Table Of ContentStudies in Computational Intelligence 1041
Vladik Kreinovich
Graçaliz Pereira Dimuro
Antônio Carlos da Rocha Costa
From Intervals
to –?
Towards a General Description
of Validated Uncertainty
Studies in Computational Intelligence
Volume 1041
SeriesEditor
JanuszKacprzyk,PolishAcademyofSciences,Warsaw,Poland
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· ·
Vladik Kreinovich Graçaliz Pereira Dimuro
Antônio Carlos da Rocha Costa
From Intervals to –?
Towards a General Description of Validated
Uncertainty
VladikKreinovich GraçalizPereiraDimuro
DepartmentofComputerScience CentrodeCiênciasComputacionais
UniversityofTexasatElPaso FederalUniversityofRioGrande
ElPaso,TX,USA RioGrande,RioGrandedoSul,Brazil
UniversidadPublicadeNavarra—UPNA
AntônioCarlosdaRochaCosta
Pamplona,Spain
ProgramadePós-GraduaçãoemFilosofia
PontifíciaUniversidadeCatólicadoRio
GrandedoSul
PortoAlegre,RioGrandedoSul,Brazil
ISSN 1860-949X ISSN 1860-9503 (electronic)
StudiesinComputationalIntelligence
ISBN 978-3-031-20568-2 ISBN 978-3-031-20569-9 (eBook)
https://doi.org/10.1007/978-3-031-20569-9
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Preface
In many real-life situations, we are interested in the physical quantities that are
difficultorevenimpossibletomeasuredirectly.Toestimatethevalueofsuchquantity
y,wemeasurethevaluesofauxiliaryquantities x ,...,x thatarerelatedto y by
1 n
a known functional relation y = f(x ,...,x ), and we then use the results x˜ of
1 n i
measuring x tofindthedesiredestimate y˜ = f(x˜ ,...,x˜ ).Duetomeasurement
i 1 n
errors,themeasuredvaluesx˜ areslightlydifferentfromtheactual(unknown)values
i
x .Inotherwords,wehaveanonzeromeasurementerrorsΔx d=efx˜ −x −i.Asa
i i i
result, our estimate y˜ is different from the actual value y = f(x ,...,x ) of the
1 n
desiredquantity.
When a measured quantity x is described by a number, then we usually know
the upper bound Δ on the absolute value of the measurement error. In this case,
after we get the measurement result x˜, the only information that we have about
t(cid:2)heactual(unkn(cid:3)own)valueofthisquantityisthatthisvaluebelongstotheinterval
x˜ −Δ,x˜ +Δ .Insituationswhenweknoweachinputsx withsuchintervaluncer-
i
tainty,wecanuseappropriatetechniques—knownasintervalcomputations—toesti-
matetheresultinguncertaintyiny.Insomereal-lifeproblems,whatweareinterested
inismorecomplexthananumber.Forexample,wemaybeinterestedinthedepen-
denceoftheonephysicalquantityx onanotheronex :wemaybeinterestedinhow
1 2
thematerialstraindependsontheappliedstress,orinhowthetemperaturedepends
on a point in 3D space; in all such cases, what we are interested in is a function.
Wemaybeinterestedinevenmorecomplexstructures:e.g.,inquantummechanics,
measuringinstrumentsaredescribedbyoperatorsinaHilbertspace,soifwewant
tohaveaprecisedescriptionofanactual(imperfect)measuringinstrument,whatwe
areinterestedinisanoperator.
For many of such mathematical structures, researchers have developed ways to
representuncertainty,butusually,foreachnewstructure,wehavetoperformalotof
complexanalysisfromscratch.Itisdesirabletocomeupwithageneralmethodology
thatwouldautomaticallyproduceanaturaldescriptionofvalidateduncertaintyforall
physicallyinterestingsituations(oratleastforasmanysuchsituationsaspossible).
Inthisbook,weproducethefoundationsforsuchamethodology;itturnsoutthat
v
vi Preface
thisproblemnaturallyleadstothetechniqueofdomainsfirstintroducedbyD.Scott
inthe1970s.
