Emerging Dimension Weights in a Conceptual Spaces Model of Concept Combination PDF
Preview Emerging Dimension Weights in a Conceptual Spaces Model of Concept Combination
Emerging Dimension Weights in a Conceptual Spaces Model of Concept Combination MarthaLewis and JonathanLawry1 Abstract. Weinvestigatethegenerationofnewconceptsfromcom- agents.Finally,section5discussesourresultsandgivesanindication binationsofpropertiesasanartificiallanguagedevelops.Todoso, offuturework. wehavedevelopedanewframeworkforconjunctiveconceptcombi- nation.Thisframeworkgivesasemanticgroundingtotheweighted sumapproachtoconceptcombinationseenintheliterature.Weim- 2 BACKGROUND plementtheframeworkinamulti-agentsimulationoflanguageevo- lution and show that shared combination weights emerge. The ex- We model concepts within the label semantics framework [7, 8], pectedvalueandthevarianceoftheseweightsacrossagentsmaybe combined with prototype theory [12] and the conceptual spaces predictedfromthedistributionofelementsintheconceptualspace, modelofconcepts[2].Prototypetheoryoffersanalternativetothe 6 asdeterminedbytheunderlyingenvironment,togetherwiththerate classical theory of concepts, basing categorization on proximity to 1 0 atwhichagentsadoptothers’concepts.Whenthisrateissmaller,the a prototype. This approach is based on experimental results where 2 agentsareabletoconvergetoweightswithlowervariance.However, human subjects were found to view membership in a concept as a thetimetakentoconvergetoasteadystatedistributionofweightsis matterofdegree,withsomeobjectshavinghighermembershipthan n longer. others[12].Fuzzysettheory[15],inwhichanobjectxhasagraded a J membership µL(x) in a concept L, was proposed as a formalism forprototypetheory.However,numerousobjectionstoitssuitability 5 1 INTRODUCTION havebeenmade[4,3,5,11,13]. 2 Humansareskilledatmakingsenseofnovelcombinationsofcon- Conceptualspacestheoryrendersconceptsasconvexregionsofa ] cepts,sotocreateartificiallanguagesforimplementationinAIsys- conceptualspace-ageometricalstructurewithqualitydimensions AI tems,wemustmodelthisability.Standardapproachestocombining andadistancemetric.Examplesare:theRGBcolourcube,pictured concepts, e.g. fuzzy set theory, have been shown to be inadequate infigure1;physicaldimensionsofheight,breadthanddepth;orthe . s [11]. Concepts formed through the combination of properties fre- tastetetrahedron.Sinceconceptsareconvexregionsofsuchspaces, c quentlyhave‘emergentattributes’[3]whichcannotbeexplicatedby thecentroidofsucharegioncannaturallybeviewedastheprototype [ decomposingthelabelintoitsconstituentparts.Wehavedevelopeda oftheconcept. 1 modelofconceptcombinationwithinthelabelsemanticsframework v asgivenin[8,10].Themodelisinspiredbyandreflectsresultsin 3 [3],inwhichmembershipinacompoundconceptcanberenderedas 6 theweightedsumofmembershipsinindividualconcepts,however, 7 itcanalsoaccountforemergentattributes,wheree.g.importancein 6 aconjunctiveconceptisgreaterthantheimportanceofanattribute 0 intheconstituentconcepts.Weimplementasimpleversionofthis . 