Introduction
Basiccontent
Fuzzylogicreferstotheuncertaintyconceptjudgmentandreasoningthinkingmodethatimitatesthehumanbrain.Forthedescriptionsystemofunknownoruncertainmodel,Aswellascontrolobjectswithstrongnonlinearityandlargelag,usefuzzysetsandfuzzyrulesforreasoning,expresstransitionalboundariesorqualitativeknowledgeandexperience,simulatehumanbrainmethods,implementfuzzycomprehensivejudgments,andreasontosolveregularfuzzyinformationthatisdifficulttodealwithbyconventionalmethods.problem.Fuzzylogicisgoodatexpressingqualitativeknowledgeandexperiencewithunclearboundaries.Itusestheconceptofmembershipfunctiontodistinguishfuzzysets,processfuzzyrelationships,simulatethehumanbraintoimplementrule-basedreasoning,andsolvethevariousproblemscausedbythelogicalfailureofthe"lawofexcludedmiddle".Identifytheproblem.
Historicaldevelopment
In1965,theAmericanmathematicianL.ZadehfirstproposedtheconceptofFuzzySet,markingthebirthofFuzzyMathematics.Theoriginallogicandmathematicsbasedonbinarylogicaredifficulttodescribeanddealwithmanyvagueobjectsintherealworld.Fuzzymathematicsandfuzzylogicessentiallydescribeandprocessfuzzyobjectsaccurately.
Inordertoestablishamathematicalmodeloffuzzyobjects,L.Zadehextendedtheconceptofordinarysetsthatonlytakethebinaryvaluesof0and1totheconceptoffuzzysetsthattakeinfinitelymanyvaluesontheinterval[0,1].Andusetheconceptof"degreeofmembership"toaccuratelydescribetherelationshipbetweenelementsandfuzzysets.Preciselybecausefuzzysetsarebasedoncontinuousinfinitelymanyvalues,fuzzylogiccanberegardedasthescienceofusingfuzzysetsofinfinitecontinuousvaluestostudyfuzzyobjects.Somebasicconceptsandmethodsoffuzzymathematicsareappliedtothefieldoflogic,resultinginbasicconceptssuchasfuzzylogicvariablesandfuzzylogicfunctions.Correspondingcomparativeresearchisalsomadeonfuzzyconnectivesandfuzzytruthtables.Chadalsocarriedoutresearchonlikelihoodinferencesuchasfuzzyhypothesisinference,andsomeoftheresultshavebeendirectlyappliedtothedevelopmentoffuzzycontrollers.
Themainsignificanceofestablishingandresearchingfuzzylogicis:
(1)Usenewideasandnewtheoriessuchasfuzzylogicvariables,fuzzylogicfunctionsandlikelihoodinferencetofindsolutionstofuzzylogic.Thebreakthroughofsexualproblemslaidatheoreticalfoundationandpointedoutthedirectionforstudyingfuzzyobjectsfromalogicalpointofview.
(2)FuzzylogicisuniqueintheautomaticcontrolprocessthatisdifficulttodescribeandprocesswiththeoriginalBooleanalgebra,binarylogicandothermathematicsandlogictools,thediagnosisofdifficultdiseases,theresearchoflarge-scalesystems,etc.Place.
(3)Intermsofmethodology,itprovidescorrectresearchmethodsforhumanresearchfromaccuracytovagueness,andfromcertaintytouncertainty.Inaddition,inthebasicresearchofmathematics,fuzzylogiccanhelpsolvesomeparadoxes.Thestudyofdialecticallogicwillalsohaveaprofoundimpact.Ofcourse,fuzzylogictheoryitselfneedstobefurthersystematized,complete,andstandardized.
Basictheory
Fuzzylogicisatautologyofbinarylogic:inmulti-valuedlogic,givenanMV-algebraA,anA-evaluationiscalculatedfromthepropositionThesetofformulasinMV-algebraicfunctions.Ifthisfunctionmapsaformulato1(or0)forallA-evaluations,thentheformulaisanA-tautology.Therefore,forinfinite-valuedlogic(suchasfuzzylogicandVukasevichlogic),weset[0,1]tobethelowersetofAtoobtain[0,1]-evaluationand[0,1]-tautology(oftenIt'scalledevaluationandtautology).ChanginventedMV-algebratostudythemulti-valuedlogicthatPolishmathematicianJan?ukasiewicz(Janukasiewicz)intervenedin1920.Chang'scompletenesstheorem(1958,1959)statesthatanyMV-algebraequationthatholdsintheinterval[0,1]alsoholdsinallMV-algebras.Throughthistheorem,itisprovedthattheinfinite-valuedVukaseviclogiccanbedescribedbyMV-algebra.Laterthesameappliestofuzzylogic.ThisissimilartotheBooleanalgebraequationthatholdsin{0,1}andholdsinanyBooleanalgebra.Booleanalgebrathereforecharacterizesstandardtwo-valuedlogic.
