From Intervals to –? Towards a General Description of Validated Uncertainty PDF
Preview From Intervals to –? Towards a General Description of Validated Uncertainty
Studies 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 The series “Studies in Computational Intelligence” (SCI) publishes new developments and advances in the various areas of computational intelligence—quickly and with a high quality. The intent is to cover the theory, applications, and design methods of computational intelligence, as embedded in the fields of engineering, computer science, physics and life sciences, as well as themethodologiesbehindthem.Theseriescontainsmonographs,lecturenotesand editedvolumesincomputationalintelligencespanningtheareasofneuralnetworks, connectionist systems, genetic algorithms, evolutionary computation, artificial intelligence, cellular automata, self-organizing systems, soft computing, fuzzy systems,andhybridintelligentsystems.Ofparticularvaluetoboththecontributors and the readership are the short publication timeframe and the world-wide distribution,whichenablebothwideandrapiddisseminationofresearchoutput. IndexedbySCOPUS,DBLP,WTIFrankfurteG,zbMATH,SCImago. AllbookspublishedintheseriesaresubmittedforconsiderationinWebofScience. · · 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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2023 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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
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