1 modelinamulti-agentsimulationoflanguageusers,andshowthat 0 theagentsconvergetosharedcombinationweights,allowingeffec- 6 tivecommunication.Theseweightsaredeterminedbythedistribu- 1 tionofobjectsintheagents’conceptualspace.Thisprovidesathe- : v oreticalgroundingtotheproposalseenintheliterature[2,3,6,16] i thatcomplexconceptscanbecharacterisedasweightedsumsofat- Figure1: The RGB cube represents colours in three dimensions of X tributes. Further, it relates the weights in the combined concept to Red,GreenandBlue.Acolourconceptsuchas‘purple’canberep- r theexternalworld.Inthispaper,wefirstlysummarisethetheoretical resentedasaregionofthisconceptualspace. a frameworkweuse(section2),andinsection3giveabriefaccount ofourmodelofconceptcombination.Insection4,weimplementa Label semantics proposes that agents use a set of labels LA = simple version of ourmodel in a multi-agent simulation ofa com- {L ,...,L }todescribeaconceptualspaceΩwithdistancemetric munityoflanguageusersinordertoexaminewhetheranhowsuch 1 n d(x,x(cid:48)).LabelsL areassociatedwithprototypesP ⊆ Ωandun- acommunityisabletoconvergetosharedcombinationweights.We i i certainthresholdsε ,drawnfromprobabilitydistributionsδ .The givesimulationmethodsandresults,andanalysethebehaviourofthe i εi thresholdε capturesthenotionthatanelementx∈Ωissufficiently i 1 University of Bristol, England, email: [email protected], close to Pi to be labelled Li. The membership of an object x in a [email protected] conceptLiisquantifiedbyµLi(x),givenby (cid:90) ∞ (cid:126)yα =(1,1,1) µ (x)=P(d(x,P )≤ε )= δ (ε )dε Li i i εi i i d(x,Pi) Labels can then be described as L =<P ,d(x,x(cid:48)),δ >. We i i εi illustratethisideainfigure2. λ 3 b a x εi λ2 2 P i λ 1 Figure4:Prototypeforα=L1∧L2∧L3andweighteddimensions inbinaryspace. x 1 Theattributes‘haswings’and‘hasfeathers’shouldbegivenmore importancethan‘flies’.Thisisbecausevariousspeciesofbirdsdo Figure 2: Prototype-threshold representation of a concept Li. The not fly, so this attribute may be relaxed whilst still allowing some- conceptual space has dimensions x and x . The concept has pro- 1 2 thingtobecategorisedasabird.Theeffectthishasistocreateellip- totype P and threshold ε . The uncertainty about the threshold is i i ticalregionsoftheconceptualspace,asillustratedinfigure5. representedbythedottedline.Elementaintheconceptualspaceis withinthethreshold,soitisappropriatetoassert‘aisL ’.Element i bisoutsidethethreshold,soitisnotappropriatetoassert‘bisLi’. (cid:126)yα =(1,1) 3 AHIERARCHICALMODELOFCONCEPT λ 2 COMPOSITION Experimentsinthepsychologicalliteratureproposethathumancon- ceptcombinationcaninmanycasesbemodelledasaweightedsum of attributes such as ‘has feathers’, ‘has a beak’ (for the concept ‘Bird’) [3]. These attributes differ from quality dimensions in con- λ 1 ceptual spaces: they tend to be binary, complex, and multidimen- Figure5: Prototype for α = L1 ∧L2 and weighted dimensions in sionalinthemselves.Wethereforevieweachattributeasalabelin