Klir G.J. Uncertainity and Information. Foundations of Generalized Information Theory
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486
BIBLIOGRAPHY
Bayes’ theorem, 66, 67
Belief measure (function), 166–169, 175,
188, 237, 399, 400, 449
Bit, 30, 68, 209, 214, 449
Body of evidence, 159, 168, 170
consonant, 176, 177
Boltzmann entropy, 92–94, 97
Boolean algebra, 102, 261, 268, 269
Boolean lattice, 116–117
Branching, 29, 74, 90, 91, 197, 205–207,
253, 254
Cardinality, 13
Cartesian product, 14, 16, 17
Category theory, 320, 350
Cauchy equation, 70
Characteristic function, 11, 12, 335,
449
Choquet capacity, 106, 115, 135, 138, 143,
185, 418
alternating, 116, 166, 449
of infinite order, 107, 143, 165, 166,
180, 416
of order k, 107, 450
of order 2, 107, 123, 180
Choquet integral, 133–135, 138, 450
Classical information theory, 7, 95, 101,
102, 356, 357, 369, 420
Classical measure, 101–103, 137, 449
Classical measure theory, 5, 9, 19, 20, 95,
101, 415
Classical set theory, 5, 9, 22, 101, 135,
266, 268, 315, 320, 415, 420
SUBJECT INDEX
487
Additive measure, 61, 449
Additivity, 47, 62, 73, 102, 103, 105, 159,
197, 201, 203, 205, 206, 209, 214, 227,
228, 235, 236, 250, 419
countable, 102
finite, 102
Aggregate uncertainty measure, 226–239,
248, 253, 255, 375, 388, 417, 421,
437–441, 449
Algebra:
Boolean, 102, 261, 268, 269
non Boolean, 261, 262
Alpha-cut (a-cut), 263, 264, 328, 350
Alpha-cut representation, 264–266, 315,
338, 340, 350, 453
Alternative rule of combination (in
DST), 173, 174, 188
Ample field, 144
Ampliative reasoning, 356, 369, 409
Approximate reasoning, 287, 293, 305,
307, 410
Approximating:
belief functions by necessity functions,
399, 410
graded possibilities by crisp
possibilities, 403–408
Approximation problems, 388–389
Average Shannon entropy, 225
Averaging operation, 269
Base variable, 295
Basic probability assignment, 167–170,
210, 449
Uncertainty and Information: Foundations of Generalized Information Theory, by George J. Klir
© 2006 by John Wiley & Sons, Inc.
Classical uncertainty theory, 8, 26, 196,
252
Commonality function, 169
Compatibility relation, 14, 316, 317, 450
Composition, 16, 284, 285
Compositional rule of inference, 294
Computing with perceptions, 305
Computational complexity, 186, 250, 399
Conditional possibility measure, 155,
156, 159, 189, 204
Conditional Shannon entropy, 80, 81, 85,
88, 97
Conditional U-uncertainty, 122, 184
Conflict, 68, 69, 196, 218–222, 239, 248,
249, 253, 450
Conflict-resolution problems, 364–369
Confusion, 219, 254
Consistency axioms, 409, 410
Consonant body of evidence, 176, 177
Constrained fuzzy arithmetic, 275–280,
347
Constraint:
equality, 275, 277, 278, 347
generalized, 423
probabilistic, 278–280, 347
Continuity, 47, 73, 104, 197, 205, 250
Convex set of probability distributions,
123, 139, 178, 183, 214, 237, 249,
334
Coordinate invariance, 47, 198
Core (of a fuzzy set), 263
Countable additivity, 102
Credal set, 139, 214, 215, 250, 255, 418
Cutworthy property, 266, 273, 284, 286,
315–317, 350, 450
Cylindric closure, 17, 41, 43, 282, 283,
316, 373–375
Cylindric extension, 17, 41, 281–283
Decision making, 1, 132
Decomposable measure, 145, 185, 186,
188, 418
Defuzzification, 298, 299, 307, 404, 450
Degree:
of membership, 19, 261, 287
of truth, 287
De Morgan laws, 13, 268, 269
Dempster’s rule of combination,
170–174, 188
Dempster-Shafer theory (DST), 143, 144,
166, 167, 169, 176, 178, 180, 187, 188,
190, 209, 218, 240–243, 253, 254, 351,
385, 399, 416, 417, 421, 430–441,
450
Dirac measure, 145, 189
Directed divergence, 94
Disaggregated total uncertainty, 239, 240,
248, 249, 254, 255, 375, 417, 450
alternative, 250, 255, 375, 450
Discord, 221, 254
Disorganized complexity, 3, 4
Disjoint sets, 13
Dissonance, 219, 254
Duality, 112, 144, 159
Entropies of order b,96
Equivalence relation, 14, 316–318, 451
Euclidean vector space, 18, 375
Expansibility, 72, 197, 205, 206, 209
Expected value, 62, 65, 133, 369, 370
