List of Figures

1.1 Hasse diagrams for T HREE and FOU R. . . . . . . . . . . .
16

2.1 Dependency graph for P 1 . . . . . . . . . . . . . . . . . . . . .
48

2.2 Dependency graph for P 2 . . . . . . . . . . . . . . . . . . . . .
48

4.1 Dependencies between single-valued fixed-point theorems. . . 113

5.1 The main classes of programs discussed in this book. . . . . . 160

7.1 The neural-symbolic cycle. . . . . . . . . . . . . . . . . . . . . 187

7.2 Unit N k in a connectionist network. . . . . . . . . . . . . . . 188

7.3 Sketch of a 3-layer recurrent network. . . . . . . . . . . . . . 191

7.4 Two 3-layer feedforward networks of binary threshold units. . 194

7.5 Transforming T P into f P . . . . . . . . . . . . . . . . . . . . . 198

7.6 The embedding of the T P -operator for Program 7.5.1. . . . . 199

7.7 The networks from Example 7.5.7. . . . . . . . . . . . . . . . 202

7.8 The embedding and approximation of a T P -operator. . . . . . 202

7.9 An approximating sigmoidal network for Program 7.5.1. . . . 204

7.10 An approximation using the raised cosine function. . . . . . . 205

7.11 An RBF network approximating a T P -operator. . . . . . . . . 206

7.12 A two-dimensional version of the Cantor set. . . . . . . . . . 207

7.13 A construction of the two-dimensional Cantor set. . . . . . . 207

7.14 A vector-based approximating network. . . . . . . . . . . . . 208

7.15 A conjunction unit for FOU R. . . . . . . . . . . . . . . . . . 216

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