Dedication......Page 6
Short Biography of Vladik Kreinovich......Page 7
Contents......Page 9
Symmetries Are Important......Page 12
References......Page 14
Constructive Mathematics......Page 17
1 Introduction......Page 18
2 Continuity......Page 21
References......Page 27
1 Introduction......Page 29
2 Cantorian Set Theory......Page 30
3 Skepticism About Cantorian Set Theory and ZFC......Page 33
3.1 Computational Facts and Laws......Page 34
3.2 Skepticism About Replacement......Page 35
3.3 Skepticism About Choice......Page 36
3.4 Skepticism About Power Sets......Page 37
3.5 Skepticism About Infinity......Page 40
4.1 Hilbert's Support for ZFC......Page 41
4.2 Der Grundlagenkrise......Page 43
5 Pedagogical Issues......Page 44
6 Language K......Page 45
6.2 Arithmetic in K......Page 46
6.4 Sets and Set Operations......Page 47
6.6 Logical Operators......Page 48
6.7 Aggregates and Set Comprehension......Page 49
6.9 Fine Points......Page 50
6.10 Typical Exercises......Page 51
7.1 Language Definition and Properties......Page 52
7.2 Finitist Semantics for P......Page 53
References......Page 54
Fuzzy Techniques......Page 55
Fuzzy Logic for Incidence Geometry......Page 56
2.1 Geometric Primitives and Incidence......Page 57
3.1 Proposed Fuzzy Logic......Page 58
3.2 Geometric Primitives as Fuzzy Predicates......Page 61
3.3 Formalization of Fuzzy Predicates......Page 63
3.4 Fuzzy Axiomatization of Incidence Geometry......Page 68
3.5 Equality of Extended Lines Is Graduated......Page 71
3.7 Equality of Extended Points and Lines......Page 72
4.1 A Euclid’s First Postulate Formalization......Page 83
4.2 Fuzzy Logical Inference for Euclid’s First Postulate......Page 85
4.3 Example......Page 89
References......Page 91
1 Introduction......Page 93
2 Past: Interval Valued Intuitionistic Fuzzy Sets—A Definition, Operations, Relations and Operators over Them......Page 94
3 Present: Interval Valued Intuitionistic Fuzzy Sets—Theory and Applications......Page 106
4 Future: Interval Valued Intuitionistic Fuzzy Sets—Open Problems and Ideas for Next Research......Page 110
References......Page 111
1 Introduction......Page 117
2 Knowledge Representation......Page 119
2.1 Fuzzy Sets and Possibility Degrees......Page 120
2.3 Linguistic Summaries......Page 121
2.4 Fuzzy Ontologies......Page 122
2.5 Similarity Measures......Page 123
References......Page 124
How to Enhance, Use and Understand Fuzzy Relational Compositions......Page 126
1.2 Fuzzy Relational Compositions......Page 127
1.3 Compositions Based on Fuzzy Quantifiers......Page 128
1.4 Excluding Features in Fuzzy Relational Compositions......Page 130
2.1 Taxonomical Classification—Setting up the Problem......Page 131
2.2 Results of the Taxonomic Classification and Discussion......Page 133
3.1 Direct Combination of Fuzzy Quantifier and Excluding Features......Page 134
3.2 Fuzzy Relational Compositions Using Grouping of Features......Page 135
3.3 Experiments......Page 137
4 Conclusion and Future Work......Page 138
References......Page 139
1 Introduction......Page 142
2.1 Łukasiewicz Logic and MV-Algebras......Page 143
2.2 Multilayer Perceptrons......Page 144
3 Łukasiewicz Equivalent Neural Networks......Page 145
4.1 Input Selection and Polynomial Completeness......Page 148
4.2 On the Number of Hidden Layers......Page 151
References......Page 152
1 Introduction......Page 155
1.1 Delays and Their Impact on Systems......Page 156
2 Motivation......Page 160
3 Delay Modeling......Page 161
4 Environment......Page 163
4.1 Implementation......Page 164
