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[null,"1.","Record Nr.",null,"UNINA9910632469303321"]
[null,null,"Autore",null,"Kreinovich Vladik"]
[null,null,"Titolo",null,"From intervals to -? : towards a general description of validated uncertainty / / Vladik Kreinovich, Graçaliz Pereira Dimuro, Antônio Carlos da Rocha Costa"]
[null,null,"Pubbl/distr/stampa",null,"Cham, Switzerland : , : Springer, , [2023]"]
[{"$nulls":4},"©2023"]
[null,null,"ISBN",null,"3-031-20569-3"]
[null,null,"Descrizione fisica",null,"1 online resource (125 pages)"]
[null,null,"Collana",null,"Studies in computational intelligence ; ; Volume 1041"]
[null,null,"Disciplina",null,"006.3"]
[null,null,"Soggetti",null,"Computational intelligence"]
[{"$nulls":4},"Uncertainty (Information theory)"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[{"$nulls":3},"Livello bibliografico","Monografia"]
[{"$nulls":3},"Nota di bibliografia","Includes bibliographical references."]
[null,null,"Nota di contenuto",null,"Intro -- Preface -- Contents -- 1 Motivation and Outline -- 1.1 Why Computers? -- 1.2 Why Interval Computations? -- 1.3 Why Go Beyond Intervals? -- 1.4 Outline -- References -- 2 A General Description of Measuring Devices: Plan -- 3 A General Description of Measuring Devices: First Step-Finite Set of Possible Outcomes -- 3.1 Every Measuring Device Has Finitely Many Possible Outcomes -- 3.2 Not All Marks on a Scale Can Be Physically Possible -- 3.3 We Need a Theory -- 3.4 We Need a Theory that Also Described a Measuring Device -- 3.5 We Want a Theory that Is ``Full'' in Some Natural Sense -- 3.6 A Seemingly Natural Definition of a Full Theory is Not Always Adequate -- 3.7 What Exactly Is a Theory? -- 3.8 What Kind of Statements Are We Allowing? -- 3.9 What Exactly Is a Full Theory -- 3.10 The Existence of a Full Theory Makes the Set of All Physically … -- 3.11 Conclusion: Algorithmically Listable Set of Physically Possible Outcomes -- 3.12 Example 1: Interval Uncertainty -- 3.13 Example 2: Counting -- 3.14 Example 3: ``Yes''-``No'' Measurements -- 3.15 Example 3a: Repeated ``Yes''-``No'' Measurements -- 3.16 Example 4: A Combination of Several Independent Measuring Instruments -- References -- 4 A General Description of Measuring Devices: Second Step-Pairs of Compatible Outcomes -- 4.1 How Do We Describe Uncertainty: Main Idea -- 4.2 Comment on Quantum Measurements -- 4.3 Some Pairs of Outcomes Are Compatible (Close),  Some Are Not -- 4.4 The Existence of a Full Theory Makes the Set of All Compatible Pairs "]
[{"$nulls":4},"of Outcomes Algorithmically Listable -- 4.5 Conclusion: Algorithmically Listable Set of Compatible Pairs of Outcomes -- 4.6 Description in Terms of Existing Mathematical Structures -- 4.7 Example 1: Interval Uncertainty -- 4.8 Example 2: Counting -- 4.9 Example 3: ``Yes''-``No'' Measurements."]
[{"$nulls":4},"4.10 Example 3a: Repeated ``Yes''-``No'' Measurements -- 4.11 Example 4: A Combination of Several Independent Measuring Instruments -- 4.12 Computational Complexity of the Graph Representation of a Measuring Device: General Case -- 4.13 Computational Complexity of the Graph Representation of a Measuring Device: Case of the Simplest Interval Uncertainty -- 4.14 Computational Complexity of the Graph Representation of a Measuring Device: General Case of Interval Uncertainty -- 4.15 Computational Complexity of the Graph Representation of a Measuring Device: Lower Bound for the Case of the General Interval Uncertainty -- 4.16 Computational complexity of the Graph Representation of a Measuring Device:  Case of Multi-D Uncertainty -- 4.17 Computational Complexity of the Graph Representation of a Measuring Device:  General Case of Localized Uncertainty -- References -- 5 A General Description of Measuring Devices: Third Step-Subsets of Compatible Outcomes -- 5.1 From Pairs to Subsets -- 5.2 Is Information About Compatible Pairs Sufficient? -- 5.3 Information About Compatible Pairs Is Sufficient  For Intervals -- 5.4 Information About Compatible Pairs is Not Sufficient in the General Case -- 5.5 The Existence of a Full Theory Makes the Family of All Compatible … -- 5.6 Conclusion: Algorithmically Listable Family of Compatible Sets of Outcomes -- 5.7 Description in Terms of Existing Mathematical Structures: Simplicial Complexes -- 5.8 Resulting Geometric Representation of a Measuring Device -- 5.9 Towards Description in Terms of Existing Mathematical Structures: Domains -- 5.10 How to Reformulate the Above Description of a Measuring Device in Terms of Domains? -- 5.11 Example 1: Interval Uncertainty -- 5.12 Example 2: Counting -- 5.13 Example 3: ``Yes''-``No'' Measurements -- 5.14 Example 4: A Combination of Several Independent Measuring Instruments."]