ElPaso,USA VladikKreinovich
RioGrande,Brazil/Pamplona,Spain GraçalizPereiraDimuro
PortoAlegre,Brazil AntônioCarlosdaRochaCosta
Contents
1 MotivationandOutline ........................................ 1
1.1 WhyComputers? ......................................... 1
1.2 WhyIntervalComputations? ............................... 2
1.3 WhyGoBeyondIntervals? ................................ 3
1.4 Outline ................................................. 3
References .................................................... 4
2 AGeneralDescriptionofMeasuringDevices:Plan ............... 7
3 A General Description of Measuring Devices: First
Step—FiniteSetofPossibleOutcomes ........................... 9
3.1 Every Measuring Device Has Finitely Many Possible
Outcomes ............................................... 9
3.2 NotAllMarksonaScaleCanBePhysicallyPossible ......... 9
3.3 WeNeedaTheory ........................................ 10
3.4 WeNeedaTheorythatAlsoDescribedaMeasuringDevice .... 11
3.5 WeWantaTheorythatIs“Full”inSomeNaturalSense ....... 11
3.6 ASeeminglyNaturalDefinitionofaFullTheoryisNot
AlwaysAdequate ........................................ 12
3.7 WhatExactlyIsaTheory? ................................. 13
3.8 WhatKindofStatementsAreWeAllowing? ................. 13
3.9 WhatExactlyIsaFullTheory .............................. 14
3.10 The Existence of a Full Theory Makes the Set of All
PhysicallyPossibleOutcomesAlgorithmicallyListable ........ 15
3.11 Conclusion:AlgorithmicallyListableSetofPhysically
PossibleOutcomes ....................................... 16
3.12 Example1:IntervalUncertainty ............................ 17
3.13 Example2:Counting ..................................... 17
3.14 Example3:“Yes”–“No”Measurements ..................... 18
3.15 Example3a:Repeated“Yes”–“No”Measurements ............ 19
vii
viii Contents
3.16 Example 4: A Combination of Several Independent
MeasuringInstruments .................................... 20
References .................................................... 21
4 A General Description of Measuring Devices: Second
Step—PairsofCompatibleOutcomes ........................... 23
4.1 HowDoWeDescribeUncertainty:MainIdea ................ 23
4.2 CommentonQuantumMeasurements ....................... 23
4.3 SomePairsofOutcomesAreCompatible(Close),Some
AreNot ................................................. 24
4.4 The Existence of a Full Theory Makes the Set of All
CompatiblePairsofOutcomesAlgorithmicallyListable ....... 25
4.5 Conclusion:AlgorithmicallyListableSetofCompatible
PairsofOutcomes ........................................ 25
4.6 DescriptioninTermsofExistingMathematicalStructures ...... 26
4.7 Example1:IntervalUncertainty ............................ 26
4.8 Example2:Counting ..................................... 26
4.9 Example3:“Yes”–“No”Measurements ..................... 27
4.10 Example3a:Repeated“Yes”–“No”Measurements ............ 27
4.11 Example 4: A Combination of Several Independent
MeasuringInstruments .................................... 28
4.12 ComputationalComplexityoftheGraphRepresentation
ofaMeasuringDevice:GeneralCase ....................... 28
4.13 ComputationalComplexityoftheGraphRepresentation
of a Measuring Device: Case of the Simplest Interval
Uncertainty .............................................. 29
4.14 ComputationalComplexityoftheGraphRepresentation
of a Measuring Device: General Case of Interval
Uncertainty .............................................. 30
4.15 ComputationalComplexityoftheGraphRepresentation
of a Measuring Device: Lower Bound for the Case
oftheGeneralIntervalUncertainty ......................... 30
4.16 ComputationalcomplexityoftheGraphRepresentation
ofaMeasuringDevice:CaseofMulti-DUncertainty .......... 31
4.17 ComputationalComplexityoftheGraphRepresentation
of a Measuring Device: General Case of Localized
Uncertainty .............................................. 32
References .................................................... 33
5 A General Description of Measuring Devices: Third
Step—SubsetsofCompatibleOutcomes ......................... 35
5.1 FromPairstoSubsets ..................................... 35
5.2 IsInformationAboutCompatiblePairsSufficient? ............ 35
5.3 Information About Compatible Pairs Is Sufficient
ForIntervals ............................................. 36
Contents ix
5.4 InformationAboutCompatiblePairsisNotSufficient
intheGeneralCase ....................................... 36
5.5 TheExistenceofaFullTheoryMakestheFamilyofAll
CompatibleSetsofOutcomesAlgorithmicallyListable ........ 37
5.6 Conclusion: Algorithmically Listable Family
ofCompatibleSetsofOutcomes ............................ 38
5.7 DescriptioninTermsofExistingMathematicalStructures:
SimplicialComplexes ..................................... 38
5.8 ResultingGeometricRepresentationofaMeasuringDevice .... 39
5.9 TowardsDescriptioninTermsofExistingMathematical
Structures:Domains ...................................... 40
5.10 HowtoReformulatetheAboveDescriptionofaMeasuring
DeviceinTermsofDomains? .............................. 42
5.11 Example1:IntervalUncertainty ............................ 42
5.12 Example2:Counting ..................................... 43
5.13 Example3:“Yes”–“No”Measurements ..................... 43
5.14 Example 4: A Combination of Several Independent
MeasuringInstruments .................................... 44
5.15 ComputationalComplexityoftheSimplicialComplex
RepresentationofaMeasuringDevice:AGeneralCase ........ 45
5.16 ComputationalComplexityoftheSimplicialComplex
RepresentationofaMeasuringDevice:CaseofInterval
Uncertainty .............................................. 46
5.17 ComputationalComplexityoftheSimplicialComplex
RepresentationofaMeasuringDevice:CaseofMulti-D
Uncertainty .............................................. 46
5.18 ComputationalComplexityoftheSimplicialComplex
Representation of a Measuring Device: General Case
ofLocalizedUncertainty .................................. 47
References .................................................... 47
6 A General Description of Measuring Devices: Fourth
Step—ConditionalStatementsAboutPossibleOutcomes .......... 49
6.1 SubsetsofCompatibleOutcomesDoNotAlwaysGive
ACompleteDescriptionofaMeasuringDevice .............. 49
6.2 WhatWeDoWeNeedtoAddtotheSubsetsDescription
toCapturetheMissingInformationAboutaMeasuring
Device? ................................................. 50
6.3 The Existence of a Full Theory Makes the Set of All
TrueConditionalStatementsAlgorithmicallyListable:
AnArgument ............................................ 51
6.4 FamilyofConditionalStatements:NaturalProperties .......... 52
6.5 Conclusion: Algorithmically Listable Family
ofConditionalStatements ................................. 53