atwodimensionalbinaryspace,togetherwiththethresholdεforα. acontinuousconceptualspaceΩi andcombinetheselabelsinabi- Thepoint(0,1),indicatedbyanopencircle,canbeconsideredtobe naryconceptualspace{0,1}n illustratedinfigure3.Inthisbinary aninstanceoftheconceptforwhichy istheprototype. conceptualspace,aconjunctionofsuchlabelsα=(cid:86)n ±L maps α i=1 i toabinaryvector(cid:126)y takingvalue1forpositivelabelsL and0for α i Thedistancemetricbetweentwovectors(cid:126)y,(cid:126)y(cid:48)inthebinaryspace negatedlabels¬L (figure4). i {0,1}n is written H ((cid:126)y,(cid:126)y(cid:48)) and defined as a weighted city block (cid:126)λ metric. Then,undercertainconstraints,membershipinthecombinedcon- {0,1}n ceptα=(cid:86)n ±L isequaltotheweightedsumofmembershipin i=1 i eachoftheL .Thisisstatedinthefollowingtheorem: i Theorem1(CompoundConcepts). LetL beattributesdescribed i bymembershipfunctionsµ (x )inconceptualspacesΩ .Letαbe aconjunctionofsuchattribLuitesi(ortheirnegation):α=(cid:86)in ±L . i=1 i If we combine these labels in a binary space {0,1}n with α˜ =< (cid:126)y ,H ,U(0,λ )>whereλ = (cid:80)n λ ,thenwemaycalculate α (cid:126)λ T T i=1 i themembershipinαinthespaceΩ ×...×Ω by: Ω1 Ω2 Ωn 1 n n µ ((cid:126)x)=(cid:88) λi µ (x ) Figure3:Combininglabelsinabinaryspace α λT ±Li i i=1 where(cid:126)x∈Ω ×...×Ω ,x ∈Ω . Wetreatmembershipinαinthebinaryconceptualspacewithin 1 n i i thelabelsemanticsframework.Soαisdescribedinthebinaryspace Wemayfurthercombinesuchcompoundconceptsθ,ϕinahigher byα˜ =<(cid:126)yα,d((cid:126)y,(cid:126)y(cid:48)),δ>asbefore,andµα(y)=(cid:82)d∞(y,yα)δε(ε)dε. levelbinaryspace.Then,againundercertainconstraints,θ˜•ϕ˜can Sincethedimensionsofthespaceareweighted,someareconsidered beexpressedinthecontinuousspaceasaweightedsumofθandϕ. more important than others. The presence or absence of some at- tributesmayberelaxed.Forexample,‘Bird’mightbecharacterised Theorem2(ConjunctionofCompoundConcepts). Letθ˜•ϕ˜ =< by(amongothers)theattributes‘hasfeathers’,‘haswings’,‘flies’. {(1,1)},H ,δ>, where θ and ϕ are compound concepts as de- w(cid:126) scribedintheorem1,sothatθischaracterisedbymembershipfunc- x∈Ω ×Ω =[0,1]2.Thespeakeragentassertsoneof 1 2 tionµ ((cid:126)x)=(cid:80)n λθi µ (x )for(cid:126)x∈Ω ×...×Ω ,x ∈Ω , andϕθissimilarlyi=c1haλrθaTcte±riLsied.iThen 1 n i i α1 =L1∧L2 α =L ∧¬L 2 1 2 µ ((cid:126)x)=(cid:88)n (w1λϕTλθi +w2λθTλϕi)µ ((cid:126)x) α3 =¬L1∧L2 θ˜•ϕ˜ i=1 wTλθTλϕT ±Li α4 =¬L1∧¬L2 where(cid:126)x∈Ω ×...×Ω ,x ∈Ω . 1 n i i where the membership in the compound concept is determined by Theseresultsshowthatundertheconstraintε∼U(0,λ ),com- T theweightedsumofthemembershipsintheconstituentconcepts. bininglabelsinaweightedbinaryconceptualspaceleadsnaturally tothecreationofcompoundconceptsasweightedsumsofindividual µ (x)=λµ (x )+(1−λ)µ (x ) α1 L1 1 L2 2 labels,reflectingresultsin[3].Animportantaspectoftheseresults µ (x)=λµ (x )+(1−λ)(1−µ (x )) isthatnon-compositionalphenomenaareseen,suchasemergentat- α2 L1 1 L2 2 µ (x)=λ(1−µ (x ))+(1−λ)µ (x ) tributes.Theseareattributesthatbecomemoreimportantinthecon- α3 L1 1 L2 2 junctionoftwoconceptsthanineitheroftheconstituentconcepts. µ (x)=λ(1−µ (x ))+(1−λ)(1−µ (x )) α4 L1 1 L2 2 A specific example seen in [3] is that the attribute‘talks’ becomes moreimportantintheconjunction‘BirdsthatarePets’thanineither ‘Birds’or‘Pets’.Relaxingtheconstraintε∼U(0,λT)allowsusto Theconceptαi assertedisthatforwhichµαi(x)ismaximal.Note account for phenomena such as emergent attributes and overexten- thatthisimpliesthatifα assertedthenµ (x)≥0.5. i αi sionofconcepts.Inthecurrentpaper,however,weconcentrateona simpleweightedsumcombinationandexaminethepropertiesofthis 4.1.2 Updatingalgorithm typeofcombinationinmulti-agentsimulations. Wecomparetwodifferentupdatingalgorithms.Thefirstimplements theideathatthelisteneragentupdatesitsconceptswhentheagent’s 4 CONVERGENCEOFDIMENSIONWEIGHTS belief in the appropriateness of a compound label, µ (x), is less ACROSSAPOPULATION αi thanthereliabilityofthespeakerasmeasuredbyaweightw∈[0,1]. Soifµ (x)≤wthelisteneragentupdatesitslabelset. Weimplementasimpleversionofthehierarchicalmodelofconcept αi The update consists in moving the dimension weight λ towards combination in a multi-agent simulation of agents playing a series thevalueAwhichsatisfiesµ (x)=w,where oflanguagegames,similartothoseusedin[14].Weinvestigatehow θ agentsusingcompoundconceptsαinaconceptualspaceconvergeto w−µ (x ) A= L2 2 asharedweightingoftheconstituentconceptsLi.Agentsdoindeed µ±L1(x1)−µ±L2(x2) converge to a shared weighting, which is dependent on the distri- bution of objects in the environment and also on the rate at which However,itispossibleforA < 0orA > 1,inwhichcasesweset theymovetowardsotheragents’concepts.Section4.1describesthe A=0orA=1,givingus: methodsandsimulationset-up.Section4.2givessimulationresults, andsection4.3givesananalysisoftheresults. 1 if µ±L1w(−x1µ)L−2µ(±x2L)2(x2) >1 A= 0 if w−µL2(x2) <0 4.1 Methods w−µL2(x2) othµe±rwL1is(xe1)−µ±L2(x2) µ±L1(x1)−µ±L2(x2) Consider agents each with labels L1 =<P1,d(x,y),δ1>∈ Ω1, Thelisteneragentupdatestheweightλtoλ(cid:48)by: L =<P ,d(x,y),δ >∈ Ω . We assume that agents combine 2 2 2 2 thesetwolabelsasinsection3-i.e.,inabinaryspace{0,1}2 with λ(cid:48) =λ+h(A−λ) weight vector (cid:126)λ = (λ ,λ )(cid:62) and where the threshold in the bi- 1 2 Wemeasuretheconvergencebetweenλ acrosstheagentsasthe naryspaceεhasdistributionδ = U(0,λ ),whereλ = λ +λ . i T T 1 2 standarddeviation(SD)oftheλ . Then,membershipintheconjunctionL ∧L inthespaceΩ ×Ω i 1 2 1 2 We also introduce a second updating algorithm. This is similar may be calculated as a weighted sum of membership in the indi- tothefirstwiththeexceptionthatthecriterionforupdatingisthat vidualspaces.W.l.o.g.weassumethatλ = 1.Wethereforehave T µ ((cid:126)x)(cid:54)=w.Thismeansthatifanagentwithlowreliabilitymakes µL1∧L2((cid:126)x)=λµL1(x1)+(1−λ)µL2(x2). anαiassertion,thelisteneragentshiftsthedimensionweightλtore- To investigate how these weights are to be determined, we run ducetheappropriatenessoftheassertionα . simulationsinwhichagentswithequallabelsbutrandomlyinitiated i weightsengageinaseriesofdialoguesaboutelementsintheconcep- tualspace,adjustingtheirweightsaftereachdialogueiscompleted. 4.1.3 SimulationDetails AgentshavelabelsL =L =<1,d,U[0,1]>,wheredisEuclidean 1 2 4.1.1 Assertionalgorithm distance,todescribetheconceptualspaceΩ=Ω1×Ω2 =[0,1]2.In thiscase,wehaveµ (x)=x .Simulationsarerunwith10agents Li i AgentsareequippedwithsharedlabelsL ∈Ω =[0,1]andL ∈ for2,000timestepswithincrementh = 10−3 unlessotherwisein- 1 1 2 Ω =[0,1],andaweightλ.Ateachtimestepagentsarepairedinto dicated.Ateachtimestep,eachagenttalkstoeveryotheragent,ina 2 speakerandlisteneragents.Eachpairofagentsisshownanelement randomisedorder.Simulationresultsareaveragedacross25runs. Parametersvariedarethedistributionofelementswithinthecon- Thefinalvalueofλmayalsobedependentonthereliabilityofthe ceptualspaceandthereliabilityoftheagents.So,forexample,one agentsasparameterisedbyw.Forsomedistributions,thevalueofw simulation might include elements sampled uniformly across the doesnotaffectthevalueofλtowhichagentsconverge.Inothers,the whole conceptual space, whereas another might include elements valueofwalterstheweightingλ.Thisisillustratedinfigure7. sampled uniformly from one half of the space. This difference in distributionleadstodifferencesinthecombinationweights. 1 SD λ 4.2 SimulationResults 0.8 4.2.1 Updatingmodel1:µ ((cid:126)x)<w αi 0.6 Within this updating model, the listener agent updates only when µ ((cid:126)x)<w.Wefindthatwhentheagentreliabilityw>0.5,agents αi 0.4 areabletoconvergetoasharedweightλ.Theweighttowhichagents convergeisdependentonthedistributionofobjectsintheenviron- ment,andthereliabilitywoftheagents.Whenw=1forallagents, 0.2 andx ∼U[0,1],x ∼U[0,0.5],sothatelementsareencountered 1 2 withinhalfofthetotalspaceΩ = [0,1]2,theagentsconvergetoa valueofλ=0.5,asseeninfigure??. 00. 5 0.6 0.7 0.8 0.9 1 ze w Whenthedistributionofelementsintheenvironmentischanged, (a)x ∼ U[0,1],x ∼ U[0,0.5].Valuesof λ may converge to a different value. For example, changing the 1 2 wvaryasindicated. distribution of the elements in the space to x ∼ U[0.25,0.75], 1 x ∼ U[0,0.5],withw = 1forallagents,resultsinafinalvalue 1 2 SD ofλ=0.25,illustratedinfigure6. λ 0.8 1 SD Mean λ 0.6 0.8 0.4 0.6 0.2 0.4 0 0.2 0.5 0.6 0.7 0.8 0.9 1 w (b) x ∼ U[0.25,0.75], x ∼ U[0,0.5]. 0 1 2 0 500 1000 1500 2000 Valuesofwvaryasindicated. Time Figure7:SDandmeanλattimet=2000fordifferentvaluesofw. (a)SDandmeanλovertime. Forx ∼ U[0,1],x ∼ U[0,0.5],providedthatw > 0.5,agents 1 2 1 converge to λ = 0.5 (LHS). In contrast, for x ∼ U[0.25,0.75], 1 x ∼U[0,0.5],agentsconvergetovaryingvaluesdependingonthe 2 0.8 valueofw(RHS). 0.6 λ 4.2.2 Updatingmodel2:µαi((cid:126)x)(cid:54)=w 0.4 We also introduce a second updating model, in which the listener agentupdateswheneverµ ((cid:126)x)(cid:54)=w.Althoughthismodelisslightly αi 0.2 lessrealistic,itismoreamenabletoanalysis,andsoweusethisas astartingpointfortheanalysisofthesesystems.Again,eachsetof simulationsisrun25timesandresultsareaveraged.Figure8com- 0 0 500 1000 1500 2000 parestheresultsofsimulationsrunusingthetwodifferentassertion Time models,wherex ∼U[0.25,0.75],x ∼U[0,0.5]. 