Application
Fuzzylogiccanbeusedtocontrolhouseholdappliancessuchaswashingmachines(itsensestheloadanddetergentconcentrationandadjuststheirwashingcycleaccordingly)andairconditioners.
Basicapplicationscanbecharacterizedassubrangesofcontinuousvariables,oftentriangularortrapezoidalinshape.Forexample,thetemperaturemeasurementofananti-lockbrakecanhavemultipleindependentmembershipfunctions(membershipfunction)thatdefineaspecifictemperaturerangethatarerequiredtocorrectlycontrolthebrake.Eachfunctionmapsthesametemperaturetoatruevalueintherangeof0to1andisanon-concavefunction(otherwiseitmaybeclassifiedascolderifthetemperatureishigherinacertainpart).Thesetruevaluescanthenbeusedtodeterminehowthebrakesshouldbecontrolled.
InFigure1,cold,warm,andhotarefunctionsofthemappedtemperaturerange.Apointonthisscalehasthree"truthvalues"—oneforeachfunction.Forthespecifictemperatureshown,thesethreetruevaluescanbeinterpretedasdescribingthetemperatureas"quitecold","somewhatwarm"and"nothot".
Usually,trapezoidisused,buttheattributionfunctionoftriangleisusedforfuzzyregressionanalysis.
Fuzzylogic(4photos)
FuzzylogicusuallyusesIF/THENrules,orconstructsequivalentthingssuchasfuzzyincidencematrix.Theruleisusuallyexpressedinthefollowingform:
IFfuzzyvariableISfuzzysetTHENaction
Forexample,averysimpletemperatureregulatorusingafan:
IFtemperatureISverycoldTHENstopthefan
IFtemperatureIScoldTHENdecelerationfan
IFtemperatureISnormalTHENkeepthecurrentlevel
IFtemperatureISHotTHENaccelerationfan
Notethatthereisno"ELSE".Allrulesareevaluatedbecausethetemperaturecanbe"cold"and"normal"atthesametimetovaryingdegrees.
ThereareAND,OR,andNOToperatorsinBooleanlogicinfuzzylogic.Theyareusuallydefinedasminimum,maximum,andcomplement;whentheyaredefinedinthisway,theyarecalledZadehoperatorsbecausetheywerefirstdefinedinZadeh'soriginalpaper.Forfuzzyvariablesxandy:
NOTx=(1-truth(x))xANDy=minimum(truth(x),truth(y))xORy=maximum(truth(x),truth(y))canalsouseotheroperatorscalledhedgeswhichareclosertonaturallanguage.Generaladverbssuchas"very"or"alittle"canusemathematicalformulastomodifytheconnotationofaset.
Programminglanguage
Inapplication,theprogramminglanguageProLogisverysuitableforimplementingfuzzylogicduetoitsdatabasefacilitythatsetsup"rules"thatareinterrogatedbydeductivelogic.Thiskindofprogrammingiscalledlogicprogramming.
Researchobject
Toclarifytheresearchobjectoffuzzylogic,youmustfirstknowthelogicalresearchobject,becausefuzzylogicisonlyadevelopmentbasedonclassicallogic.Branchdiscipline.Aslongastheresearchobjectoflogicisclarified,thentheresearchobjectoffuzzylogicwillbeeasytounderstand.Sowhatexactlyistheresearchobjectoflogic?Therearevariousanswerstothisquestion."
Theobjectsoflogiccanbedividedintothefollowingviewpointsfromabroadperspective:
(1)Logicisthestudyofthinking;
(2)Logicisthestudyoftheobjectiveworld;
(3)Logicisthestudyoflanguage;
(4)Logicisthestudyofthevalidityoftheformofreasoning."
