Weighted logics for artificial intelligence : an introductory discussion

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rchive

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OULOUSE

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Dubois, Didier and Godo, Lluis and Prade, Henri

Weighted logics for artificial intelligence : an introductory discussion.

(2014) International Journal of Approximate Reasoning, vol. 55 (n° 9).

pp. 1819-1829. ISSN 0888-613X

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Weighted

logics

for

artificial

intelligence

–

an

introductory

discussion

Didier Dubois

a

,

Lluís Godo

b

,

∗

_,

_{Henri Prade}

a

a_IRIT,_Université_Paul_{Sabatier, 118}_route_de_Narbonne,₃₁₀₆₂_Toulouse_cedex_{9, France} b_IIIA_{- CSIC,}_Campus_de_la_UAB_s/n,₀₈₁₉₃_{Bellaterra, Spain}

a

b

s

t

r

a

c

t

Keywords: Weightedlogics Gradedtruth Uncertainty Preferences Similarity

Beforepresentingthecontentsofthespecialissue,weproposeastructuredintroductory overview ofalandscape ofthe weighted logics (in ageneralsense) that can be found intheArtificialIntelligenceliterature,highlightingtheirfundamentaldifferencesandtheir applicationareas.

1. Introduction

Inthelastdecades,andespeciallyinArtificialIntelligence(AI),therehasbeenanexplosionoflogicalformalismscapable ofdealingwithavarietyofreasoningtasksthatrequireanexplicitrepresentationofquantitativeorqualitativeweights asso-ciatedwithclassicalormodallogicalformulas(inoneformoranother).Thesemanticsoftheweightsrefertoalargevariety ofintendedmeanings:beliefdegrees,preferencedegrees,truthdegrees,trustdegrees,etc.Examplesofsuchweighted for-malismsincludeprobabilisticorpossibilisticuncertaintylogics,preferencelogics,fuzzydescriptionlogics,differentformsof weighted orfuzzylogicprogramsundervarioussemantics,weightedargumentationsystems,logicshandlinginconsistency withweights,logicsforgradedBDIagents,logicsoftrustandreputation,logicsforhandlinggradedemotions,etc.

The underlying logics range from fully compositional systems, like systems of many-valued or fuzzy logic, to non-compositional oneslikemodal-likeepistemiclogics forreasoningaboutuncertainty, probabilisticorpossibilisticlogics, or evensome combinationofthem.Sometimestheweights arenotexplicitandtheformalismsusetotalorpartialorderings instead.

Inthisshortpaperwepresentanintroductorydiscussionorganizingalandscapeofweightedlogics(inageneralsense) thatcanbefoundintheliterature,highlightingtheirdifferencesandapplicationareas.Inparticularweoverviewthemain approachesin AIthatdeal withgradednotions ofuncertainty,truth, preferencesandsimilarity, wediscusswhat arethe logicalissuesbehindthem,andfinallywealsopointoutnewemergingareasforgradedsettings.ThisisdoneinSections2 to4.

OuraimisnottoprovideafullandexhaustiveoverviewofgradedformalismsinAI,butratheraninformeddiscussion ofthemaingeneralissuesandapproaches,thatmayhelpthereaderpositionthepapersinthisspecialissue,whosecontent ispresentedinSection5.

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2. Typicalgradednotions

OneheavilyentrenchedtraditioninAI,especiallyinknowledgerepresentationandreasoningistorelyonBooleanlogic. However,manyepistemicnotionsincommonsensereasoningareperceivedasgradualratherthanall-or-nothing.Neglecting thisaspectmayleadtoinsufficientlyexpressiveframeworksandmaycausesomeconfusion.Suchnaturallygradual(inour opinion)notionsarereviewedbelow.

2.1. Gradedtruth

Truthis akey notion inthephilosophy ofknowledge thatis oftenviewedasBoolean inessence. Yetin thescope of informationstorageand management,this absoluteview becomesquestionable. Whenrepresenting knowledge,one may choose a language whose primitives are Boolean, but it is also possible to use one whose primitives are many-valued entities.Indeed,asclaimedquiteearlyby DeFinetti [29] commentingŁukasiewiczlogic,decidingthat apropositionisan entitythat canonlybetrueorfalseisamatter ofconvention,asitisa matterofchoosingtherangeofa (propositional) variable.Inthissense,truth isan onticnotion,asoneparticipatingto thedefinitionofaproposition. One maytake into accounttheideathatinsomecontextsthetruthofaproposition(understoodasitsconformitywithaprecisedescriptionof thestateofaffairs)isamatterofdegree.Forinstance,iftheheightofJohnisknownonemightconsiderthattheproposition “Johnistall”isnotalwaysmodeledasbeingjusttrueorfalse.Thisistheviewheldbyfuzzylogic[10].Howeverthisnotion ofgradedtruthisrestrictedtoan informationprocessingperspective andhasnothingto contributetothefoundationsof formalphilosophy.

Ifthe truth setcontains intermediary truth degrees, one issue iswhetheror not we can keepthe truth-functionality assumptionthatisthekeyfeatureofclassicallogic.Mathematically, theanswerisyesasdemonstratedby thelargesetof multiple-valuedlogics that arenowavailable. Howeverthereare alotofunresolvedissuesabout many-valuedlogicsand theirapplicationstoAIsuchas

•

Whyaretheresofewpapersusingmultiple-valuedlogicsasarepresentationofgradualpropertiesinAI?

•

Howtochooseamongthemanyavailabletruthfunctionalmany-valuedsystems?

•

Doestruth-functionalityalwaysmakesensefromanappliedknowledgerepresentationperspective?

•

Howdoesthenotionoftruth-functionalmany-valuedtrutharticulateswithstudiesofvagueness?