5 Results......Page 169
7 Future Work......Page 180
References......Page 181
1 Introduction......Page 183
2 Some Generalizations of Fuzzy Sets—An Overview......Page 184
3 Aggregation Functions on Intervals......Page 188
4 Aggregation Functions for Picture Fuzzy Sets......Page 190
5 Concluding Remarks......Page 192
References......Page 193
1 Introduction......Page 199
2 Formulation of the IWA......Page 200
3 Computing the IWA......Page 201
4 Properties of the IWA......Page 203
5 Algorithms for Finding the Switch Points......Page 204
6 Centroid Type-Reduction of an IT2 Fuzzy Set......Page 205
7 Type-Reduction in IT2 Fuzzy Systems......Page 206
8 Remarks......Page 209
9 Centroid Type-Reduction for GT2 FSs......Page 211
10 Type-Reduction in a GT2 Fuzzy System......Page 212
References......Page 214
Fuzzy Answer Set Programming: From Theory to Practice......Page 216
1 Introduction......Page 217
2 Modeling Problems as Logic Programs......Page 218
3 Syntax and Semantics of FASP......Page 219
4 Solving FASP Programs......Page 221
4.1 Non-disjunctive Programs......Page 222
4.2 Disjunctive Programs......Page 224
5 An Application of FASP in Biological Network Modeling......Page 225
References......Page 229
1 Introduction......Page 232
2 Generic Structure of FCM and Its Development......Page 234
3 Design FCMs Based on Experts......Page 238
4.1 Enhancement, Generalization of Individual Units and New Topologies (Architectures)......Page 240
4.2 Timed Fuzzy Cognitive Maps......Page 242
5.1 Competitive Fuzzy Cognitive Maps with Case Based Reasoning......Page 243
5.2 Timed Fuzzy Cognitive Maps with Hidden Markov Models......Page 244
6 Applications Areas......Page 245
7 Main Future Directions......Page 246
References......Page 247
Interval Computations......Page 250
Rigorous Global Filtering Methods with Interval Unions......Page 251
1.1 Context......Page 252
1.2 Interval Unions and Related Work......Page 253
1.3 Contribution......Page 254
2 Interval Unions......Page 255
3 Interval Union and CSPs......Page 256
3.1 The Forward-Backward Constraint Propagation......Page 257
3.2 The Interval Union Newton Operator......Page 259
3.3 Gap Filling......Page 261
4 GloptLab......Page 262
5.1 The COCONUT Test Set......Page 264
5.2 Forward-Backward Constraint Propagation......Page 265
5.3 Interval Union Newton Method......Page 266
References......Page 268
1 Introduction......Page 270
2 How Can One Measure the Complexity of Computation Problems?......Page 271
2.2 Encoding the Input and Output of Computation Problems by Strings......Page 272
2.3 Some Discrete Complexity Classes......Page 273
2.4 Polynomial Time Computable Real Numbers and NP-Real Numbers......Page 276
3.1 The General Range Computation Problem for Polynomials......Page 277
3.2 Linear Functions......Page 278
3.4 The Lower Complexity Bound of Gaganov......Page 279
5 On the Complexity of the Range Computation Problem for a Fixed Sequence of Polynomials......Page 281
5.1 Noncomputable Sequences Versus Polynomial Time Computable Sequences......Page 282
5.3 The Problem for a Fixed Sequence of Polynomials of Degree at Least 2......Page 283
5.4 The Problem for a Fixed Sequence of Polynomials and a Fixed Sequence of Interval Boxes......Page 284
5.5 Summary......Page 285
6 Proofs......Page 286
References......Page 294
1 Introduction......Page 295
3 M-matrices and H-matrices......Page 297
4 Inverse Nonnegative Matrices......Page 299
5 Totally Positive Matrices......Page 300
6 P-Matrices......Page 302
7 Diagonally Interval Matrices......Page 303
