 

  42. Penczek, W., Szreter, M.: SAT-based unbounded model checking of timed automata. Fundamenta Informaticae 85, 425–440 (2008)

    MATH  MathSciNet  Google Scholar 

  43. Penczek, W., Woźna, B., Zbrzezny, A.: Bounded model checking for the universal fragment of CTL. Fundamenta Informaticae 51, 135–156 (2002)

    MATH  MathSciNet  Google Scholar 

  44. Woźna, B., Zbrzezny, A., Penczek, W.: Checking reachability properties for timed automata via SAT. Fundamenta Informaticae 55, 223–241 (2003)

    MATH  MathSciNet  Google Scholar 

  45. Pawlak, Z.: Information systems – theoretical foundations. Information Systems 6, 205–218 (1981)

    Article  MATH  Google Scholar 

  46. Pawlak, Z.: Information Systems: Theoretical Foundations (in Polish). Wydawnictwo Naukowo-Techniczne, Warsaw (1983)

    Google Scholar 

  47. Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer, Dordrecht (1991)

    MATH  Google Scholar 

  48. Pawlak, Z.: Rough set elements. In: [103], vol. 1, pp. 10–30 (1998)

    Google Scholar 

  49. Bazan, J.G.: Hierarchical classifiers for complex spatio-temporal concepts. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 474–750. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  50. Fensel, D.: Ontologies: A Silver Bullet for Knowledge Management and Electronic Commerce. Springer, Berlin (2003)

    Google Scholar 

  51. Nguyen, S.H., Nguyen, H.S.: Improving rough classifiers using concept ontology. In: Ho, T.-B., Cheung, D., Liu, H. (eds.) PAKDD 2005. LNCS (LNAI), vol. 3518, pp. 312–322. Springer, Heidelberg (2005)

    Google Scholar 

  52. Nguyen, S.H., Nguyen, T.T., Nguyen, H.S.: Ontology driven concept approximation. In: Greco, S., Hata, Y., Hirano, S., Inuiguchi, M., Miyamoto, S., Nguyen, H.S., Słowiński, R. (eds.) RSCTC 2006. LNCS (LNAI), vol. 4259, pp. 547–556. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  53. Skowron, A., Stepaniuk, J.: Ontological framework for approximation. In: Ślęzak, D., Wang, G., Szczuka, M.S., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 718–727. Springer, Heidelberg (2005)

    Google Scholar 

  54. Staab, S., Studer, R. (eds.): Handbook on Ontologies. Springer, Heidelberg (2004)

    Google Scholar 

  55. Gomolińska, A.: Variable-precision compatibility spaces. Electronical Notices in Theoretical Computer Science 82, 1–12 (2003), http://www.elsevier.nl/locate/entcs/volume82.html

    Google Scholar 

  56. Gomolińska, A.: Approximation spaces based on relations of similarity and dissimilarity of objects. Fundamenta Informaticae 79, 319–333 (2007)

    MATH  MathSciNet  Google Scholar 

  57. Pawlak, Z.: A treatise on rough sets. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV. LNCS, vol. 3700, pp. 1–17. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  58. Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)

    MATH  MathSciNet  Google Scholar 

  59. Słowiński, R., Greco, S., Matarazzo, B.: Dominance-based rough set approach to reasoning about ordinal data. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 5–11. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  60. Słowiński, R., Vanderpooten, D.: Similarity relation as a basis for rough approximations. In: Wang, P.P. (ed.) Advances in Machine Intelligence and Soft Computing, vol. 4, pp. 17–33. Duke University Press (1997)

    Google Scholar 

  61. Yao, Y.Y., Wong, S.K.M.: A decision theoretic framework for approximating concepts. Int. J. of Man–Machine Studies 37, 793–809 (1992)

    Article  Google Scholar 

  62. Yao, Y.Y., Wong, S.K.M., Lin, T.Y.: A review of rough set models. In: Lin, T.Y., Cercone, N. (eds.) Rough Sets and Data Mining: Analysis of Imprecise Data, pp. 47–75. Kluwer, Dordrecht (1997)

    Google Scholar 

  63. Ziarko, W.: Variable precision rough set model. J. Computer and System Sciences 46, 39–59 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  64. Ziarko, W.: Probabilistic rough sets. In: Ślęzak, D., Wang, G., Szczuka, M.S., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 283–293. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  65. Zadeh, L.A.: Outline of a new approach to the analysis of complex system and decision processes. IEEE Trans. on Systems, Man, and Cybernetics 3, 28–44 (1973)

    MATH  MathSciNet  Google Scholar 

  66. Zadeh, L.A.: Fuzzy sets and information granularity. In: Gupta, M., Ragade, R., Yager, R. (eds.) Advances in Fuzzy Set Theory and Applications, pp. 3–18. North-Holland, Amsterdam (1979)