Asforthelastquestion, seee.g.[22].Ontheother hand,the mostpopularmany-valuedlogics inAIseemto bethose with3(Kleene[80]),4(Belnap[11])or5(equilibriumlogic[96])truthvalues,withaviewtohandleepistemicnotionssuch asignorance,contradiction,negationasfailureordefaultknowledge,followingalongtraditiondatingbacktoŁukasiewicz andKleene [57].However, thesystematicuseoftruth-tablesforbasicconnectivesintheseapproacheslooksquestionable assuchnotions haveastrong epistemicflavor closertoideasofuncertainty,whiletruth isan onticnotion.Forinstance, Kleene suggestedthat the thirdtruth value could mean “unknown”, andthisview has beentaken forgranted by many. However,“unknown”canbeopposedto“certainlytrue”and“certainlyfalse”,notto(ontologically)trueandfalse.Thishas ledtoconfusing debates thatcannot besolved withoutlettingtherepresentationofuncertaintyandmodalitiesenterthe picture[35].However,usualepistemiclogics withtheirrelationalsemanticslookfarawayfromKleene logic.Yet,asimple fragmentofa KDlogic,wheremodalitiesonlyappearinfront ofliterals,cancaptureKleene logicandotherthree-valued logicsofpartialknowledge[24].Thisrestrictionontheexpressivepower(inKleenelogic,itisnotpossibletorepresentthe factthat p

∨

q isunknown)isthepricepaidforthetruth-functionalityoflogicsoftheunknown.

2.2.Uncertainty

Uncertaintymodelingpertainstotherepresentationofanagent’sbeliefs.Thereareseveralsourcesofuncertainty:

•

Therandom variationofaclass ofrepeatableeventsleavesan agent unableto predicttheevent that corresponds to theoutcomeofthenexttrial.

•

Thesheerlackofinformationmaycauseanagentbeinguncertainabouttheanswertoaquestion.

•

Thesimultaneouspresenceofinconsistentpiecesofinformationduetotoomanysourcesmayequallypreventanagent fromassertingthetruthorthefalsityofastatement.

TherearetwotraditionsinAIforrepresentinguncertainty,thatstillneedtobereconciledtoalargeextent:

•

Thenon-gradedBooleantraditionof(monotonic)epistemiclogicsthatrelyonthemodalformalism,andincludesome exception-tolerantnon-monotoniclogics.

•

Thegradedtraditiontypicallyrelyingondegreesofprobability.Measuresofuncertaintyaimatformalizingthestrength ofourbeliefs inthe truth (occurrence)ofsome propositions (events)by assigning to thosepropositions adegree of belief[72].

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Whateverthetradition,itmustbestressedthatbeliefisahigherordernotionw.r.t.truth,thatis,astatementpertainingto beliefencapsulatesaBooleanpropositioninsideabeliefqualifier:thetruthdegreeofthestatement“Ibelieve p”doesnot refertothetruthofp,buttothebeliefitself:itisthedegreeofbeliefinp,irrespectivelyofp beingtrueornot.Fromthe mathematical pointofview,ameasure ofuncertaintyisafunction thatassigns toeach propositionorevent(understood hereasaformulainaspecificlogicallanguage)avaluefromagivenscale,usuallytherealunitinterval

[

0

,

1

]

,undersome suitableconstraints.Awell-knownexampleisgivenbyprobabilitymeasureswhichtrytocaptureourdegreeofconfidence intheoccurrenceofeventsbyadditive[0,1]-valuedassignments.Itcanbeshownthatsuchanuncertaintycalculuscannot be compositionalwithrespecttoalllogicalconnectiveswithoutleadingto someformoftrivialization oftheconstruction

[45].Forinstance,probabilitiesareonlycompositionalwithrespecttonegation.Incontrast,wehaveseenthatdegreesof truthcanbetruth-functionalandthisassumptionstillleadstosophisticatedlogicalsystems.

Itisnoteasy toreconciletheprobabilistic andthelogicalviewofbeliefs(see[27]foraspecialissueonthistopic).At themostelementarylevelasetofBooleanformulasisofteninterpretedasa beliefbasecontainingpropositionsan agent believes. IntheBayesian probabilistictradition,informationis representedbya singleprobability distributionon possible worldspossiblyencodedasaBayesnet.Thetwoapproachesareatodds,irrespectivelyofthechoiceofBooleanvs.gradual representationsofbelief:

•

Thelogicalapproachleavesroomforincompleteinformation.

•

The Bayesianapproach [97] seemsto be veryinformationdemanding asthelack ofbeliefina propositionisalways equatedwiththebeliefinitsnegation.

Ontheotherhand,anaturalgradedextensionoftheelementarylogicalapproachiscapturedbypossibilisticlogic[46,47]

wheredegreesofuncertaintymaybecapturedbymeansofameretotalorderingofpossibleworldsandsomepropositions canbemorebelievedthanothers.Thisplausibilityorderingcanbeencodedasanumericalpossibilitydistributionifneeded. Itpreservesthepropertythatifanagentbelievestwopropositions(toanydegree),thisagentalsobelievestheirconjunction. Thisproperty(typicalofepistemiclogic)isviolatedinprobabilitytheory.

The use ofa modallanguage enablesthe syntactic expression ofpartial ignorance, andexplicitpatterns ofreasoning fromit,butthesemanticsofepistemiclogicintermsofaccessibilityrelationsdoesnotfitwiththenumericaltraditionof representingbeliefs.However,specializingthissemanticstomereepistemicstatesandrestrictingitslanguage toepistemic sentences restoresthe linkwithuncertaintytheories [7]. Puttingtogether the probabilistic andsuch simplifiedepistemic logicapproachestobeliefleadstoreasoninginthesettingofimpreciseprobabilities[114]whichexplicitlyattachdegrees of belief anddegrees of plausibility to propositions [91]. Such degrees are graded versions of necessity and possibility modalities;besides,possibilitydistributionscanencodespecialcasesofconvexprobabilityfamilies[43].

Aniceby-product ofreconcilingprobabilityandepistemicorpossibilisticlogics istoofferalogical handlingof condi-tioning:pushingprobabilitydowntotheBooleancontext,indeFinetti style,leadstothethree-valuedlogicofconditional objects[44],thatisalsoasemanticsfornon-monotoniclogics;addingweightstothisconstructionbridgesthegapbetween logicalrepresentationsofbeliefandbothprobabilityandpossibilitytheories[13].

It remains theissue of choosinga proper scale for gradingbeliefs in a given applicationcontext, namelyhow much numerical isit useful to be? Aunified framework forreasoning about uncertainty leavesusthe choice betweenvarious gradedrepresentations:ordinal,qualitative(withafinitevaluescale),integer-based(aswithSpohnkappafunctions[106]) orreal-valued.

2.3. Preferences

Preferences are clearly not Boolean most of the time. Artificial Intelligence has developed a Boolean framework for decision-making problems based on constraint propagation and satisfaction. While many problems are amenable to a constraint-basedformulation,itmakeslittlesensetoignorethegradualnatureofpreferences.