[{"$nulls":4},"5.15 Computational Complexity of the Simplicial Complex Representation … -- 5.16 Computational Complexity of the Simplicial Complex Representation of a Measuring Device: Case of Interval Uncertainty -- 5.17 Computational Complexity of the Simplicial Complex Representation of a Measuring Device: Case of Multi-D Uncertainty -- 5.18 Computational Complexity of the Simplicial Complex  Representation of a Measuring Device: General Case  of Localized Uncertainty -- References -- 6 A General Description of Measuring Devices: Fourth Step-Conditional Statements About Possible Outcomes -- 6.1 Subsets of Compatible Outcomes Do Not Always Give A Complete Description of a Measuring Device -- 6.2 What We Do We Need to Add to the Subsets Description to Capture the Missing Information  About a Measuring Device? -- 6.3 The Existence of a Full Theory Makes the Set of All True Conditional Statements Algorithmically Listable: An Argument -- 6.4 Family of Conditional Statements: Natural Properties -- 6.5 Conclusion: Algorithmically Listable Family  of Conditional Statements -- 6.6 Description in Terms of Existing Mathematical Structures: Deduction Relation -- 6.7 Description in Terms of Existing Mathematical Structures: Domains -- 6.8 Example 1: Interval Uncertainty -- 6.9 Example 2: Counting -- 6.10 Example 3: ``Yes''-``No'' Measurements -- 6.11 Example 4: A Combination of Several Independent Measuring Instruments -- 6.12 Computational Complexity of the Domain Representation of a Measuring Device:  A General Case -- 6.13 Computational Complexity "]
[{"$nulls":4},"of the Domain Representation of a Measuring Device:  Case of Interval Uncertainty -- 6.14 Computational Complexity of the Simplicial Complex Representation of a Measuring Device:  Case of Convex Multi-D Uncertainty."]
[{"$nulls":4},"6.15 Computational Complexity of the Domain Representation of a Measuring Device:  General Case of Localized Uncertainty -- References -- 7 A General Description of Measuring Devices: Fifth Step-Disjunctive Conditional Statements About the Possible Outcomes -- 7.1 Addition of Conditional Statements Does Not Always Lead to a Complete Description of a Measuring Device -- 7.2 What We Do We Need to Add to the Conditional Statements Description to Capture the Missing Information About a Measuring Device? -- 7.3 The Existence of a Full Theory Makes the Set  of All True Disjunctive Conditional Statements Algorithmically Listable -- 7.4 Family of True Disjunctive Conditional Statements: Natural Properties -- 7.5 Conclusion: Algorithmically Listable Family  of Disjunctive Conditional Statements -- 7.6 Description in Terms of Existing Mathematical Structures: Sequent Calculus -- 7.7 Description in Terms of Existing Mathematical Structures: Boolean Vectors -- 7.8 Example -- 7.9 Description in Terms of Existing Mathematical Structures: Boolean Algebra -- 7.10 Example -- 7.11 Description in Terms of Existing Mathematical Structures: Domains -- 7.12 Example -- 7.13 Is This a Final Description of Validated Uncertainty? -- 7.14 Example 1: Interval Uncertainty -- 7.15 Example 2: Counting -- 7.16 Example 3: ``Yes''-``No'' Measurements -- 7.17 Example 4: A Combination of Several Independent Measuring Instruments -- 7.18 Computational Complexity of the Boolean Representation of a Measuring Device:  A General Case -- 7.19 Computational Complexity of the Boolean Representation of a Measuring Device:  Case of Interval Uncertainty -- 7.20 Computational Complexity of the Boolean Representation of a Measuring Device:  Case of Convex Multi-D Uncertainty -- 7.21 Computational Complexity of the Domain Representation of a Measuring Device: General Case of Localized Uncertainty."]
[{"$nulls":4},"References -- 8 A General Description of Measuring Devices: Summary -- 8.1 Summary -- 8.2 Measuring Device: A Final Description -- 9 Physical Quantities: A General Description -- 9.1 General Idea -- 9.2 From the General Idea to a Formal Description -- 9.3 Set of Possible Outcomes: The Notion of a Projection -- 9.4 Pairs of Compatible Outcomes: The Notion of a Projection -- 9.5 Subsets of Compatible Outcomes: The Notion of a Projection -- 9.6 Definition Reformulated in Domain Terms -- 9.7 General Domains and Boolean Vectors: The Notion of a Projection -- 9.8 The Family of All Measuring Devices Measuring A Given … -- 9.9 Physical Quantity as a Projective Limit of Measuring Devices -- 9.10 Example -- 9.11 Within This Definition, The Fact that Every Outcome … -- 9.12 Different Sequences of Measurement Results May Correspond to the Same Value of the Measured Quantity -- 9.13 Case of Graphs -- 9.14 Within This Definition, The Fact that simI Describes Exactly Compatible Pairs Is Now a Theorem -- 9.15 Case of Simplicial Complexes -- 9.16 Within This Definition, The Fact that calSI Describes Exactly Compatible Subsets Is Now A Theorem -- 9.17 Cases of Conditional Statements and Boolean Vectors -- 9.18 Examples: A Brief Introduction -- 9.18.1 Example 1: Interval Uncertainty Leads to Real Numbers -- 9.19 Conclusion -- 9.20 Example 2: Counting Leads to Natural Numbers -- 9.21 Example 3: ``Yes''-``No'' Measurements Lead to Truth Values -- 9.22 Example 4: A Combination of Several Independent Physical Quantities -- References -- 10 Properties of Physical Quantities -- 10.1 A Useful Auxiliary Result: We Can Always Restrict Ourselves to a Sequence of "]
[{"$nulls":4},"Measuring Devices -- 10.1.1 From the Physical Viewpoint, It is Important to Consider the Most General Families of Measuring Devices."]
[{"$nulls":4},"10.1.2 From the Purely Mathematical Viewpoint (of Proving Results), it is Desirable to Consider the Simpler Case of Sequences."]