    Google Scholar 

  67. Gomolińska, A.: Judgement of satisfiability under incomplete information. In: Czaja, L., Szczuka, M. (eds.) Proc. 18th Workshop on Concurrency, Specification and Programming (CS& P 2009), Kraków Przegorzały, September 2009, vol. 1. Warsaw University, Warsaw, pp. 164–175 (2009)

    Google Scholar 

  68. Gomolińska, A.: A graded meaning of formulas in approximation spaces. Fundamenta Informaticae 60, 159–172 (2004)

    MATH  MathSciNet  Google Scholar 

  69. Gomolińska, A.: On rough judgment making by socio-cognitive agents. In: Skowron, A., et al. (eds.) Proc. 2005 IEEE/WIC/ACM Int. Conf. on Intelligent Agent Technology (IAT 2005), Compiègne, France, September 2005, pp. 421–427. IEEE Computer Society Press, Los Alamitos (2005)

    Chapter  Google Scholar 

  70. Gomolińska, A.: Satisfiability and meaning of formulas and sets of formulas in approximation spaces. Fundamenta Informaticae 67, 77–92 (2005)

    MATH  MathSciNet  Google Scholar 

  71. Gomolińska, A.: Satisfiability of formulas from the standpoint of object classification: The RST approach. Fundamenta Informaticae 85, 139–153 (2008)

    MATH  MathSciNet  Google Scholar 

  72. Greco, S., Matarazzo, B., Słowiński, R.: Handling missing values in rough set analysis of multi-attribute and multi-criteria decision problems. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) RSFDGrC 1999. LNCS (LNAI), vol. 1711, pp. 146–157. Springer, Heidelberg (1999)

    Google Scholar 

  73. Grzymała-Busse, J.W.: Characteristic relations for incomplete data: A generalization of the indiscernibility relation. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV. LNCS, vol. 3700, pp. 58–68. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  74. Kryszkiewicz, M.: Rough set approach to incomplete information system. Information Sciences 112, 39–49 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  75. Lipski, W.: Informational systems with incomplete information. In: Proc. 3rd Int. Symp. on Automata, Languages and Programming, pp. 120–130. Edinburgh University Press, Edinburgh (1976)

    Google Scholar 

  76. Stefanowski, J., Tsoukiàs, A.: Incomplete information tables and rough classification. Computational Intelligence 17, 545–566 (2001)

    Article  Google Scholar 

  77. Rissanen, J.: Modeling by shortest data description. Automatica 14, 465–471 (1978); See also An introduction to the MDL Principle, http://www.mdl-research.org/jorma.rissanen

    Article  MATH  Google Scholar 

  78. Gomolińska, A.: Construction of rough information granules. In: [82], pp. 449–470 (2008)

    Google Scholar 

  79. Inuiguchi, M., Hirano, S., Tsumoto, S. (eds.): Rough Set Theory and Granular Computing. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  80. Nguyen, H.S., Skowron, A., Stepaniuk, J.: Granular computing: A rough set approach. Computational Intelligence 17, 514–544 (2001)

    Article  MathSciNet  Google Scholar 

  81. Pedrycz, W. (ed.): Granular Computing: An Emerging Paradigm. Physica, Heidelberg (2001)

    MATH  Google Scholar 

  82. Pedrycz, W., Skowron, A., Kreinovich, V. (eds.): Handbook of Granular Computing. John Wiley & Sons, Chichester (2008)

    Google Scholar 

  83. Skowron, A., Stepaniuk, J.: Towards discovery of information granules. In: Żytkow, J.M., Rauch, J. (eds.) PKDD 1999. LNCS (LNAI), vol. 1704, pp. 542–547. Springer, Heidelberg (1999)

    Google Scholar 

  84. Skowron, A., Swiniarski, R., Synak, P.: Approximation spaces and information granulation. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets III. LNCS, vol. 3400, pp. 175–189. Springer, Heidelberg (2005)

    Google Scholar 

  85. Gomolińska, A.: Possible rough ingredients of concepts in approximation spaces. Fundamenta Informaticae 72, 139–154 (2006)

    MATH  MathSciNet  Google Scholar 

  86. Polkowski, L., Skowron, A.: Rough mereology in information systems. A case study: Qualitative spatial reasoning. In: [104], pp. 89–135 (2001)

    Google Scholar 

  87. Stepaniuk, J.: Knowledge discovery by application of rough set models. In: [104], pp. 137–233 (2001)

    Google Scholar 

  88. Leśniewski, S.: Foundations of the General Set Theory 1 (in Polish), Moscow. Works of the Polish Scientific Circle, vol. 2 (1916); Also in [89], pp 128–173