The traditioninpreferencemodelinghasbeentouseeitherorder relations(total orpartial) ornumericalutility func-tions,albeitwithlittleattentiontotheissueofpreferencerepresentationinpractice.Onthecontrary,ArtificialIntelligence hasfocused oncompactlogicalorgraphicalrepresentationsofpreferencesonmulti-dimensional(often Booleandomains)

[33].In thissituation, an interpretation representsan optiondescribed by Booleanattributes. Forinstance,CP-nets have exploitedananalogywithBayesnetstodesignagraphicalstructureencodingordinalpreferenceswhenlocaldecision vari-ablesareBoolean.HoweverCP-netsarefarfromcapturingallpossibleorderingrelationsbetweenpossibleworldsandthe notionofpreferencehandledby CP-netsisall-or-nothing.Moregenerallogicallanguageswhere apreferencerelation be-tweenformulasappearsinthelanguagehavebeenused,thataremoreexpressive.Howeverthequestionofthemeaningof comparingtwologicalformulasintermsofitsconsequencesonapreferenceorderingbetweeninterpretationsisnot obvi-ous,andseveralproposalsexistthatareatoddswitheachother.Forinstancedowemeanthatallmodelsofthepreferred formulashouldbepreferredtoallmodelsoftheotherformula?Orjusttheirbestmodels?See,e.g.[76].

One alternative is to useweights attached toBoolean formulas. Such a weight mayreflect the imperativenessofthe satisfactionoftheassociatedproposition,thenviewedasagoaltoreach(aprioritizedconstraint).Whenthisweightisused as apenalty forthe interpretations that violate theformula, the weight can be viewedasa lower bound of anecessity measure[14]:thisisjustanotheruseofpossibilisticlogic.However,other approachesexist,e.g.,[52,20],wheretheweight

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isarewardwhensatisfyingtheformula,and[82]wheredesiresorpreferenceshaveautilitariansemantics.Moregenerally atthesemanticlevelonemayfocusontheleastpreferredinterpretationsthatviolatetheformulaoronthebestpreferred onesthat satisfyit.Oronthecontrary,aninterpretationmaybeconsideredallthebetter(resp.worse)asthesumofthe rewards(resp.penalties)attachedtoformulasitsatisfies(resp.violates)ishigher(resp.smaller).Analternativetotheuse ofweightsistheintroductionofapreferencerelationinsidetherepresentationlanguage,asin,e.g.,[113,109,15].

Intheabovepreferencerepresentationframework,neitherthepresenceofseveralcriterianorthepossibilityof uncer-taintyisconsidered.Inthiscase,theremaybetwokindsofweights:

•

Weightsexpressingpreferencesofoptionsoverotheroptions.

•

Weightsexpressingthelikelihoodofeventsorimportanceofgroupsofcriteria.

Inthe case ofdecisionunder uncertainty, one way out is to usetwo sets of formulas, one forthe knowledge baseand oneforthepreferencebase,andtoarticulatesome kindofinference techniqueexploitingbothbasesso astoencode the optimizationof a givencriterion mixing uncertaintyandutility [36]. Thisapproach looks more problematicformultiple criteriadecision-making, where value scales maynot be commensurate, andimportance weights pertain to yet another dimensionoftheevaluationprocess.Still,alogicalcounterparttoSugenointegrals,afamilyofqualitativemultiplecriteria aggregationoperators,hasbeenrecentlyobtained[49].

2.4.Similarity

The use of the idea of similarity in reasoning may refer two different points of view: either one doesnot want to distinguishbetweenobjectsthatare foundtobe similar,thus buildinga partition,oronewants totakeadvantage ofthe existenceofametricstructureonasetofobjectsandexploititinsomeway.Inthefirstcase,weperformagranulationof theuniverseofdiscourse,whileinthesecondcaseweareinterestedinextrapolationorinterpolation.

Similarityisoftenagradednotion,especiallywhenitisrelatedtotheideaofdistance.Itmayrefertoaphysicalspace asinspatialreasoning,ortoanabstractspaceusedfordescribingsituations,ase.g.,incase-basedreasoning.

Therepresentationofspatial relationsbetweenregions oftenreliesonfirstordertheoriesbasedon afamilyofpartial preordersbetweenregions,calledmereologies.Althoughtheterm“mereology”usually referstothe ideaofparthoodasa basicnotion,theideaofconnectionmaybealsousedasaprimitivenotion.Thereareeightbasicrelationsbetweenregions knownasRCCrelations (RCCstands for“RegionConnectionCalculus”) [25]. Onemayeven startwitha fuzzyconnection relation (which might be defined froma distance or a pointwisecloseness relation), and then define a “part of”, oran “overlap”fuzzyrelationbetweenregionsforinstance,andobtainagradedextensionoftheRCCcalculus[102].Modallogics are alsoused forrepresenting spatial information.Spatial interpretations of modalitieshave beenprovided forcapturing variousspatialconcepts qualitativelywithatopologicalorgeometric flavorsuchasnearnessordistance,forexample.See

[53]forareviewofthelogic-basedrepresentationsofmereotopologiesinclassicalormodallogics,andinfuzzyandrough setssettings,aswellasmodallogicrepresentationsofgeometries.

Roughsets [95] provide a formal setting forthe “granulation”of a universeof discourse partitionedinto equivalence classesofelements thatarefoundindistinguishable(because theyshareexactlythesamepropertiesamongtheonesthat areconsidered).Theycomefromtheideaofinformationsystemswheretheavailabledescriptionsofobjectsviaattributes arenot sufficienttodistinguish betweenthem (moreattributeswouldbe needed).Onlyupperandlowerapproximations ofsubsetsofobjectscan be defined.Such clusteringbetweenindistinguishableobjectsalso leadstoeliminate inessential attributes.Modallogicshavebeenproposedforreasoningwithlowerandupperapproximationsofsetsofmodelsinsucha setting[56,32].Clearlytheequivalencerelationmaybeturnedintoafuzzyrelation,givingbirthtogradedroughsetcalculi

[42,99].Inthatcasethepartitionbecomesafuzzycoveringofthesetofobjects.Thereisto-datealargetechnicalliterature puttingroughsets,fuzzysetsandfuzzysimilarityrelationstogether.

