Fuzzy logic

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.Zadehextendedtheconceptofordinarysetsthatonlytakethebinaryvalues​​of0and1totheconceptoffuzzysetsthattakeinfinitelymanyvalues​​ontheinterval[0,1].Andusetheconceptof"degreeofmembership"toaccuratelydescribetherelationshipbetweenelementsandfuzzysets.Preciselybecausefuzzysetsarebasedoncontinuousinfinitelymanyvalues,fuzzylogiccanberegardedasthescienceofusingfuzzysetsofinfinitecontinuousvalues​​tostudyfuzzyobjects.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).Thesetruevalues​​canthenbeusedtodeterminehowthebrakesshouldbecontrolled.

InFigure1,cold,warm,andhotarefunctionsofthemappedtemperaturerange.Apointonthisscalehasthree"truthvalues"—oneforeachfunction.Forthespecifictemperatureshown,thesethreetruevalues​​canbeinterpretedasdescribingthetemperatureas"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.

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