ThisisasummarymadebythefamousdomesticlogicscholarChenBo.Inthebook,ChenBoanalyzedtheabovefourviewpointsonebyone,andpointedouttheadvantagesanddisadvantagesofvariousviewpoints.Finally,heputforwardhisownview,hebelievedthattheresearchobjectoflogicisthevalidityofreasoningform.Thisviewisalsorecognizedinthefirstchapter"WhatisLogic"writtenbyLiXiaowuin"NineChaptersofPhilosophyofLogic"editedbyZhangQingyu.Inlayman'sterms:theobjectoflogicresearchisthecorrectnessofreasoning.Strictlyspeaking(moreacademically),theobjectoflogicresearchisthevalidityoftheformofreasoning.
Theviewthattheobjectoflogicresearchisthevalidityoftheformofreasoninghasbeenrecognizedbymostscholarsandexperts,andIhavenoobjectiontothisview.Afterclarifyingtheresearchobjectoflogic,IcanenterthequestionIwanttotalkabout.Whatistheresearchobjectoffuzzylogic?Here,Iwanttodiscussfromthefollowingaspects:
(1)Thebackgroundoffuzzylogic.Human'sunderstandingofnaturecanberoughlydividedintotwocategories.Oneisprecisephenomena,whichcanbedescribedinpreciselanguage.Forexample,2+2=4;GuiyangCityisthecapitalofGuizhouProvince;MoutaiisChina’snationalliquor,andsoon.Itcanbeseenthatthesephenomenaallhaveprecisedefinitionsandproperties.However,intherealworld,thereisanotherphenomenonthatisdifficulttoaccuratelydescribeanddefine.Forexample,Huaxiisabeautifulplace(whatexactlyisbeautifulscenery?):Hisfatherisatallman(howtallisatallman?);TeacherZhangisamiddle-agedperson(howoldisamiddle-agedpersondefinedas?)?),andmanymore.Therearecountlesssuchphenomena.Correspondingtothe"precisionphenomenon"wecallitthe"fuzzyphenomenon".Inordertouserigorousscientificmethodstostudyfuzzyphenomenaandanalyzefuzzyproperties,fuzzymathematicscameintobeing.Andfuzzylogicisoneofthebranchdisciplinesderivedfromfuzzymathematics.
(2)Theresearchobjectoffuzzylogic.Asmentionedearlier,theresearchobjectoflogicisthevalidityoftheformofreasoning,andwhenitcomestofuzzylogic,itsresearchobjectisthevalidityoffuzzyreasoning.Sowhatisfuzzyreasoning?Whatisthedifferenceandconnectionbetweenfuzzyreasoningandprecisereasoning?Theseissueswillbediscussedbelow.
First,let'stakealookatwhatfuzzyreasoningis.Likeexactreasoning,fuzzyreasoningisalsocomposedofbasiclogicalelementssuchasconceptsandjudgments,butfuzzyreasoninghasitsownuniquewayofreasoning.Theconclusionsderivedbyfuzzyreasoningarenotabsolutelytrueandfalse.Itsconclusionscanonlybedescribedbymembershipdegree.Forexample,theteacherZhanginthepreviousexampleisamiddle-agedperson.Thisisaverytypicalfuzzyjudgmentsentence.HerewejustTheabsolutetruthandfalsehoodintraditionallogiccan’tbeusedtodescribetheconceptofmiddle-agedpeople.Forexample,40-year-oldistrueformiddle-agedpeople.Isittruethat41-year-oldismiddle-agedandisregardedasfalse?BecauseinbinarylogicThereareonlytwoconclusions,trueandfalse.Forthispowerlessprobleminbinarylogicbutcanbeeasilysolvedinfuzzylogic,weuseChadnotationtodescribethiscase.Chadnotationexpressesalltheelementsinthefuzzysetthroughthesumoffractions.Anditsdegreeofmembership,wherethedenominatorrepresentstheelement,andthenumeratorrepresentsthedegreeofmembership.Intheaboveexample,wecanexpressitas(A)=(0.5/Mr.Zhang),whichmeansthatMr.Zhangisamiddle-agedpersonandonly0.5intermsofdegree.Hereweputasidetheabsolutetruthandfalsehood.However,thefuzzyphenomenonhasalsobeenaccuratelydescribed.Thereasonfortheaccuracyofthefuzzyphenomenonismainlyforthefuzzyreasoningtoberealizedonthemachine.