Extrapolationandinterpolationreasoningarebasedontheideaofclosenessbetweeninterpretations.Thus,forinstance, ifthesetofmodelsofa proposition p failsto beincludedinthe setofmodelsofa propositionq,butremains included in the set of interpretations that are close to models of q, one may say that p

→

q is “close to being true”. This has beenadvocated by different authors [101,79] (andcontrasts withnon-monotonic reasoningwhere one requires that the preferred/normalmodelsofp beincludedinthemodelsofq).Thecloseness(orproximity)relationbetweenthe interpreta-tionsgivesbirthtoagradedconsequencerelation,whichisthebasisforalogicofsimilaritydedicatedtointerpolation[37], andcapturesfuzzy logic-basedapproximate reasoning.In asimilar spirit, a logicallowing toreasonabout thesimilarity withrespecttospecificsetsofprototypeshasbeenrecentlyproposed[111].

Intheaboveapproaches,similarityisgraded.Morequalitativeapproacheshavebeenproposed,usingcomparative rela-tions.Thelogic

CSL

[105]isbasedonamodalbinaryoperatorthatisusedfordenotingthesetofinterpretationsthatare closerto p thantoq.Theunderlyingdistance-basedsemanticscan berestatedintermsofpreferentialstructuresusinga ternaryrelationexpressingthat“allthepointsinaregionz areatleastasclosetoregionx thantoregion y”[2].

Anotherqualitative approach,withoutanygrade,reliesatthesemanticlevelontheconceptual spacesframework[63]

wherethepossibilityofexpressingspatial-likelocalizationsuch as“beinginbetween”(bymeansofaternaryrelation),or “parallelism” (bymeans ofa quaternaryrelation) providesa basis forcapturinginterpolative andextrapolative reasoning

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respectively [103]. Thisproposalcomes closeto logicalreasoning withanalogical proportions [98],which are quaternary statementsoftheform“a istob asc istod”(involvingBoolean,orgraded,properties),butremainsmorecautious.

Lastly,letusalsomentionthatapartfromreasoningabout similarity,whatmaybetermedreasoningwith similarityhas been alsoproposed as asemantic basis innon-monotonic reasoning[65],information updating (which relieson ternary comparativeclosenessrelation)[78],orindistance-basedinformationfusion,where,e.g.,Hammingdistancesarecomputed withrespecttosetofinterpretations[81].Anotherviewofinformationfusion,recentlyproposedin[104],reliesontheidea thatinconsistencycanbeoftenresolvedbyenlargingthesetsofmodelsoftheinformationtobefused,thankstosimilarity relations.

3. Logicalissues

Inthissectionouraimistolaybarethemaincontrastingfeaturesoflogicalformalismsdealingwithgradeduncertainty andgradedtruth.

3.1. Uncertaintysemantics

Letusassume anagent istoreasonabout whattheworldisandassumethateach ofthepossiblestatesoftheworld isdescribedby acompleteBooleantruthevaluationofagiven(finite)setofatomicpropositionsVar.Solet

Ω

denotethe set of Booleantruth-evaluations w

: L

→ {

0

,

1

}

of formulasfrom a propositional language

L

builtfrom thefinite set of variablesVar andwiththeusualconnectives

∧

,

∨

,

→

and

¬

.Itiswellknownthattheconnectives

∧

,

∨

and

¬

endow

L

, modulologicalequivalence,withastructureofaBooleanalgebra.

Westartbyconsideringthecasewheretheagenthascompleteinformationabouttheworld,soheknowsthattheactual worldsisw0

∈ Ω

.Inthiscasethereisnouncertaintyatall,infact,knowingwhattheworldis,theagentisabletoascertain

thetruthstatusofeverypossibleproposition.Thiscorrespondstoconsidertheagent’sepistemicstateasbeingrepresented bythepair

(Ω,

E

)

where

Ω

reflectstheagent’slanguage(thatgoverntheprecisioninthedescriptionofpossiblestatesof theworld)andE

= {

w0

}

isasingleton.

A firstformofuncertainty appearswhen theagent’sinformationonly allowshim toknowforcertain that theactual world w0 is in some given subset E

⊆ Ω

. This is the typical case where the agent has a theory T (a set of formulas)

describing whatheknows about theworld. Theepistemic state oftheagent isthen representedas

(Ω,

E

)

where E isa non-emptysubsetofinterpretations,indeedthesetofmodelsofT ,andtheagentisonlyabletodeterminethetruthstatus ofsomepropositions,butnotforsomeothers.Indeed,aproposition

ϕ

isknowntobetrue ifE

|H

ϕ

(orequiv.T

⊢

ϕ

),itis knowntobefalse whenE

|H ¬

ϕ

anditisunknown otherwise,i.e.whenboth E

6|H

ϕ

and E

6|H ¬

ϕ

.Thereforeinthissetting, propositionsmaybeinthreedifferentepistemicstatuses.

A morerefined representationiswhen theagent mayassociateweights to interpretationsrelatedto thelikelihood of each interpretation ofdescribing the actual world. Inthis case, theepistemic state can be represented asa pair

(Ω,

µ

)

, where

µ

: Ω → [

0

,

1

]

attachesaweighttoeach possibleworld.Intheprobabilistic model(see e.g.earlyworksbyNilsson

[94]),

µ

=

p isaprobabilitydistributionon

Ω

,hencerequiring

P{

p

(

w

)

|

w

∈ Ω}

=

1,thatallowstorankthelikelihoodof anypropositionofbeingtrue accordingtoits probability measure P

(

ϕ

)

=

P{

p

(

w

)

|

w

∈ Ω,

w

|H

ϕ

}

.When p

(

w

)

∈ {

0

,

1

}

forall w

∈ Ω

,thentheepistemicstatereducestoasingletonE

= {

w0

}

where w0 istheuniqueworldsuchthatp

(

w0

)

=

1.