1 2 (b)Behaviourofindividualλiovertime(results When the updating condition requires that µαi((cid:126)x) (cid:54)= w, agents fromjustonesimulation). convergetoasharedweightλevenwhentheirreliabilityw ≤ 0.5. However,thevaluesofλtowhichagentsconvergearenotthesame Figure6:x1 ∼U[0.25,0.75],x2 ∼U[0,0.5],w=1forallagents. asthoseconvergedtounderthefirstassertionalgorithm. anotherα wouldbemaximal).Wecanthereforesplituptheconcep- 1 j µ (x) < w tualspaceintoquadrantscorrespondingtowhereeachα isasserted, α i µ (x) = w displayed in figure 9. Each quadrant where αi is asserted is called 0.8 α R . i 1 0.6 D S 0.8 R R 3 1 0.4 0.6 x 2 0.2 0.4 R4 R2 0 0.2 0.5 0.6 0.7 0.8 0.9 1 w 0 (a)SDattimet = 2000fordifferentvaluesof 0 0.2 0.4 0.6 0.8 1 wunderthetwodifferentupdatingalgorithms. x 1 Figure9:Eachassertionαiismade1when(cid:126)x=(x1,x2)fallsinRi µ (x) < w α µ (x) = w Toinvestigatethevaluetowhichtheλ converge,welookatthe 0.8 α i quantity (A−λ). If (A−λ) is positive, λ will increase, and if it isnegative,λwilldecrease.Wethereforelookatthecircumstances λ0.6 thatleadtoλincreasingordecreasing.Wedothisonacasebycase an basisdependingonwhichassertionisbeingmade. e M 0.4 Case:α isasserted 1 0.2 Whenα isasserted,µ (x ) = x > 0.5,µ (x ) = x > 0.5. 1 L1 1 1 L2 2 2 WewishtoknowwhenA>λ 0 0.5 0.6 0.7 0.8 0.9 1 w−µ (x ) w−x w A= L2 2 = 2 ≥λ µ (x )−µ (x ) x −x (b)Meanλattimet = 2000fordifferentval- L1 1 L2 2 1 2 uesofwunderthetwodifferentupdatingalgo- λisupdatedwhenµ =λx +(1−λ)x <w,i.e.when rithms. α1 1 2 w−x ≥λ(x −x ) Figure8:x1 ∼ U[0.25,0.75],x2 ∼ U[0,0.5],valuesofw varyas 2 1 2 indicated.Thepopulationofagentsconvergestodifferentvaluesof Thenif(x −x ) > 0thenA ≥ λ,andtheupdateisapositive 1 2 λdependingontheupdatingalgorithmused. increment.Otherwisethereverseholds,i.e.A≤λandtheupdateis anegativeincrement. These results show that the combination weights that the agents convergetoaredependentonfirstlythedistributionofelementsin Case:α isasserted 2 the conceptual space, as determined by the environment, and sec- ondlythereliabilityoftheagentsinthespace.Wegoontogivean Whenα2isasserted,λisupdatedwhenµα2 =λx1+(1−λ)(1− analysisoftheseresults,andtoprovesomeresultsrelatingthevalues x2)<w,i.e.when ofthefinalweightstothedistributionofobjectswithintheconcep- tualspace. w−(1−x2)≥λ(x1+x2−1) Thenif(x +x −1) > 0,A ≥ λ,andtheupdateisapositive 1 2 increment.Otherwisethereverseholds,i.e.A≤λandtheupdateis 4.3 Analysis anegativeincrement. Wewishtoanalysetheresultsseeninsection4.2inordertopredict the value to which λ will converge and also the extent to which it Case:α isasserted 3 will converge across agents as measured by the standard deviation Whenα isasserted,λisupdatedwhenµ ((cid:126)x)=λ(1−x )+(1− oftheλiacrossagents.Todoso,westartwithsomeanalysisofthe 3 α3 1 λ)x <w,i.e.when particular space and labels we have used before