    Google Scholar 

  89. Surma, S.J., Srzednicki, J.T., Barnett, J.D. (eds.): Stanisław Leśniewski Collected Works. Kluwer/Polish Scientific Publ., Dordrecht/Warsaw (1992)

    Google Scholar 

  90. Polkowski, L., Skowron, A.: Rough mereology. In: Raś, Z.W., Zemankova, M. (eds.) ISMIS 1994. LNCS (LNAI), vol. 869, pp. 85–94. Springer, Heidelberg (1994)

    Google Scholar 

  91. Polkowski, L., Skowron, A.: Rough mereology: A new paradigm for approximate reasoning. Int. J. Approximated Reasoning 15, 333–365 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  92. Polkowski, L., Skowron, A.: Towards adaptive calculus of granules. In: [133], vol. 1, pp. 201–228 (1999)

    Google Scholar 

  93. Drwal, G., Mrózek, A.: System RClass – software implementation of a rough classifier. In: Kłopotek, M.A., Michalewicz, M., Raś, Z.W. (eds.) Proc. 7th Int. Symp. Intelligent Information Systems (IIS 1998), Malbork, Poland, Warsaw, PAS Institute of Computer Science, June 1998, pp. 392–395 (1998)

    Google Scholar 

  94. Gomolińska, A.: On certain rough inclusion functions. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 35–55. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  95. Gomolińska, A.: Rough approximation based on weak q-RIFs. In: Peters, J.F., et al. (eds.) Transactions on Rough Sets X. LNCS, vol. 5656, pp. 117–135. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  96. Polkowski, L.: A note on 3-valued rough logic accepting decision rules. Fundamenta Informaticae 61, 37–45 (2004)

    MATH  MathSciNet  Google Scholar 

  97. Polkowski, L.: Rough mereology in analysis of vagueness. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 197–205. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  98. Xu, Z.B., Liang, J.Y., Dang, C.Y., Chin, K.S.: Inclusion degree: A perspective on measures for rough set data analysis. Information Sciences 141, 227–236 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  99. Łukasiewicz, J.: Die logischen Grundlagen der Wahrscheinlichkeitsrechnung. In: [132], pp. 16–63 (1970); First published Kraków (1913)

    Google Scholar 

  100. Zadeh, L.A.: Fuzzy logic = computing with words. IEEE Trans. on Fuzzy Systems 4, 103–111 (1996)

    Article  Google Scholar 

  101. Zhao, Y., Yao, Y.Y., Luo, F.: Data analysis based on discernibility and indiscernibility. Information Sciences 177, 4959–4976 (2007)

    Article  MATH  Google Scholar 

  102. Bazan, J.G., Skowron, A., Swiniarski, R.: Rough sets and vague concept approximation: From sample approximation to adaptive learning. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 39–62. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  103. Polkowski, L., Skowron, A. (eds.): Rough Sets in Knowledge Discovery, vol. 1-2. Physica, Heidelberg (1998)

    Google Scholar 

  104. Polkowski, L., Tsumoto, S., Lin, T.Y. (eds.): Rough Set Methods and Applications: New Developments in Knowledge Discovery in Information Systems. Physica, Heidelberg (2001)

    MATH  Google Scholar 

  105. Stepaniuk, J.: Approximation spaces in multi-relational knowledge discovery. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 351–365. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  106. Bazan, J.G., Nguyen, S.H., Nguyen, H.S., Skowron, A.: Rough set methods in approximation of hierarchical concepts. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 346–355. Springer, Heidelberg (2004)

    Google Scholar 

  107. Nguyen, S.H., Bazan, J.G., Skowron, A., Nguyen, H.S.: Layered learning for concept synthesis. In: Peters, J.F., Skowron, A., Grzymała-Busse, J.W., Kostek, B.z., Świniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 187–208. Springer, Heidelberg (2004)

    Google Scholar 

  108. Peters, J.F.: Approximation spaces for hierarchical intelligent behavioral system models. In: Dunin-Kȩplicz, B., Jankowski, A., Skowron, A., Szczuka, M. (eds.) Monitoring, Security, and Rescue Techniques in Multiagent Systems, pp. 13–30. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  109. Stone, P.: Layered Learning in Multi-agent Systems: A Winning Approach to Robotic Soccer. The MIT Press, Cambridge (2000)

    Google Scholar 

  110. Synak, P., Bazan, J.G., Skowron, A., Peters, J.F.: Spatio-temporal approximate reasoning over complex objects. Fundamenta Informaticae 67, 249–269 (2005)