8 Nonnegative Matrices......Page 304
9 Inverse M-matrices......Page 305
10 Parametric Matrices......Page 307
References......Page 308
1 Introduction......Page 311
2 Generic Algorithm......Page 314
3.1 Herbrand Expansion......Page 317
3.2 Shared Quantities......Page 319
3.4 When Is the Second Phase Not Necessary?......Page 320
4 Necessary Conditions......Page 321
5 Seeking Local Optima of a Function......Page 322
6 Example Heuristics......Page 323
References......Page 325
1 Remarks on the History of Interval Arithmetic......Page 329
2 High Speed Interval Arithmetic by Exact Evaluation of Dot Products......Page 330
3 From Closed Real Intervals to Connected Sets of Real Numbers......Page 331
4 Computing Dot Products Exactly......Page 332
5 Early Super Computers......Page 337
6 Conclusion......Page 338
References......Page 339
1 Introduction......Page 341
2 Set Inversion for Nonlinear Gaussian Estimation......Page 342
3 Linearization Method......Page 347
4 Test-Cases......Page 348
4.2 Test-Case 2......Page 349
4.3 Test-Case 3......Page 352
5 Conclusion......Page 354
References......Page 356
1 Introduction......Page 358
2.1 Condition Number of a Problem......Page 360
2.2 Amplification Factor for Interval Computations......Page 361
3 Summation......Page 362
4 Solving Linear Systems......Page 365
5 Univariate Nonlinear Equations......Page 368
6 Conclusion and Future Work......Page 370
References......Page 371
1 Problem Statement......Page 373
2 Idea of the Solution......Page 374
3 Implementation of the Idea......Page 378
4 Tolerable Solution Set for Interval Linear Systems of Equations......Page 381
5 Recognizing Functional and Its Application......Page 384
6 Formal (algebraic) Approach......Page 391
References......Page 395
Uncertainty in General and its Applications......Page 397
1 Introduction......Page 398
3 Probabilities......Page 400
3.1 What if Alice Does Not Walk?......Page 404
3.3 The Case of n53 Steps in Random Walks......Page 405
3.4 The Probability to Guess the Other Party's Number......Page 406
5 Conclusions......Page 407
References......Page 408
1 Introduction......Page 409
2 Global Processes......Page 412
3 Local Processes......Page 416
4 Example......Page 419
References......Page 423
1 Introduction......Page 425
2 Definition of Selectors......Page 428
3 Deterministic Selectors and Downward Self-Reducibility......Page 431
4 Probabilistic Selectors for Languages Beyond PSPACE......Page 433
References......Page 436
1 Introduction......Page 438
2 Background......Page 441
3 Analysis of Magnetic Data......Page 446
4 Conclusions......Page 453
References......Page 454
1 Formulation of the Problem......Page 456
2 Main Idea......Page 458
3 Main Result: Formulation and Discussion......Page 460
4 Proof......Page 461
References......Page 464
1 Vladik Kreinovich and His Wife. Autumn, 1986......Page 466
2.1 Observable Causality Implies Lorentz Group......Page 467
2.2 Approximately Measured Causality Implies the Lorentz Group......Page 468
4 Time Machine......Page 469
4.2 Time Machine Construction Using Resilient Leaf in 5-Dimensional Hyperspace M5......Page 470
5 Antigravitation......Page 474
References......Page 475
Bilevel Optimal Tolls Problems with Nonlinear Costs: A Heuristic Solution Method......Page 476
1 Introduction......Page 477
2 The Toll Optimization Problem......Page 479
3 Linear-Quadratic Bilevel Program Reformulation......Page 481
4 The Heuristic Algorithms......Page 482
5.1 Algorithm 1......Page 484
5.2 Algorithm 2......Page 485