giving some more 2 generalresults. w−x ≥λ(1−x −x ) Withintheresultsinsection4.2,agentshavelabelsL =L =< 2 1 2 1 2 1,U[0,1] > to describe the conceptual space Ω = [0,1]2. In this Thenif(1−x −x ) > 0,A ≥ λ,andtheupdateisapositive 1 2 case,wehaveµLi(x)=xi.Eachassertionαiismadeexactlywhen increment.Otherwisethereverseholds,i.e.A≤λandtheupdateis eachcomponentµ±L1(x) > 0.5,µ±L2(x) > 0.5(sinceotherwise anegativeincrement. Case:α isasserted Thenconsiderthebehaviouroftheexpectedvalueofλovertime: 4 Whenα isasserted,λisupdatedwhenµ ((cid:126)x)=λ(1−x )+(1− E(λ )=E(λ |(cid:126)x∈R+)p++E(λ |(cid:126)x∈R−)p− 4 α4 1 t+1 t+1 t+1 λ)(1−x2)<w,i.e.when =E(λ (1−h)+h|(cid:126)x∈R+)p+ t w−(1−x2)≥λ(x2−x1) +E(λt(1−h)|(cid:126)x∈R−)p− Then if (x2 − x1) > 0, A ≥ λ, and the update is a positive =(1−h)E(λt)+p+h increment.Otherwisethereverseholds,i.e.A≤λandtheupdateis Ast→∞,wehave anegativeincrement. Eachofthesecasescanberepresentedgraphically,sincethecon- E(λ)=E(λ)(1−h)+p+h ditionsaredeterminedbythelinesx1 =x2andx1 =1−x2.This =p+ isillustratedinfigure10.Wecallareasofthespacewhereapositive updateismadepositiveregionsandareasofthespacewherenegative updatesaremadenegativeregions. Wehavethereforerelatedthevalueofλtowhichagentsconverge tothedistributionofelements(cid:126)xintheconceptualspaceΩ. 1 Theorem 4. Suppose agents are equipped with labels L , L and 1 2 0.8 - - combineandupdatetheselabelaccordingtothelanguagegamede- scribed.Thentheexpectedvalueofλ,E(λ)=E(A). + + 0.6 Proof. ToobtainE(λ),consider: x 2 0.4 λt+1 =λt(1−h)+hA + + 0.2 - - E(λt+1)=E(λt(1−h)+hA) =(1−h)E(λ )+hE(A) t 0 Thenast→∞,wehave 0 0.2 0.4 0.6 0.8 1 x E(λ)=E(λ)(1−h)+hE(A) Figure10:Thetypeofupdatemade1when(cid:126)x=(x1,x2)fallsineach areaofthespace =E(A) We can now prove some results concerning the final value of λ acrossthepopulationofagents. Thisresultshowsthattheweightinggiventothecompoundcon- ceptsα maybedirectlypredictedfromthedistributionofelements i Theorem 3. Suppose that agents have labels L1 = L2 =< intheconceptualspaceandthemembershipfunctionsusedtocate- 1,U(0,1) > and all agents have weight 1. Agents update their gorisetheconstituentlabelsL . j conceptsaccordingtothelanguagegamedescribedusingupdating Ifweconsiderupdatingmodel2,whereagentsupdatewhenever model1.Supposethatthereisaprobabilityp+of(cid:126)xfallinginapos- µ ((cid:126)x) (cid:54)= w,wemayalsoobtainanexpressionforthevarianceof αi itiveregionandprobabilityp− = 1−p+ of(cid:126)xfallinginanegative theλ acrossagents. i region.Thentheexpectedvalueofλconvergestop+ Theorem5. SupposeagentshavelabelsL ,L andthatagentsup- 1 2 Proof. Whenever(cid:126)xfallsinthepositiveregion,A≥1andhencewe dateaccordingtoupdatingmodel2:µαi((cid:126)x)(cid:54)=w.Thenthevariance setA=1.Forexample,suppose(cid:126)x∈R .Then oftheλacrossagentsisVar(λ)= h Var(A) 1 2−h 1−x Proof. WeobtainVar(λ)byfirstlycalculatingE(λ2): A= 2 x1−x2 E(λ2 )=E((λ (1−h)+hA)2) t+1 t Now1−x2 ≥0sothesignofAisdeterminedbyx1−x2.If(cid:126)xfalls =E((1−h)2λ2+2h(1−h)λ A+h2A2)) t t inthepositivequadrantthenx >x