    MATH  MathSciNet  Google Scholar 

  111. Pawlak, Z., Polkowski, L., Skowron, A.: Rough sets and rough logic: A KDD perspective. In: [104], pp. 583–646 (2001)

    Google Scholar 

  112. Aamodt, A., Plaza, E.: Case-based reasoning: Foundational issues, methodological variations, and system approaches. AI Communications 7, 39–52 (1994)

    Google Scholar 

  113. Aha, D.W., Kibler, D., Albert, M.K.: Instance-based learning algorithms. Machine Learning 6, 37–66 (1991)

    Google Scholar 

  114. Bazan, J.G.: Discovery of decision rules by matching new objects against data tables. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 521–528. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  115. Cover, T.M., Hart, P.E.: Nearest neighbor pattern classification. IEEE Trans. on Information Theory 13, 21–27 (1967)

    Article  MATH  Google Scholar 

  116. Duda, R.O., Hart, P.E., Stork, R.: Pattern Classification. John Wiley & Sons, New York (2002)

    Google Scholar 

  117. Dzeroski, S., Lavrac, N. (eds.): Relational Data Mining. Springer, Berlin (2001)

    MATH  Google Scholar 

  118. Greco, S., Matarazzo, B., Słowiński, R.: Dominance-based rough set approach to case-based reasoning. In: Torra, V., Narukawa, Y., Valls, A., Domingo-Ferrer, J. (eds.) MDAI 2006. LNCS (LNAI), vol. 3885, pp. 7–18. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  119. Grzymała-Busse, J.W.: LERS – a system for learning from examples based on rough sets. In: Słowiński, R. (ed.) Intelligent Decision Support. Handbook of Applications and Advances of the Rough Sets Theory, pp. 3–18. Kluwer, Dordrecht (1992)

    Google Scholar 

  120. Grzymała-Busse, J.W.: LERS – A data mining system. In: [15], pp. 1347–1351 (2005)

    Google Scholar 

  121. Grzymała-Busse, J.W.: Rule induction. In: [15], pp. 255–267 (2005)

    Google Scholar 

  122. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edn. Springer Science + Business Media, LLC, New York (2009)

    MATH  Google Scholar 

  123. Michalski, R.S.: Inferential theory of learning as a conceptual basis for multistrategy learning. Machine Learning 11, 111–151 (1993)

    MathSciNet  Google Scholar 

  124. Mitchell, M.: Analogy-making as Perception: A Computer Model. The MIT Press, Cambridge (1993)

    Google Scholar 

  125. Mitchell, M.: Analogy-making as a complex adaptive system. In: Segel, L.E., Cohen, I.R. (eds.) Design Principles for the Immune System and Other Distributed Autonomous Systems, pp. 335–360. Oxford University Press, New York (2001)

    Google Scholar 

  126. Stefanowski, J.: On rough set based approaches to induction of decision rules. In: [103], vol. 1, pp. 500–529 (1998)

    Google Scholar 

  127. Stepaniuk, J., Hońko, P.: Learning first-order rules: A rough set approach. Fundamenta Informaticae 61, 139–157 (2004)

    MATH  MathSciNet  Google Scholar 

  128. Vapnik, V.: Statistical Learning Theory. John Wiley & Sons, New York (1998)

    MATH  Google Scholar 

  129. Wojna, A.G.: Analogy-based reasoning in classifier construction. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets IV. LNCS, vol. 3700, pp. 277–374. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  130. Polkowski, L.: Rough Sets: Mathematical Foundations. Physica, Heidelberg (2002)

    MATH  Google Scholar 

  131. Torra, V., Narukawa, Y.: Modeling Decisions: Information Fusion and Aggregation Operators. Springer, Heidelberg (2007)

    Google Scholar 

  132. Borkowski, L. (ed.): Jan Łukasiewicz – Selected Works. North Holland/Polish Scientific Publ., Amsterdam/Warsaw (1970)

    Google Scholar 

  133. Zadeh, L.A., Kacprzyk, J. (eds.): Computing with Words in Information/Intelligent Systems. Physica, Heidelberg (1999)

    Google Scholar 

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  1. Department of Mathematics, Białystok University, Akademicka 2, 15267, Białystok, Poland

    Anna Gomolińska

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  1. Anna Gomolińska
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  1. University of Manitoba, Department of Electrical and Computer Engineering, , , ,, Winnipeg, R3T 5V6, Manitoba, Canada

    James F. Peters

  2. Institute of Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Andrzej Skowron

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Gomolińska, A. (2010). Satisfiability Judgement under Incomplete Information. In: Peters, J.F., Skowron, A. (eds) Transactions on Rough Sets XI. Lecture Notes in Computer Science, vol 5946. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11479-3_5

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