5.3 The Algorithm for Calculating the ARSBs......Page 486
5.4 The Procedure for Finding the Jacobian Matrices......Page 488
5.5 Filled Function Algorithm......Page 489
6 Numerical Results......Page 490
7 Conclusions......Page 496
References......Page 510
1.1 Preliminaries......Page 512
1.2 Challenges of k-Fold Cross Validation......Page 514
1.3 k-Fold Cross Validation Process......Page 516
2.1 Shannon Function......Page 517
2.2 Alternative Algorithms......Page 519
2.3 Interactive Hybrid Algorithm......Page 521
3.1 Case Study 1: Linear SVM and LDA in 2-D on Modeled Data......Page 522
3.2 Case Study 2: LDA and Visual Classification in 4-D on Iris Data......Page 527
3.3 Case Study 3: GLC-AL and LDA in 9-D on Wisconsin Breast Cancer Diagnostic Data......Page 529
4 Discussion and Conclusion......Page 531
Appendix 1: Base GLC-L Algorithm......Page 533
Appendix 2: Algorithm GLC-AL for Automatic Discovery of Relation Combined with Interactions......Page 535
References......Page 537
1 Introduction......Page 539
2 What is a Conditional Event and Why?......Page 540
3.1 A Non Boolean Structure for Conditional Events......Page 542
3.2 The Product Space Approach to Conditional Events......Page 544
4 Implications of Conditional Event Algebras......Page 546
References......Page 547
1 Formulation of the Problem......Page 550
2 Analysis of the Problem......Page 552
3 A General Approach to Reaching Stationarity......Page 555
References......Page 557
1 Introduction......Page 558
1.1 Research Process......Page 559
2.2 Copula Method......Page 560
2.3 Value at Risk (VaR)......Page 561
2.5 Kernel Density Estimation......Page 562
3 Computation of Value at Risk......Page 563
3.1 Definition......Page 564
3.2 Delta Normal Model......Page 565
3.3 Monte Carlo Simulation......Page 566
4.1 Analytical Method......Page 567
4.2 VaR Computation Based on Copula Model......Page 568
4.3 Back Testing......Page 570
5.1 Normality Test and Correlation Analysis......Page 571
5.2 Augmented Dickey-Fuller (ADF) Unit Root Test......Page 572
5.3 Normality Test......Page 573
5.4 Evaluation of Marginal Distribution and Copula Parameter......Page 574
5.5 The Selection of Optimal Copula......Page 575
5.6 Tail Dependence Research......Page 578
5.7 Value at Risk Calculation......Page 579
6 Conclusion......Page 581
References......Page 583
Minimax Context Principle......Page 584
1 Contexts and Their Supports in Classical Realm......Page 585
2 Device Resolution Order......Page 586
4 Minimax Context Principle for Classical Systems......Page 587
5 TFTP and Daseinisation......Page 588
7 A Toy Model of `Topologimeter'......Page 589
References......Page 592
Neural Networks......Page 593
1 Rectified Linear Neurons: Formulation of the Problem......Page 594
2 Our Explanation......Page 596
3.1 Symmetry-Based Argument......Page 600
3.2 Complexity-Based Argument......Page 601
3.3 Fuzzy-Based Argument......Page 602
References......Page 603
1 Introduction......Page 605
2 Orthogonal and Quasiorthogonal Geometry......Page 606
3 Graph Theoretic Aspects of Quasiorthogonality......Page 609
4 Quasiorthogonal Sets in Hamming Cubes......Page 611
5 Construction of Sparse Ternary Quasiorthognal Sets......Page 613
6 Vector Space Models of Word Semantics......Page 614
7 Some Variants of Orthogonality......Page 615
References......Page 617
1 Introduction......Page 620
2 Preliminaries......Page 622
3 Infinite Heaviside Perceptron Networks......Page 625
4 Generalizing the Integral Formula......Page 628
5 Network Complexity......Page 631
References......Page 636
