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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 9 QUADRATIC 0-1 PROGRAMMING 8S 9. 1 Energy Minimization 86 9. 2 Notation and Tenninology . . . . . . . . . . . . . . . . . 87 9. 3 Minimization Technique . . . . . . . . . . . . . . . . . . 88 9. 4 An Example . . . . . . . . . . . . . . . . . . . . . . . . 92 9. 5 Accelerated Energy Minimization. . . . . . . . . . . . . 94 9. 5. 1 Transitive Oosure . . . . . . . . . . . . . . . . . 94 9. 5. 2 Additional Pairwise Relationships 96 9. 5. 3 Path Sensitization . . . . . . . . . . . . . . . . . 97 9. 6 Experimental Results 98 9. 7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 10 TRANSITIVE CLOSURE AND TESTING 103 10. 1 Background . . . . . . . . . . . . . . . . . . . . . . . . 104 10. 2 Transitive Oosure Definition 105 10. 3 Implication Graphs 106 10. 4 A Test Generation Algorithm 107 10. 5 Identifying Necessary Assignments 112 10. 5. 1 Implicit Implication and Justification 113 10. 5. 2 Transitive Oosure Does More Than Implication and Justification 115 10. 5. 3 Implicit Sensitization of Dominators 116 10. 5. 4 Redundancy Identification 117 10. 6 Summary 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 11 POLYNOMIAL-TIME TESTABILITY 123 11. 1 Background 124 11. 1. 1 Fujiwara's Result 125 11. 1. 2 Contribution of the Present Work . . . . . . . . . 126 11. 2 Notation and Tenninology 127 11. 3 A Polynomial TlDle Algorithm 128 11. 3. 1 Primary Output Fault 129 11. 3. 2 Arbitrary Single Fault 135 11. 3. 3 Multiple Faults. . . . . . . . . . . . . . . . . . . 137 11. 4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . 139 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 ix 12 SPECIAL CASES OF HARD PROBLEMS 141 12. 1 Problem Statement 142 12. 2 Logic Simulation 143 12. 3 Logic Circuit Modeling . 146 12. 3. 1 Modelfor a Boolean Gate . . . . . . . . . . . . . 147 12. 3. 2 Circuit Modeling 148 12.