In the possibilistic setting[39],

µ

=

π

is a possibility distribution, where

π

(

w

)

=

1 means that it is totally possible (plausible)that w

=

w0,

π

(

w

)

=

0 meansthat w

=

w0 istotallydisbelieved. Thenpropositionsareweightedaccordingto

thecorrespondingconjugatenecessityandpossibilitymeasures:N

(

ϕ

)

=

1

−

min

{

π

(

w

)

|

w

|H ¬

ϕ

}

,and

Π (

ϕ

)

=

max

{

π

(

w

)

|

w

|H

ϕ

}

. Again, when

π

(

w

)

∈ {

0

,

1

}

for all w

∈ Ω

,then the epistemic state definedby

π

corresponds to thepreviously consideredthree-valuedsettingwith E

= {

w

|

π

(

w

)

=

1

}

.Othermeasures canalsobeused,liketheguaranteedpossibility

1(

ϕ

)

=

min

{

π

(

w

)

|

w

|H

ϕ

}

,whichisequalto1wheneverallmodelsof

ϕ

areconsideredplausible.Thecasewhen

1(

ϕ

)

>

Π (¬

ϕ

)

expressesaformofstrongbeliefin

ϕ

[8].

More generally, one mayconsider gradedepistemic states ofthe form

(Ω,

µ

)

, where

µ

:

2Ω

_{→ [}

₀

_,

₁

_]

_is_a _monotonic

set-function, in such a way that every proposition

ϕ

can be attached a likelihood orbelief degree of beingtrue asthe evaluationw.r.t.

µ

oftheset ofmodels of

ϕ

,i.e. of

µ

({

w

∈ Ω |

w

(

ϕ

)

=

1

})

.Thisrepresentationgeneralizes theprevious probabilistic orpossibilisticmodels, toother moregeneraloneslike forinstance thosedefinedby belieffunctions, upper andlower probabilitiesorimpreciseprobabilities(seee.g. [70,71]forageneralapproachencompassing manyuncertainty models ina modallogic setting). The information cannot then be reduced to a distribution over interpretations. In the qualitative setting, a monotonic set-function can always be viewed asthe lower bound of a set of possibility measures (

µ

(

A

)

=

minn_i₌₁

Π

i

(

A

)

)whichcomesdowntorepresentingepistemicstatesexpressedby

µ

asasetofpossibility

distribu-tionsover

Ω

(e.g.correspondingtoseveralagents)[48].

3.2. Syntacticformatsforuncertainty

From a syntacticalpoint ofview, anumber offormalismscoping withgradeduncertainty havebeenproposed in the literature. Mostofthemuseamodality, eitherexplicitorimplicit,referring togradedbelief.Forillustrationpurposes,we

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mentiontwokindsoflanguages:thosebasedon Booleanuncertaintystatementsrestrictingdegreesofuncertaintyof for-mulasintoagivensubset;andthoseexpressinggradualuncertaintystatementusingmany-valuedpredicatesormodalities.1 An instance of the first approach is Halpern’s probability logic [71], where formulas express constraints among the probabilitiesof

ϕ

1

,

. . . ,

ϕ

kaslinearinequalitiesoftheforma1P

(

ϕ

1

)

+ ...

+

akP

(

ϕ

k

)

≥

b,witha1

,

. . . ,

ak

,

b beingrealnumbers

andwhereP

(

ϕ

i

)

standsforprobabilityof

ϕ

i.Thesamelanguageisusedtoreasonaboutotherkindsofuncertaintymeasures

likeplausibilitymeasures,belieffunctions, etc.,seealso [115]fora relatedgeneralapproachtoreasonunderuncertainty usinglinearconstraints.IntheSerbianschool(Markovi ´c,Ognjanovi ´candcolleagues)onprobabilitylogics[34,93,100],they usegradedmodalitiesoftheformP≥a

ϕ

,declaringthattheprobabilityof

ϕ

isatleasta,andthentheseprobabilisticatoms

arecombinedby meansofclassical connectives.Asimilarapproach isLukasiewicz’sprobabilistic logic[89] (see also[28]

foralogicallowing toexpressindependence),wheretheauthorusesexpressions oftheform

(

ϕ

)[

l

,

u

]

todenote thatthe probability of

ϕ

lies inthe interval

[

l

,

u

]

, as well as van der Hoek andMeyer’s approach [110] on graded probabilistic modalities. Likewise, in Dubois–Prade’s possibilistic logic [39,46,47], formulas are pairs

(

ϕ

,

α

)

, stating that the necessity

N

(

ϕ

)

of

ϕ

isatleast

α

.Recently,thislanguagehasalsobeengeneralizedtodealwithBooleancombinationsofpossibilistic atomsN≥α

ϕ

,inasimilarwaytothepreviousprobabilisticlogiclanguage[50].

AdifferentapproachbyHájekandcolleagues [68,67,64]hasalsobeenproposed,wherethemodality B usedtodenote gradedbelief (probability,necessity, etc.)is graded itself,that is,even if

ϕ

is Boolean,the atomicmodal expression B

ϕ

, read as“

ϕ

is believed”, is graded innature (

ϕ

can be moreor less believed, probable, necessary, etc.). In thisway, the truth-degree of B

ϕ

can be taken as, e.g., the probability (or necessity) degree of

ϕ

, and then these graded atoms are combinedusingtherulesofasuitablefuzzylogic.

3.3.Logicsofgradedtruth

Indeed,formalismsthatcopewithgradedtruth(fuzzylogics)radicallydepartfromtheformalismsforuncertainty rea-soning[45,38].Givingup the bivalenceprinciple andadoptingatruth scale withintermediate degreesbetween0(false) and1(true)(usuallytheunitrealinterval

[

0

,

1

]

) leadtoanumberofmany-valuedtruth-functionalsystemswith connec-tivesextendingtheclassicalones.Themostrelevantexamplesofformalfuzzylogicsystemsaretheso-calledt-normbased fuzzylogics [67].These correspondto logicalcalculi withthereal interval

[

0

,

1

]

asset oftruth-valuesand definedby a conjunction& andan implication

→

interpretedrespectively by acontinuoust-norm

∗

andits residuum

⇒

,andwhere negationisdefinedas

¬

ϕ

=

ϕ

→

0,with0 beingthetruth-constantforfalsity.Inthisframework,eachcontinuoust-norm

∗

uniquelydeterminesasemantical(propositional)calculusPC

(∗)

overformulasdefinedintheusualwayfromacountable setofpropositionalvariables,connectives& and

→

andtruth-constant 0 (furtherconnectives,like

ϕ

∧ ψ

as

ϕ

&

(

ϕ

→ ψ )

, can also be defined). Evaluationsof propositional variables are mappings e assigning to each propositional variable p a

truth-valuee

(

p

)

∈ [

0

,

1

]

,whichextendunivocallytocompoundformulasasfollows:

e

(

ϕ

∧ ψ ) =

min

¡

e

(

ϕ

),

e

(ψ )