Extension principle, 318–320, 350
Finite additivity, 102
Focal set (element), 168, 328
Frame of discernment, 166
Function, 16
characteristic, 11, 12, 335, 449
membership, 19–21, 260, 335
Fuzzification, 261, 315, 320, 321, 334, 338,
348, 350, 351, 416, 451
Fuzziness, 321, 322, 451
Fuzzy arithmetic, 273–280, 306
constrained, 275–280, 347
standard, 273–277, 306
Fuzzy conjunction, 287
Fuzzy complement, 266, 287, 306, 317,
322, 350, 451
Fuzzy differential calculus, 306
Fuzzy differential equations, 306
Fuzzy disjunction, 287
Fuzzy event, 334
Fuzzy implication, 293, 307, 451
Lukasiewicz, 293
Fuzzy inference rules, 297
Fuzzy intersection, 266–268, 287, 306
Fuzzy interval, 270, 295, 316, 337, 338
canonical form of, 270, 271
trapezoidal, 271
488
SUBJECT INDEX
Fuzzy logic, 286, 287, 305, 306, 419,
420
in a broad sense, 286, 287, 307, 321
in a narrow sense, 286, 287, 307
Fuzzy measure, 106, 137
Fuzzy negation, 287
Fuzzy number, 271, 306, 451
complex, 307
triangular, 271
Fuzzy partition, 337, 338, 451
Fuzzy predicate, 287
Fuzzy probability, 351
Fuzzy proposition, 287, 326
conditional, 291
probability qualified, 290, 291
truth qualified, 288, 291
Fuzzy propositional form:
canonical, 287
conditional, 291
probability-qualified, 290
truth-qualified, 288
Fuzzy qualifier, 293
Fuzzy relation, 280–286, 294, 307, 327,
451
binary, 284–286
inverse, 284
Fuzzy relation equations, 286, 294,
307
Fuzzy sets, 19, 20, 186, 260–270, 305, 403,
419, 420
convex, 315, 316, 450
interval-valued, 299, 300, 303, 307
intuitionistic, 302, 308
lattice-based (L-fuzzy), 302, 307
nonstandard, 261, 299, 417, 418
normal, 263
of level l (l ≥ 2), 301, 302, 307
of type k (k ≥ 2), 300, 301, 303, 307
rough, 302, 303, 308
standard, 19, 260–270, 299, 303, 315
Fuzzy set theory, 9, 20, 260–262, 266, 270,
287, 303, 305, 306, 420
Fuzzy set union, 266–268, 287, 306
Fuzzy system, 294, 305, 451
Games:
cooperative, 138
non-atomic, 139
General interval structures, 139
Generalized Hartley measure, 198,
209–216, 243, 248, 253, 254, 346, 375,
380, 430–436, 451
conditional, 213
Generalized Information Theory (GIT),
7–10, 22, 102, 261, 355–357, 375, 383,
415–424
Generalized measure, 103
Generalized modus ponens, 293
Generalized Shannon entropy, 216, 218,
219, 226, 239–249, 375, 442–448, 451
Generalized theory of uncertainty
(GTU), 423, 424
Genetic algorithms, 308
Gibbs’ theorem, 79, 81, 82
Graded possibility, 144, 145, 186, 206, 403
Granulation, 295–297
Hartley-like measure, 45–57, 451
Hartley-like U-uncertainty, 206, 208
Hartley measure, 27–34, 36, 38, 42, 44, 46,
56, 197–199, 203, 204, 355, 416, 417,
451
conditional, 33, 44
joint, 32, 33
marginal, 32, 33
normalized, 31
Height (of a fuzzy set), 263
Identification problems, 373
Imprecise probabilities, 115, 129–133,
135, 136, 138, 139, 143–145, 160,
178–180, 185, 416
Independence, 63, 64, 156, 159, 187
Infimum, 15, 18
Information, 6, 9, 10, 22, 38, 42, 53, 415,
420, 423, 424
algorithmic, 23
linguistic, 417
uncertainty-based, 7, 22, 23, 95, 101,
424, 453
Information gap, 418, 421
Information transmission, 34, 43, 44, 46,
80, 84, 85, 94, 97, 204, 451
normalized, 35, 83
Informativeness, 38
predictive, 87
Interaction representation, 116–119, 136,
137, 138, 225
SUBJECT INDEX 489
International Fuzzy Systems Association
(IFSA), 306
Interval:
closed, 17, 316
fuzzy, 270, 271, 295, 316, 337, 338
open, 17
semiopen, 18
Interval arithmetic, 306
Interval-valued probability distributions,
144, 145, 178, 451
conditional, 184, 188
joint, 183
marginal, 183
reachable (feasible), 179, 180, 188, 255,
280, 338, 350, 351, 416
Inverse:
of binary relation, 16
of function, 16
Inverse problems, 286, 294, 409
k-additive measures, 145, 186, 188, 418
k-monotone measure, 145, 452
Knowledge acquisition, 303
Lagrange multipliers, 370
Lambda measures (l-measures), 144,
145, 160–165, 178, 348–350, 377, 402,
403, 416
Lambda rule (l-rule), 161, 162, 185, 186,
187
Lattice, 15
Lebesgue integral, 133, 134, 137
Linguistic hedge, 269
