This monograph is devoted to theoretical and experimental study of partial reductsandpartialdecisionrulesonthebasisofthestudyofpartialcovers. The use of partial (approximate) reducts and decision rules instead of exact ones allowsustoobtainmorecompactdescriptionofknowledgecontainedindecision tables, andtodesignmorepreciseclassi?ers. Weconsideralgorithmsforconstructionofpartialreductsandpartialdecision rules, boundsonminimalcomplexityofpartialreductsanddecisionrules, and algorithms for construction of the set of all partial reducts and the set of all irreducible partial decision rules. We discuss results of numerous experiments with randomly generated and real-life decision tables. These results show that partial reducts and decision rules can be used in data mining and knowledge discoverybothforknowledgerepresentationandforprediction. Theresultsobtainedinthe monographcanbe usefulforresearchersinsuch areasasmachinelearning, dataminingandknowledgediscovery, especiallyfor thosewhoareworkinginroughsettheory, testtheoryandLAD(LogicalAnalysis ofData). The monographcan be usedunder the creationofcoursesforgraduates- dentsandforPh. D. studies. An essential part of software used in experiments will be accessible soon in RSES-RoughSetExplorationSystem(InstituteofMathematics, WarsawU- versity, headofproject-ProfessorAndrzejSkowron). We are greatly indebted to Professor Andrzej Skowron for stimulated d- cussionsand varioussupportof ourwork. We aregratefulto ProfessorJanusz Kacprzykforhelpfulsuggestions. Sosnowiec, Poland MikhailJu. Moshkov April2008 MarcinPiliszczuk BeataZielosko Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 PartialCovers, ReductsandDecisionRules . . . . . . . . . . . . . . . . 7 1. 1 PartialCovers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1. 1. 1 MainNotions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1. 1. 2 Known Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1. 1. 3 PolynomialApproximateAlgorithms. . . . . . . . . . . . . . . . . . 10 1. 1. 4 Bounds on C (?)Based on Information about min GreedyAlgorithm Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1. 1. 5 UpperBoundon C (?). . . . . . . . . . . . . . . . . . . . . . . . . . 17 greedy 1. 1. 6 Covers fortheMostPartofSetCoverProblems. . . . . . . . 18 1. 2 PartialTests and Reducts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1. 2. 1 MainNotions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1. 2. 2Relationships betweenPartialCovers and Partial Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1. 2. 3 PrecisionofGreedyAlgorithm. . . . . . . . . . . . . . . . . . . . . . . 24 1. 2. 4 PolynomialApproximateAlgorithms. . . . . . . . . . . . . . . . . . 25 1. 2. 5 Bounds on R (?)Based on Information about min GreedyAlgorithm Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1. 2. 6 UpperBoundon R (?). . . . . . . . . . . . . . . . . . . . . . . . . . 28 greedy 1. 2. 7 Tests fortheMostPartofBinaryDecisionTables. . . . . . 29 1. 3 PartialDecision Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

This monograph is devoted to theoretical and experimental study of partial reductsandpartialdecisionrulesonthebasisofthestudyofpartialcovers. The use of partial (approximate) reducts and decision rules instead of exact ones allowsustoobtainmorecompactdescriptionofknowledgecontainedindecision tables, andtodesignmorepreciseclassi?ers. Weconsideralgorithmsforconstructionofpartialreductsandpartialdecision rules, boundsonminimalcomplexityofpartialreductsanddecisionrules, and algorithms for construction of the set of all partial reducts and the set of all irreducible partial decision rules. We discuss results of numerous experiments with randomly generated and real-life decision tables. These results show that partial reducts and decision rules can be used in data mining and knowledge discoverybothforknowledgerepresentationandforprediction. Theresultsobtainedinthe monographcanbe usefulforresearchersinsuch areasasmachinelearning, dataminingandknowledgediscovery, especiallyfor thosewhoareworkinginroughsettheory, testtheoryandLAD(LogicalAnalysis ofData). The monographcan be usedunder the creationofcoursesforgraduates- dentsandforPh. D. studies. An essential part of software used in experiments will be accessible soon in RSES-RoughSetExplorationSystem(InstituteofMathematics, WarsawU- versity, headofproject-ProfessorAndrzejSkowron). We are greatly indebted to Professor Andrzej Skowron for stimulated d- cussionsand varioussupportof ourwork. We aregratefulto ProfessorJanusz Kacprzykforhelpfulsuggestions. Sosnowiec, Poland MikhailJu. Moshkov April2008 MarcinPiliszczuk BeataZielosko Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 PartialCovers, ReductsandDecisionRules . . . . . . . . . . . . . . . . 7 1. 1 PartialCovers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1. 1. 1 MainNotions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1. 1. 2 Known Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1. 1. 3 PolynomialApproximateAlgorithms. . . . . . . . . . . . . . . . . . 10 1. 1. 4 Bounds on C (?)Based on Information about min GreedyAlgorithm Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1. 1. 5 UpperBoundon C (?). . . . . . . . . . . . . . . . . . . . . . . . . . 17 greedy 1. 1. 6 Covers fortheMostPartofSetCoverProblems. . . . . . . . 18 1. 2 PartialTests and Reducts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1. 2. 1 MainNotions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1. 2. 2Relationships betweenPartialCovers and Partial Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1. 2. 3 PrecisionofGreedyAlgorithm. . . . . . . . . . . . . . . . . . . . . . . 24 1. 2. 4 PolynomialApproximateAlgorithms. . . . . . . . . . . . . . . . . . 25 1. 2. 5 Bounds on R (?)Based on Information about min GreedyAlgorithm Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1. 2. 6 UpperBoundon R (?). . . . . . . . . . . . . . . . . . . . . . . . . . 28 greedy 1. 2. 7 Tests fortheMostPartofBinaryDecisionTables. . . . . . 29 1. 3 PartialDecision Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .