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Studies in Computational Intelligence 1047 Vladik Kreinovich Towards Explainable Fuzzy AI: Concepts, Paradigms, Tools, and Techniques Studies in Computational Intelligence Volume 1047 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 Towards Explainable Fuzzy AI: Concepts, Paradigms, Tools, and Techniques VladikKreinovich DepartmentofComputerScience TheUniversityofTexasatElPaso ElPaso,TX,USA ISSN 1860-949X ISSN 1860-9503 (electronic) StudiesinComputationalIntelligence ISBN 978-3-031-09973-1 ISBN 978-3-031-09974-8 (eBook) https://doi.org/10.1007/978-3-031-09974-8 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface What this book is about and who is the intended audience. This book is an introduction to fuzzy approach to explainable AI. The intent is that the material shouldbeunderstandableevenforundergraduatestudents—and,ofcourse,graduate students,researchers,andpractitionerswillalsohopefullybenefitfromthismaterial. NeedforexplainableAI.WhatisexplainableAIandwhydoweneeditinthefirst place? ModernAItechniques—especiallydeeplearning—provide,inmanycases,very goodrecommendationswhereaself-drivingcarshouldgo,whethertogiveacompany aloan,etc.Theproblemisthatthesetechniquesarenot(yet)perfect. In some cases, the recommendations generated by an AI system are not good. Ofcourse,asthefamousMarilynMonroemoviesays,“Nobody’sperfect”.Human expertsarenotperfecteither.However,whenahumanexpert—beitabankingofficial oramedicaldoctor—makesarecommendation,heorshecan,ifasked,providean explanation.Ifyoufindtheexplanationnotsufficientlyconvincing,youcanaskfor someoneelse’sadvice. Unfortunately,recommendationsprovidedbyanAIsystem(suchasadeepneural network)usuallycomewithoutanexplanation.Sowecannotsoeasilyseparategood andbadadvice.ItisthereforedesirabletomakeAImoreexplainable. Why fuzzy techniques. Providing an explanation means finding natural language rules and ideas which are, in some reasonable sense, equivalent to the numerical resultsprovidedbytheAItools.Theproblemofconnectingnaturallanguagerules and numerical decisions is known since 1960s. Then, the need was recognized to incorporateexpertknowledgeintocontrolanddecision-making. Experts use imprecise words like “small”. For this incorporation, a special technique was invented—known as fuzzy techniques. This technique led to many successfulapplications.Itisthereforereasonabletousethesetechniquesindesigning explainableAI. What we study in this book. If we knew how to make AI explainable, teaching this class would be easier. We would just teach the corresponding algorithms and methods. v vi Preface Atpresent,explainableAIremainslargelyanultimategoal.Wedonotyetknow whichtoolswillworkbetter.So,insteadofstudyingspecifictools,itmakessenseto studythefoundationsforthesetools,sothatwewillknowwhyweneedtousethese tools,andwewillknowwhichtoolsarebetterinwhichsituations.Thiswillhelpus selectappropriatetoolsformakingcurrentAIapplicationsmoreexplainable. Firsttopic:Introductiontofuzzytechniques.Wewanttobetterunderstandhow fuzzy techniques can help with explainable AI. For this, we need to have a good understandingofthesetechniques.Wewilllearnthecorrespondingtechniquesand howtheyareusedincontrolandinotherapplications. Wewillalsotrytomakethesetechniquesthemselvesmoreexplainable.Namely wewillexplainthefirst-principlemotivationsforthesetechniques. Wewillstudyallthreemainstagesoffuzzytechniques: (cid:129) describingtheoriginalimprecisewordslike“small”innumericalterms, (cid:129) combiningthecorrespondingnumbers;todescribeBoolean(and-andor-)combi- nationsofthecorrespondingproperties,special“and”and“or”operationsareused forthis; (cid:129) “defuzzification”—transforming imprecise recommendations into a precise controlvalue. Second topic: Which version of fuzzy technique to select. In all three stages of fuzzy techniques, there are several options. Empirically, in different situations, differentoptionsworkbest.Thismakessense,sinceindifferentsituations,wehave different objectives. For example, if we launch a single drone to inspect an area, themainobjectiveistomaximizetheprobabilitythatitsmissionsucceeds.Onthe otherhand,ifwelaunchaswarmofdronestoinspectthesamearea,itisprobably OKifoneofthemdoesnotdomuch—aslongas,onaverage,theoverallmissionis successful. How do we select the best techniques? In some cases, we have finitely many parameters. So, we need to find the best values of these parameters. To find the largest and the smallest values of a function of several such variables, we can use calculus.(Donotworryifyouhaveforgottensomeofit,wewillrefresh). Inmanyothercases,however,weneedtoselectafunction—e.g.thebest“and”- and “or”-operations. There is a natural generalization of calculus that deals with such optimization problems. It is known as variational calculus, and it is actively usedincontrol.Wewilllearnthebasicsofthesetechniques.Asanexample,wewill usethistechniquetocomeupwithoptimal“and”-and“or”-operationsforthetwo above-describeddronesituations. Thirdtopic:Towardsexplainablemachinelearning.Theultimategoalistomake theresults ofmachinelearning(andotherAItechniques)explainable.Wearestill workingonthis. Meanwhile,animportanthelpwouldbetomakethemachinelearningtechniques themselvesexplainable.Atpresent,inmanycases,theonlyreasonweselectsome techniquesandsomeparametersofthesetechniquesisthatthesetechniquesempir- icallyworkwellonseveralproblems.Thisisnotasconvincingaswhenweprove Preface vii thatthesetechniquesare,insomereasonablesense,optimal.Wewillanalysedeep learningfromthisviewpoint. Finalwordbeforetheactualmaterialstarts:Enjoy! ElPaso,TX,USA VladikKreinovich December2021 Contents 1 WhyExplainableAI?WhyFuzzyExplainableAI?WhatIs Fuzzy? ........................................................ 1 1.1 WhyExplainableAI? ...................................... 1 1.2 Why Fuzzy Techniques Seem a Reasonable Approach forExplainableAI ......................................... 6 1.3 WhatIsFuzzyMethodology ................................ 8 1.4 SummaryofFuzzyMethodology ............................ 17 1.5 Exercises ................................................. 19 2 Defuzzification ................................................. 21 2.1 FormulationoftheProblem:Reminder ....................... 21 2.2 MainIdeaandtheResultingFormula ......................... 21 2.3 IntegralForm ............................................. 27 2.4 Important Comment: Centroid Defuzzification Is Not aPanacea ................................................ 29 2.5 Exercises ................................................. 32 2.6 Self-Test1 ............................................... 33 3 WhichFuzzyTechniques? ....................................... 35 3.1 WhatWeStudyinThisChapter ............................. 35 3.2 InterpolationShouldBeRobust ............................. 36 3.3 WhichInterpolationIstheMostRobust ...................... 39 3.4 “And”-and“Or”-OperationsMustBeRobustToo .............. 43 3.5 WhichIstheMostRobust“And”-Operation ................... 44 3.6 WhichIstheMostRobust“Or”-Operation .................... 47 3.7 GroupRobustnessVersusIndividualRobustness ............... 50 3.8 WhichInterpolationIstheMostIndividuallyRobust ........... 51 3.9 TheMostIndividuallyRobust“And”-Operation ............... 56 3.10 RobustnessVersusIndividualRobustness:Example ............ 58 3.11 TheMostIndividuallyRobust“Or”-Operation ................. 61 3.12 GeneralConclusion ........................................ 63 3.13 Exercises ................................................. 64 ix x Contents 4 SoHowCanWeDesignExplainableFuzzyAI:Ideas .............. 67 4.1 MachineLearningRevisited ................................ 67 4.2 Exercises ................................................. 71 4.3 Self-Test2 ............................................... 71 5 HowtoMakeMachineLearningItselfMoreExplainable .......... 73 5.1 How Can We Make Machine Learning Itself More Explainable:Idea .......................................... 73 5.2 SelectionofanActivationFunction .......................... 74 5.3 SelectionofPooling ....................................... 78 5.4 WhatAboutFuzzy? ....................................... 79 5.5 Exercises ................................................. 80 5.6 Self-Test3 ............................................... 80 6 FinalSelf-Test ................................................. 83 AppendixA:TermsUsedintheBook(inAlphabeticOrder) ........... 85 AppendixB: WhyDoWeNeed...?(inAlphabeticOrder) ............. 89 AppendixC:SolutionstoExercises ................................. 93 AppendixD:SolutionstoSelf-Tests ................................. 111 AppendixE: AdditionalReadings ................................... 129