Secondly,discusstheeffectiveness.ChenBomadeamoreincisivesummaryofthevalidityofreasoningandputforwardfiverequirements.Hebelievesthatwhetherareasoningiseffectiveshouldmeetthefollowingfiveconditionsatthesametime:(1)Fidelity.(2)Contentrelevance.(3)Independence.(4)Subjectneutralityoruniversalapplicability.(5)Simplicity.AlthoughChenBoproposedsuchaframework,itisalmostimpossibleforanykindoflogicalreasoningtomeettheabovefivecriteriaatthesametime.HereIonlyexpresssomesimpleviewsontheeffectivenessoffuzzylogic.Thereasoningcommonlyusedinfuzzylogicincludesfuzzyhypotheticalreasoningandfuzzyconditionalreasoning.Amongthem,fuzzyhypotheticalreasoningisthemostrepresentative.Thedefinitionoffuzzyhypotheticalreasoningis:itisknownthatfuzzypropositionA(majorpremise)containsfuzzypropositionB.IfthereisafuzzypropositionA1(smallpremise)thatisnotexactlythesameasA,thenthecorrespondingconclusioncanbederived.Wecallthisreasoningprocessfuzzyhypotheticalreasoning.Forexample:
(1)Ifthefoodyoueatisrichinnutrients,yourbodywillbegood;thenifthefoodyoueatisrichinnutrients,whatwillyourbodybelike?
(2)IfChinawasverystronginthelateQingDynasty,itwouldnotbebulliedbytheimperialistcountries;thenifChinawasnotverystronginthelateQingDynasty,itwouldnotbebulliedbytheimperialistcountries?
DuetovaguehypothesesThelargeandsmallpremisesofreasoningarefuzzy,soitsconclusionsarealsofuzzy.Thisiscompletelydifferentfromtheaccuracyrequiredbytraditionallogic.Sohowshouldfuzzyreasoningbeaccuratelydescribedsothatitcanberecognizedbymachines?Wecandiscussitfromtwoaspects:humanexperienceandfuzzymathematics.
Thesignificanceofcreatingandresearchingfuzzylogic
(1)Usingnewideasandnewtheoriessuchasfuzzylogicvariables,fuzzylogicfunctions,andlikelihoodThebreakthroughlaidthetheoreticalfoundationandpointedoutthedirectionforthestudyoffuzzyobjectsfromthelogicalpointofview.
(2)FuzzylogicisuniqueintheautomaticcontrolprocessthatisdifficulttodescribeandprocesswiththeoriginalBooleanalgebra,binarylogicandothermathematicsandlogictools,thediagnosisofdifficultdiseases,theresearchoflarge-scalesystems,etc.Place.
(3)Intermsofmethodology,itprovidescorrectresearchmethodsforhumanresearchfromaccuracytovagueness,andfromcertaintytouncertainty.
Inaddition,inthebasicresearchofmathematics,fuzzylogichelpstosolvesomeparadoxes.Thestudyofdialecticallogicwillalsohaveaprofoundimpact.Ofcourse,fuzzylogictheoryitselfneedstobefurthersystematized,complete,andstandardized.
Otherexamples
Ifaperson’sheightis1.8meters,considerhimastall:
IFmaleIStrueANDheight>=1.8THENis_tallIStrue
IFmaleIStrueANDheight>=1.8THENis_shortISfalse
Buttheabovedefinitionisunrealistic.Therefore,underthefuzzyrules,thereisnoobviousdistinctionbetweentallandshort:
IFheight>=mediummaleTHENis_shortISagreesomehow
IFheight>=mediummaleTHENis_tallISagreesomehow
Inthecaseofblur,thereisnoheightlike1,83meters,onlytheblurvalue,suchasthefollowingassignment:
dwarfmale=[0,1.3]
msmallmale=(1.3,1.5)
mediummale=(1.5,1.8)
tallmale=(1.8,2.0)
giantmale>2.0mFortheconclusion,therearenotjusttwovalues,butfive:
agreenot=0
agreelittle=1
agreenot=0
agreelittle=1
p>agreesomehow=2
agreealot=3
agreefully=4
Inabinaryor"fragile"situation,theheightApersonwhois1.79metersmaybeconsideredshort.Iftheheightofanotherpersonis1.8metersor2.25meters,thesepeopleareconsideredtall.
ThisfragileexampleisdeliberatelydifferentfromthevagueExample.Wecan’tputinthepremise
IFmale>=agreesomehowAND...becausegenderisoftenconsideredtobebinaryinformation.Soit’snotascomplicatedasheight.