¢

e

(

ϕ

&ψ )

=

e

(

ϕ

)

∗

e

(ψ )

e

(

ϕ

→ ψ ) =

e

(

ϕ

)

⇒

e

(ψ )

Aformula

ϕ

isasaidtobea1-tautologyofPC

(∗)

ife

(

ϕ

)

=

1 foreachevaluatione.Letthesetofall1-tautologiesofPC

(∗)

be denotedasTAUT

(∗)

. The mainaxiomatic systemsof fuzzylogic,like Łukasiewiczlogic (Ł),Gödel logic(G) orProduct logic(

Π

),syntacticallycapturedifferentsetsoftautologiesTAUT

(∗)

,accordingtothechoiceofthet-norm

∗

,seee.g.[69, 67,66,23].Indeedonealwayshasresultsoftheform:

ϕ

is provable in the logic Ł iff

ϕ

∈

TAUT

(∗

Ł

)

(1)

where

∗

Łcanbechosenasx

∗

Ły

=

max

(

0

,

x

+

y

−

1

)

(ŁukasiewiczlogicŁ),x

∗

Gy

=

min

(

x

,

y

)

(GödellogicG)andx

∗

Πy

=

x

·

y

(Productlogic

Π

).

Itisworth noticingthat,incontrastwithclassical logic,thealgebraic structuresofthesetofformulasmodulo logical equivalenceinthesesystemsoffuzzylogicarenolongerBooleanalgebrasbutweakerstructureslikeMV-algebras,prelinear Heyting algebras, etc.Strictly speaking,a formulain a manyvaluedlogic isinterpreted asa function fromtheCartesian productofcopiesofthetruth-set tothetruthset.Eachmany-valuedlogiccapturesaclassoffunctionsattheonticlevel, andthereisnoepistemicingredientinsuchlogics,whichmayexplainwhytheyarenotmuchusedinArtificialIntelligence. Also,inall thesesystems,theimplicationcapturesthe truth-ordering,sinceif

⇒

istheresiduum ofaleft-continuous t-norm

∗

(i.e., x

⇒

y

=

max

{

z

∈ [

0

,

1

]

|

x

∗

z

≤

y

}

), then x

⇒

y

=

1 iff x

≤

y, andhence e

(

ϕ

→ ψ )

=

1 iff e

(

ϕ

)

≤

e

(ψ )

. Therefore,aformula

ϕ

→ ψ

actuallyrepresentsthat

ψ

isatleastastrueas

ϕ

,thisistosay,t-normbasedfuzzylogicsare logicsofcomparative truth.Toexplicitlydealwithtruth-degreesinthereasoning,onemayintroducetruth-constantsr,e.g. forallrationalvaluesinr

∈ [

0

,

1

]

.Thenaformular

→

ϕ

expressesthatthetruth-degree of

ϕ

isatleastr (seee.g.[54]),

1 _There_is_a_third_approach_using_statements_like_“_ϕ_is_more_likely_than_ψ_”,_that_relies_on_conditional_logics_in_the_style_of_Lewis_[83]_,_that_takes_another pointofviewonlogicsofuncertainty.

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whichsuggeststhatsuchmany-valuedlogics,castattheepistemiclevel,mayaccountforweightedlogicsofuncertaintyof thepreviouskind(usinggradedmodalities).

From an epistemic point of view, in a fuzzy logic setting, the states of the world are described by complete

[

0

,

1

]

-evaluationsofatomicformulas.Let

Ω

′ bethesetoftheseevaluations,i.e.

Ω

′

= {

w

|

w

:

Var

→ [

0

,

1

]}

.Notethat the set

Ω

ofBoolean interpretations isindeeda subsetof

Ω

′.Because oftruth-functionality,a completelyinformed scenario thuscorrespondsnowtoaprecisemany-valuedtruth-valueassignment w0

∈ Ω

′toallpropositionalvariables.Analogously

to the Boolean case, differentkinds of incompleteinformation about the world translateshere to differentmany-valued generalizations ofepistemic states.In generalthey are ofthe form

(Ω

′

,

µ

′

)

,where

µ

′

_{: [}

₀

_,

₁

_]

Ω′

→ [

0

,

1

]

isa generalized uncertaintymeasureoverfuzzysetsofinterpretations(seee.g.[59,58,60]forsomelogicalformalismsthatareabletodeal withuncertaintyovernon-Boolean(fuzzy)events).

4. Newareasforgradedsettingsandconclusionofthisoverview

Beyondthehandlingoftheideasoftruth,uncertainty,preference, andsimilaritythatmayrequiregradedsettingsfora properhandling,thefieldofdeonticreasoningisanotherareawhereoneencountersbasicnotionswhosestrengthsmaybe usefully compared,stratifiedintolayers, orevengraded: onemaythinkofgradedobligations,orpermissionsforinstance

[84,30]. One may also distinguish betweenweak and strong permissions [77]. Priorities, or preference relations among worlds,aimingatorderingworldsfromthemostidealonestotheleastidealones,mayhelpsolvingdilemmas[21].

Besides,thereareseveraldomainsofactiveresearchinArtificialIntelligencenowadayswhereseveralofthebasicnotions mentionedabovecanbeencountered,possiblycombinedwithothernotionsthatalsoneedtobegraded.Letusreviewthem briefly.

•

So-called BDIagents [18] aresupposed to havebeliefsabout the world, desires, fromwhichthey elicitateintentions that are feasibledesires. Clearly beliefsmaybe pervaded withuncertainty,desiresmay bemodeled ascollections of goals withdifferentprioritylevels, feasibilitymay bea matter ofcost,leading to ordered/graded intentions,see,e.g.

[19].Themodeling ofdesiresintermsofguaranteed possibilitymeasures,alreadyatwork intheprevious reference, hasbeenfurtherdiscussedandinvestigatedin[40,41].

•

Modelingthetrustthatcanbeassociatedwithinformationsourcesoragents,aswellasrelatednotionssuchasdistrust orreputation, isan importantissueinpractice. Manyproposalsexist –modalornumerical –wheregradedness has been introduced invarious ways[85,88,31,112,12].Still, there isnot yet afully clearview ofhow graded trust may relatetobeliefsanduncertainty.

