Analysis and Control of Boolean Networks: A Semi-tensor Product Approach

Communications and Control Engineering......Page 2
Analysis and Control of Boolean Networks: A Semi-tensor Product Approach......Page 4
9780857290960......Page 5
Preface......Page 6
Contents......Page 12
Notation......Page 16
1.1 Statements......Page 18
1.2 Implication and Equivalence......Page 22
1.3 Adequate Sets of Connectives......Page 25
1.4 Normal Form......Page 28
1.5 Multivalued Logic......Page 31
References......Page 35
2.1 Multiple-Dimensional Data......Page 36
2.2 Semi-tensor Product of Matrices......Page 46
2.3 Swap Matrix......Page 54
2.4 Properties of the Semi-tensor Product......Page 58
2.5 General Semi-tensor Product......Page 66
References......Page 70
3.1 Structure Matrix of a Logical Operator......Page 72
3.2 Structure Matrix for k-valued Logic......Page 76
3.3 Logical Matrices......Page 80
References......Page 82
4.1 Solution of a Logical Equation......Page 84
4.2 Equivalent Algebraic Equations......Page 85
4.3 Logical Inference......Page 95
4.4 Substitution......Page 101
4.5 k-valued Logical Equations......Page 102
4.6.1 Matrix Expression of Route Logic......Page 106
4.6.2 Failure Location......Page 109
4.6.3 Cascading Inference......Page 114
References......Page 117
5.1 Introduction to Boolean Networks......Page 120
5.2 Dynamics of Boolean Networks......Page 121
5.3 Fixed Points and Cycles......Page 125
5.4 Some Classical Examples......Page 136
5.5 Serial Boolean Networks......Page 141
5.6 Higher Order Boolean Networks......Page 143
5.6.1 First Algebraic Form of Higher Order Boolean Networks......Page 145
5.6.2 Second Algebraic Form of Higher Order Boolean Networks......Page 154
References......Page 156
6.1 Boolean Control Networks......Page 158
6.2 Semi-tensor Product Vector Space vs. Semi-tensor Product Space......Page 160
6.3 Cycles in Input-State Space......Page 163
6.4 Cascaded Boolean Networks......Page 168
6.5 Two Illustrative Examples......Page 171
References......Page 178
7.1 Reconstructing Networks......Page 180
7.2 Model Construction for General Networks......Page 188
7.3 Construction with Known Network Graph......Page 193
7.4 Least In-degree Model......Page 194
7.5 Construction of Uniform Boolean Network......Page 198
7.6 Modeling via Data with Errors......Page 201
References......Page 204
8.1 State Spaces of Boolean Networks......Page 206
8.2 Coordinate Transformation......Page 208
8.3 Regular Subspaces......Page 213
8.4 Invariant Subspaces......Page 221
8.5 Indistinct Rolling Gear Structure......Page 224
References......Page 229
9.1 Control via Input Boolean Network......Page 230
9.2 Subnetworks......Page 237
9.3 Controllability via Free Boolean Sequence......Page 239
9.4 Observability......Page 244
References......Page 248
10.1 What Is a Realization?......Page 250
10.2 Controllable Normal Form......Page 252
10.3 Observable Normal Form......Page 256
10.4 Kalman Decomposition......Page 259
10.5 Realization......Page 263
References......Page 265
11.1 Boolean Matrices......Page 266
11.2 Global Stability......Page 270
11.3 Stabilization of Boolean Control Networks......Page 278
References......Page 290
12.1 Problem Formulation......Page 292
12.2 Y-friendly Subspace......Page 293
12.3 Control Design......Page 300
12.4 Canalizing Boolean Mapping......Page 306
12.5 Solving DDPs via Constant Controls......Page 309
References......Page 312
13.1 Decomposition of Control Systems......Page 314
13.2 The Cascading State-space Decomposition Problem......Page 315
13.3 Comparable Regular Subspaces......Page 320
13.4 The Parallel State-space Decomposition Problem......Page 322
13.5 Input–Output Decomposition......Page 325
References......Page 328
14.1 A Review of k-valued Logic......Page 330
14.2 Dynamics of k-valued Networks......Page 333
14.3 State Space and Coordinate Transformations......Page 337
14.4 Cycles and Transient Period......Page 341
14.5 Network Reconstruction......Page 342
14.6 k-valued Control Networks......Page 347
14.7 Mix-valued Logic......Page 357
References......Page 362
15.1 Input-State Transfer Graphs......Page 364
15.2 Topological Structure of Logical Control Networks......Page 368
15.3 Optimal Control of Logical Control Networks......Page 373
15.4 Optimal Control of Higher-Order Logical Control Networks......Page 378
References......Page 386
16.1 The Input-State Incidence Matrix......Page 388
16.2 Controllability......Page 391
16.3 Trajectory Tracking and Control Design......Page 395
16.4 Observability......Page 396
16.5 Fixed Points and Cycles......Page 399
16.6 Mix-valued Logical Systems......Page 400
References......Page 405
17.1 What Is Identification?......Page 406
17.2 Identification via Input-State Data......Page 407
17.3 Identification via Input–Output Data......Page 410
17.4.1 General Algorithm......Page 413
17.4.2 Numerical Solution Based on Network Graph......Page 417
17.4.3 Identification of Higher-Order Systems......Page 420
17.5 Approximate Identification......Page 421
References......Page 424
18.1 Strategies with Finite Memory......Page 426
18.2 Cycle Strategy......Page 429
18.3 Compounded Games......Page 432
18.4 Sub-Nash Solution for Zero-Memory Strategies......Page 434
18.5 Nash Equilibrium for μ-Memory Strategies......Page 436
18.6 Common Nash (Sub-Nash) Solutions for μ-Memory Strategies......Page 438
References......Page 446
19.1 Markov Chains......Page 448
19.2 Vector Form of Random Boolean Variables......Page 456
19.3 Matrix Expression of a Random Boolean Network......Page 459
19.4 Some Topological Properties......Page 464
References......Page 467
A.1 Computation of Logical Matrices......Page 468
A.2 Basic Functions......Page 470
A.3 Some Examples......Page 475
B. Proofs of Some Theorems Concerning the Semi-tensor Product......Page 480
References......Page 483
Index......Page 484