From Intervals to –? Towards a General Description of Validated Uncertainty

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 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
	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
	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 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
	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
	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 Measuring Devices
		10.1.1 From the Physical Viewpoint, It is Important to Consider the Most General Families of Measuring Devices
		10.1.2 From the Purely Mathematical Viewpoint (of Proving Results), it is Desirable to Consider the Simpler Case of Sequences
	10.2 When Is the Indistinguishability  Relation sim Transitive: A Criterion
		10.2.1 In Some Cases, the Indistinguishability Relation  Is Transitive, in Some Cases, It is Not
		10.2.2 How to Check Transitivity: Formulation of the Problem
		10.2.3 If Indistinguishability Is Transitive, Then from Each i, We Can Effectively Find j for Which ajsimj bj  And bjsimj cj Imply aisimi ci
	10.3 How to Extend This Result to General Results  and Techniques: Topology, Compactness, Etc.
		10.3.1 Observations
		10.3.2 Notion of Convergence
		10.3.3 Compactness Result
		10.3.4 Discussion: Possible Applications to the Inverse Problem
		10.3.5 Additional Topology-Like Definition: Closed Set
		10.3.6 Additional Topology-Like Definition: Open Set
	10.4 When Can We Reduce Simplicial Complex Description to a Graph Description?
	Reference
11 Future Work
	11.1 More Efficient Ways to Take Conditional Knowledge Into Consideration When Describing the Measured Quantity
	11.2 Extension to Probabilistic Uncertainty
	11.3 Extension to the Case When Do Not Have a Full Theory
	11.4 Formalizing the Notion of Possibility
	References