•

Argumentationisanotherarea wheretheidea ofstrengthseems naturallyassociatedwitharguments[5],asrecently investigated [51,61,26,6,74]. However, it is likely that a uniform view of strength is not applicable here, since the strengthofanargumentmayrefertotheuncertaintypervadingthepiecesofinformationonwhichitisbased[4,73,3]

andonthereliabilityofthesource(s)oftheargument,aswellasontherhetoricformoftheargument(e.g.itslength); moreovertheargumentitselfmayrefertoagradualviewoftruth(whenstatingforinstancethat“thehigherthefever the more certain thechild should remain inbed”). The handlingof such strengths mayalso depend on thekind of problemtowhichargumentationisapplied:persuasion,negotiation,ordeliberation,forinstance.

•

Emotions,such assurprise, fearetc. havebeenrecentlymodeledbymeans ofdefinitionsinmodallogic[1].Itseems natural here again, to have these modalities complemented withgrades, leading to hybrid notions that might be a compoundinvolvingmorebasicnotionssuchasuncertainty,preference,orsimilarity[107,87,86].

•

In weightedsocial networks, thedegreesofconnection betweenactorsmaybe described ina gradedlogical setting, suchasmany-valuedmodallogics[55].

In thisintroductionwe havebriefly discussed severalissues inthe mainapproachesat workto representandreason withfundamentalnotionsinAI,wheretheuseofdegreesappearstobenatural,suchastruth,uncertainty,preferences,or similarity (butalso trust,permission, obligation, desires, etc.).Specifically, the basicdifference betweengradedtruth and other gradednotionshasbeenhighlighted.Whiletheformerimpliesachangeattheonticlevel(fromtwotruthvaluesto multipletruth-values),withoutanyreferencetoepistemicknowledgeorignorance,thelatterrelatestointensionalnotions that(usually)applytoBooleanpropositions,liketheirepistemic(belief)status,orhowtheycomparetootherpropositions in terms of preference, utility, similarity, etc. This is reflected on the kind of formal models that support thesegraded notions, many-valued truth-functionalmodels in theformer case, Kripke-likemodels andgraded modalities inthe latter case.

Actually,many-valuedlogics havebeenseriously criticizedatthephilosophicallevelbecauseoftheconfusionbetween truth-valuesontheonehandanddegreesofbelief,orvariousformsofincompleteinformation,ontheother hand,a con-fusionthatevengoesbacktopioneersincludingŁukasiewicz(e.g.,theideaofpossible asathirdtruth-value).Actually,due to thisissueandthenumericalflavoroffuzzylogic,there isalongtradition ofmutualdistrust betweenArtificial Intelli-genceandfuzzylogic.Therearetwowaystofillthisgap.Oneistousefuzzylogicstoencodegradedepistemicmodalities that canaccountforvariousuncertaintytheories;thesegradedmodalitiesmaybeappliedtoBooleanformulas,yieldinga two-tiered logic.The other way isto show how reasoningabout knowledge anduncertainty can alsobe definedon top of fuzzy/gradual propositionsby augmenting fuzzylogic withepistemic modalities.Recent worksalong this linemaybe

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consideredasfirststepstowardsareconciliationbetweenpossibilitytheoryandothertheoriesofbeliefaswellwithfuzzy logic(inthesenseofarigoroussymbolicsettingtoreasonaboutgradualnotions),seee.g.[16,59].

Finally,wewouldlike tomentionthat, althoughwe havemainlyfocused onlogical issues,manyoftheconcerns dis-cussedherehavealsoechoesincloselyrelatedareaslikedescriptionlogics,logicprogrammingandanswersetprogramming whentheycometohandleuncertainty,preferencesorfuzziness,seeforinstance[90,108,62,75,92,9,17]foravarietyoflogic programmingapproachescopingwithgradeduncertaintyand/ortruth.

5. Contentsofthespecialissue

This specialissue organized by the last two authors of this introduction,originates in an international workshop on Weighted Logics for Artificial Intelligence (WL4AI’12), co-located withthe 20th European Conference on Artificial Intel-ligence, andheld in Montpellier (France) on August 28, 2012. It includes expanded versions of papers presentedat the workshopaswell aspaperssubmitted in response toa call forpapers onthe workshoptopic publishedinthisjournal. Allacceptedcontributionshavelargelybenefitedofuseful reviewers’commentswhichhelpedtheauthorsimprovingtheir paperssubstantiallythroughseveralroundsofrevision.

The topicsof these papersillustrate several ofthe many concerns just surveyed. Indeed the special issuegathers 13 contributionswhichcovera broadvarietyoftopics:probabilistic logics,three-valuedlogics,descriptionlogics, answerset programming,argumentation,deonticnorms,causality,andinformationfusion.

Thetwo first papersdeals withprobabilistic logics. Thefirst one “Hierarchiesofprobabilistic logics”by N.Ikodinovi ć, Z. Ognjanovi ć,A.Perovi ć,andM.Raskovi ć,studieslogicswithprobabilityoperatorsexpressingthattheprobabilityofa for-mulaisatleasts (wheres isarationalnumber),orthattheprobabilityisinasetF rangingoverdifferentrecursivefamilies ofrecursivesubsetsofthesetofrationalsintheunitinterval,leadingtodistinctlogicsintermsofexpressivity.Thesecond paper“Conditional p-adicprobability logic”by A.Ili ćStepi ć,Z.Ognjanovi ć,N.Ikodinovi ć,presentsa proof-theoretical ap-proachto p-adicvaluedconditionalprobabilisticlogicsthatextendclassicalpropositionallogicwithconditionalprobability operatorsandwhereformulasareinterpretedinKripke-likemodelsthatarebasedon p-adicprobabilityspaces.

The third paper, “Borderline vs. unknown – Comparingthree-valued representations of imperfect information”, deals withother basicconcerns. Theirauthors, D. Ciucci,D. Dubois andJ. Lawrycomparethe expressivepower ofelementary representationformatsforvague,incompleteorconflictinginformation,namelyBooleanvaluationpairs,orthopairsofsets ofvariables,Booleanpossibilityandnecessitymeasures,three-valuedvaluations,andsupervaluations,makingexplicittheir connections with strong Kleene logic and with Belnaplogic of conflicting information. The paper aims at clarifying the similaritiesanddifferencesbetweentruth-functionalandnon-truth-functionalapproaches.