Linguistic uncertainty, 321
Linguistic variable, 294–296, 423, 452
Locally inconsistent subsystems,
364–369, 408
Log-interval scales, 392–394, 396, 398,
402, 403, 410
Lower probability function, 112–115,
119, 130, 132, 137, 138, 161, 178, 179,
196, 237, 280, 416, 452
conditional, 122, 184
joint, 122, 183
marginal, 121
Marginal problem, 385, 386
Measure:
additive, 61, 449
classical, 101–103, 137, 449
decomposable, 145, 185, 186, 188,
418
fuzzy, 106, 137
generalized, 103
k-additive, 145, 186, 188, 418
k-monotone, 145, 452
monotone, 103–107, 117–119, 135, 137,
139, 143–148, 160, 167, 185, 196, 262,
351, 452
nonadditive, 106, 137
probability, 62, 64, 146, 162, 166, 174,
175, 334
regular, 104
semicontinuous, 104
subadditive, 105, 453
Sugeno, 144, 160, 187, 453
superadditive, 105, 453
2-monotone, 145
•-monotone, 145
Measurement, 105
Measurement unit, 30, 46, 68
Measure of fuzziness, 321–326, 350, 351,
452
Membership function, 19–21, 260, 335
constructing, 303, 308
trapezoidal, 339
Möbius representation, 108, 136–138,
149, 150, 153, 165, 186, 196, 237, 329,
346, 452
Möbius transform, 108, 119, 138, 168,
214, 416
Modal logic, 56, 188
Modifier, 269, 306
Monotone measure, 9, 10, 103–107,
117–119, 135, 137, 139, 143–148, 160,
167, 185, 196, 262, 351, 452
Monotonicity, 30, 47, 74, 103–105, 197,
205, 206, 211–213, 215, 250, 254
Natural language, 20, 305, 307, 321, 334,
420
Necessity, 26, 56
Necessity measure (function), 26, 56,
106, 144, 147–149, 158, 159, 176, 187,
330, 399, 400, 452
Nested sets, 14, 263, 328, 452
Neural network, 187, 304, 308
Nonadditive measure, 106, 137
490
SUBJECT INDEX
Noninteraction, 33, 34, 63–66, 123–125,
138, 156, 159, 177, 202, 203, 210,
214
Nonspecificity, 28, 53, 196, 209, 239, 243,
248, 249, 254, 255, 385, 452
identification, 41, 42
predictive, 37
Normalized Shannon entropy, 83
Normalization, 30, 75, 147, 158, 159, 197,
205, 206, 209, 329
North American Fuzzy Information
Processing Society (NAFIPS),
306
Optimization problems, 375
Ordinal scales, 394–398
Organized complexity, 3, 4
Organized simplicity, 3, 4
Partial ordering, 15, 316, 317, 452
Partition, 14, 336, 452
fuzzy, 337, 338, 451
Plausibility measure (function), 166–169,
188, 452
joint, 169
Possibility, 7, 26, 56, 423
comparative, 160
graded, 144, 145, 186, 206, 403
similarity-based, 160
Possibility—l-measure transformations,
402–403
Possibility function, 26, 27, 56, 452
basic, 26, 146, 149, 159, 449
conditional, 155, 156, 159, 189, 204
joint, 153–155
marginal, 153–155
Possibility measure, 106, 144–149, 158,
159, 176, 185, 187, 402, 403, 416
Possibility profile, 149–153, 187, 198–200,
326, 452
greatest, 152
smallest, 152
Possibility theory:
classical, 26, 144, 252
fuzzy-set interpretation of, 160,
326–331, 351
generalized, 144, 145
graded, 144, 145, 148, 158–160, 176,
186, 253, 326, 399, 410, 423
Potential surprise, 186
Power set:
crisp, 12
fuzzy, 262
Prediction, 37, 87
Principle:
of information preservation, 357,
388
of maximum entropy, 356, 369–372,
408, 409, 421
of maximum nonspecificity, 373,
375–381, 410
of maximum uncertainty, 356–358, 369,
375, 383, 410
of minimum cross-entropy, 372, 409,
410
of minimum entropy, 356,
of minimum information loss, 357, 367,
368
of minimum nonspecificity, 364
of minimum uncertainty, 356–359,
408
of requisite generalization, 356–358,
383–387
of uncertainty invariance, 356–358,
387, 388, 399, 402–404, 410
Probability, 5, 7, 26, 61, 95, 187, 420, 422,
423
Probability box (p-box), 188, 189
Probability consistency, 227, 228,
235
Probability density function, 65, 334
conditional, 65
joint, 65
marginal, 65
Probability distribution, 62
Probability distribution function, 62, 64,
65, 146, 159, 167, 334
conditional, 63, 159
joint, 63
marginal, 63
Probability granule, 339, 351
Probability measure, 62, 64, 146, 162, 166,
174, 175, 334
Probability of fuzzy event, 335, 351
Probability-possibility consistency index,
396
Probability-possibility transformations,
390–398, 410
SUBJECT INDEX 491