## Rough Sets: Theoretical Aspects of Reasoning about DataTo-date computers are supposed to store and exploit knowledge. At least that is one of the aims of research fields such as Artificial Intelligence and Information Systems. However, the problem is to understand what knowledge means, to find ways of representing knowledge, and to specify automated machineries that can extract useful information from stored knowledge. Knowledge is something people have in their mind, and which they can express through natural language. Knowl edge is acquired not only from books, but also from observations made during experiments; in other words, from data. Changing data into knowledge is not a straightforward task. A set of data is generally disorganized, contains useless details, although it can be incomplete. Knowledge is just the opposite: organized (e.g. laying bare dependencies, or classifications), but expressed by means of a poorer language, i.e. pervaded by imprecision or even vagueness, and assuming a level of granularity. One may say that knowledge is summarized and organized data - at least the kind of knowledge that computers can store. |

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### Contents

THEORETICAL FOUNDATIONS | 1 |

12 Knowledge and Classification | 2 |

14 Equivalence Generalization and Specialization of Knowledge | 6 |

Exercises | 7 |

IMPRECISE CATEGORIES APPROXIMATIONS AND ROUGH SETS | 9 |

23 Approximations of Set | 10 |

24 Properties of Approximations | 11 |

25 Approximations and Membership Relation | 15 |

73 Semantics of Decision Logic Language | 83 |

74 Deduction in Decision Logic | 85 |

75 Normal Forms | 88 |

76 Decision Rules and Decision Algorithms | 89 |

77 Truth and Indiscernibility | 91 |

78 Dependency of Attributes | 94 |

79 Reduction of Consistent Algorithms | 95 |

710 Reduction of Inconsistent Algorithms | 98 |

26 Numerical Characterization of Imprecision | 16 |

27 Topological Characterization of Imprecision | 17 |

28 Approximation of Classifications | 22 |

29 Rough Equality of Sets | 24 |

210 Rough Inclusion of Sets | 27 |

Summary | 29 |

References | 30 |

REDUCTION OF KNOWLEDGE | 33 |

33 Relative Reduct and Relative Core of Knowledge | 35 |

34 Reduction of Categories | 38 |

35 Relative Reduct and Core of Categories | 41 |

Summary | 42 |

References | 43 |

DEPENDENCIES IN KNOWLEDGE BASE | 45 |

43 Partial Dependency of Knowledge | 47 |

Summary | 48 |

Exercises | 49 |

KNOWLEDGE REPRESENTATION | 51 |

52 Examples | 52 |

53 Formal Definition | 55 |

54 Significance of Attributes | 58 |

55 Discernibility Matrix | 60 |

Summary | 62 |

References | 63 |

DECISION TABLES | 68 |

63 Simplification of Decision Tables | 71 |

Summary | 77 |

Exercises | 78 |

References | 79 |

REASONING ABOUT KNOWLEDGE | 81 |

72 Language of Decision Logic | 82 |

711 Reduction of Decision Rules | 101 |

712 Minimization of Decision Algorithms | 106 |

Summary | 110 |

References | 111 |

APPLICATIONS | 116 |

83 Simplification of Decision Table | 119 |

84 Decision Algorithm | 129 |

85 The Case of Incomplete Information | 130 |

Exercises | 131 |

DATA ANALYSIS | 133 |

93 Derivation of Control Algorithms from Observation | 138 |

94 Another Approach | 146 |

95 The Case of Inconsistent Data | 150 |

Summary | 159 |

References | 162 |

DISSIMILARITY ANALYSIS | 164 |

103 Beauty Contest | 172 |

104 Pattern Recognition | 174 |

105 Buying a Car | 180 |

Summary | 187 |

SWITCHING CIRCUITS | 188 |

113 MultipleOutput Switching Functions | 196 |

Summary | 202 |

References | 203 |

MACHINE LEARNING | 205 |

123 The Case of an Imperfect Teacher | 212 |

124 Inductive Learning | 215 |

Summary | 219 |

225 | |

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### Common terms and phrases

Algebraic Analysis Artificial Intelligence attribute values Bull CHAPTER classification Computer Science concepts condition and decision condition attributes considered consistent core values decision attributes decision class Decision Logic decision rules decision table defined degree of dependency denoted Dependency of Attributes discussed Dry-cough elementary categories equivalence classes equivalence relations Expert Systems formal formulas Grzymala-Busse Hence imprecision inconsistent IND(P independent Indiscernibility Relations indispensable Information Systems input variable instance Intuitionistic Logic kiln knower's knowledge base knowledge representation system learner machine learning Math means Methodologies for Intelligent minimal decision algorithms Mrozek North Holland Novotny objects Orlowska Pawlak Polish Acad positive region PQ-algorithm problem Proc properties Proposition propositional calculus Rauszer Removing attribute Rough Set Approach Rough Set Theory Rybnik semantic set of attributes Skowron Slowinski Switching Functions Table 9 truth table universe University of Kansas upper approximation values of attributes values of condition Wasilewska Ziarko