Thethree nextpapers dealwithdescriptionlogics, afamilyofknowledge representationformalismsforhandling tax-onomies.M.Cerami,A.García-CerdañaandF.Estevadefinefuzzydescriptionlogicsonthebasisoffirst-ordermany-valued fuzzylogics.Theirpaper“Onfinitelyvaluedfuzzydescriptionlogics”isbasedonthefuzzylogicofafiniteBL-chaininthe senseofHájek.Thepaperaddressesthehierarchyoffuzzyattributivelanguagesanddiscussesreasoningtasksand compu-tationalcomplexity.Thesecondpaper“Consistencyreasoninginlattice-basedfuzzydescriptionlogics”byS.Borgwardtand R. Peñalozaisalso about fuzzydescription logics withasemantics based oncomplete DeMorganlatticesandone basic reasoningtask, namely deciding whetheran ontology representing aknowledge domain is consistent.The paperfocuses onthefuzzydescriptionlogicL-

SHI

,providingatableaux-basedalgorithmfordecidingconsistencywhentheunderlying lattice isfinite,andidentifies decidable andundecidableclasses offuzzydescription logics overinfinite lattices. Thelast descriptionlogicpaper“Completion-basedgeneralizationinferencesforthedescriptionlogic

E LOR

withsubjective proba-bilities”byA.Ecke,R.PeñalozaandA.-Y.TurhanstudiesthelogicProb-

E LOR

thatextends

E LOR

withprobabilities,and presentsacompletion-basedalgorithmforpolynomialtimereasoninginarestrictedversionofProb-

E LOR

.Thisalgorithm isextendedinordertoprovidetoolsforbuildingontologies automaticallyfromexamples bycomputingthemostspecific concept that generalizes a givenindividual into a concept description, andthe leastcommon subsumer that generalizes severalconceptdescriptionsintoone.Thefeasibilityoftheapproachisdemonstratedempiricallyonaprototype.

Thepaper“ComplexityoffuzzyanswersetprogrammingunderŁukasiewiczsemantics”,by M.Blondeel,S.Schockaert, D. VermeirandM.DeCock,illustratesanotheractiveareaofresearch,namelyanswersetprogramming(ASP),here gener-alizedtofuzzyASPinwhichpropositionsmaybegraded.The paperanalyzesthecomputational complexityoffuzzyASP underŁukasiewiczsemantics,andshowsthatthecomplexityofthemainreasoningtasksislocatedatthefirstlevelofthe polynomialhierarchy,evenfordisjunctiveFASPprogramsforwhichreasoningisclassicallylocatedatthesecondlevel,and areductionfromreasoningwithsuchfuzzyASPprogramstobilevellinearprogrammingisestablished.

The two next papers deal with argumentationsystems, another hot topicin artificial intelligence nowadays.In their paper “Valued preference-based instantiation of argumentation frameworks with varied strength defeats”, S. Kaci and Ch. Labreuche consider an extension of Dung’s abstract argumentation framework where the degree to which an argu-mentdefeats anotherargumentcanbe graded,andinstantiateitby apreference-basedargumentationframework witha valuedpreferencerelation(which maybe derived froma weightfunction). Theyshow undersome conditionsthat there areless situationsinwhichadefensebetweenargumentsholdswithavaluedpreferencerelationcomparedtoa Boolean preferencerelation.Thepaper“Postulatesforlogic-basedargumentationsystems”,byL.Amgoud,focusesonargumentation systemsthatarebasedondeductivelogicsandthatuseDung’ssemantics.Itproposesanddiscussesfiverationality postu-latesthatsuchsystemsshould satisfy:consistencyandclosureunderthe consequenceoperator oftheunderlyinglogicof

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thesetofconclusionsofargumentsofeachextension,closureundersub-argumentsandexhaustivenessoftheextensions, andafreeprecedencepostulateensuringthatthefreeformulasoftheknowledgebase(i.e.,theonesthatarenotinvolved ininconsistency)areconclusionsofargumentsineveryextension.Postulatesforweightedargumentationsystemsarefinally addressed.

The paper “Reasoning about norms under uncertaintyin dynamic environments”by N. Criado, E.Argente, P. Noriega, and V. Botti proposes a multi-context graded BDI architecture that models norm-autonomous agents able to deal with uncertainty in dynamic environments. A degree of saliency is attachedto norms, and beliefs,desires andintentions are graded.Thisarchitecturehasbeenexperimentallyevaluatedonafire-rescuescenarioandresultsarereportedinthepaper. In“Aweightedcausaltheoryforacquiringandutilizingopen knowledge”,J.JiandX.Chenintroduceaweighted coun-terparttoMcCainandTurner’snon-monotoniccausaltheories,theactionmodelofarobot/agentbeingspecifiedasacausal theory.Theauthorssolveanabduction problemforfindingthemostsuitablesetofcausalrulesfromopen-source knowl-edgeresources,sothata planforaccomplishingtherobot taskcanbecomputedusingtheactionmodelandtheacquired knowledge. Theweightsassociated tohypothetical causalrulesare usedforcomparingcompetingexplanationswhich in-ducecausalmodelssatisfyingthetask.Thisisillustratedwithaservicerobotexample.

The lasttwo papersdiscusstoolsforinformationfusion.In “Sum-basedweighted beliefbasemerging: from commen-surable to incommensurable framework”, S. Benferhat, S. Lagrue, and J. Rossitprovide a postulate-based analysis ofthe sum-based merging operator when sources tomerge are incommensurable(i.e., they do notshare the samemeaning of uncertaintyscales).Theyshowthatthisoperatorcanbecharacterizedeitherintermsofaninfinitesetofcompatiblescales, orbyaParetoorderingonasetofpropositionallogicinterpretations.Finally,T.NakamaandE.Ruspini,in“Combining de-pendentevidentialbodiesthatsharecommonknowledge”,establishaformulaforcombiningdependentbodiesofevidence that are conditionally independentgiven their shared knowledge. Theyshow that theDempster–Shafer formulaand the conjunctiveruleofSmets’TransferableBeliefModelcanberecoveredasspecialcasesoftheircombinationformula.

Hopefullythisspecialissue,by providinganoverviewofweightedlogics inAIandgatheringarichsampling ofworks illustratingtheinterestofintroducinggradesinmanyknowledgerepresentationandreasoningproblems,willcontributeto abetterunderstandingoftheseissues,andtotheproperuseofthedifferentpossibleapproaches.

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