so 1 2 =(1−h)2E(λ2)+2h(1−h)E(λ )E(A)+h2E(A2) t t 1−x A= 2 ≥1 x −x 1 2 Wemaythencalculate SoA=1andtheupdateisλ =λ +h(1−λ ) n+1 n n If(cid:126)xfallsinthenegativequadrantthenx1 <x2so Var(λt+1)=E(λ2t+1)−(E(λt+1))2 A= 1−x2 ≤0 =(1−h)2E(λ2t)+2h(1−h)E(λt)E(A) x1−x2 +h2E(A2)−(E(λ)(1−h)+hE(A))2 SoA = 0andtheupdateisλ = λ −hλ .Similararguments =(1−h)2E(λ2)+2h(1−h)E(λ )E(A) n+1 n n t t canbemadefortheotherquadrants,givingusthefollowingupdating +h2E(A2)−(1−h)2(E(λ ))2 t rule: λ =(cid:40)λt(1−h) if(cid:126)x∈R− −2h(1−h)E(λt)E(A)−h2(E(A))2 t+1 λt(1−h)+h if(cid:126)x∈R+ =(1−h)2Var(λt)−h2Var(A) Ast→∞,wehave Toillustratetheseresults,weransimulationsofthelanguagegame with1000agentseachwithL =L =<1,U(0,1)>,w=1,x ∼ 1 2 1 Var(λ)=(1−h)2Var(λ)−h2Var(A) U[0.25,0.75],x ∼U[0,0.5],andh∈{10−2,10−3,10−4,10−5}. 2 h Figure11showsthevaluesofE(λt)overtimeobtainedfromsimula- = Var(A) 2−h tions,togetherwiththepredictedvalueofE(λt)andalsotheresting stateE(λ). Wecanfurtherexaminehowquicklythemeanandvarianceofλ h = 10−2 h = 10−3 i 0.5 0.5 approachthatofA.WeobtainanexpressionforE(λ )andVar(λ ) Simulations t t intermsoftandsolve.Thespeedatwhichtheλi convergetothe 0.45 PRreesdtiinctge dState 0.45 restingstateisdependentonh. 0.4 0.4 Theorem 6. Suppose agents are equipped with labels L1, L2 and λE()0.35 0.35 combine and update these label according to the language game 0.3 0.3 described, using updating model 2. Then the number of timesteps 0.25 0.25 until |E(λ ) − E(A)| ≤ (cid:15) for some small (cid:15) is t ≥ (log((cid:15)) − t log(|E(λ0)−E(A)|))/log(1−h), and the number of timesteps 0.20 10 20 30 40 50 0.20 10 20 30 40 50 until|Var(λ )− h Var(A)|≤(cid:15)is t 2−h h = 10−4 h = 10−5 (log((cid:15))−log(|Var(λ )− h Var(A)|)) 0.5 0.5 t≥ 0 2−h 2log(1−h) 0.45 0.45 0.4 0.4 Proof. We firstly obtain an expression for E(λ ) in terms of t, h, t E(A)andE(λ0) λE()0.35 0.35 0.3 0.3 0.25 0.25 E(λ )=E(λ )(1−h)+hE(A) t t−1 =E(λ )(1−h)t+E(A)(cid:88)t (h(1−h)t−1) 0.20 10 T2i0meste3p0s 40 50 0.20 10 T2i0meste3p0s 40 50 0 k=1 Figure11: Predicted value of E(λ) and actual value of E(λ) over =E(λ0)(1−h)t+E(A)(1−(1−h)t) timefordifferentvaluesofh.TherateatwhichE(λ)approachesits restingstateisslowerforsmallerh. Now,consider Figure12showsthevaluesofVar(λ )overtimeobtainedfrom t simulations,togetherwiththepredictedvalueofVar(λ )andalso t therestingstateVar(λ). (cid:15)≥|E(λ )−E(A)| t Theseresultsillustratethattherateatwhichagentsconvergetothe =|E(λ0)−E(A)|(1−h)t restingstateisdependentonthevaluehbywhichagentsadoptothers t≥(log((cid:15))−log(|E(λ )−E(A)|))/log(1−h) agents’ viewpoints. Larger values of h allow faster convergence to 0 therestingstate,howeverthatrestingstatewillhavealargervariance To calculate the number of timesteps needed until Var(λ) has whenhislarger. reacheditsrestingstate,weagainobtainanexpressionforVar(λ ) t intermsoft,h,Var(A)andVar(λ ). 0 5 DISCUSSION Varλ =(1−h)2Var(λ t−1))−h2Var(A) t ( Characterising concepts as a weighted sum of attributes is seen =Var(λ )(1−h)2t+Var(A)(cid:88)t h2(1−h)2t throughout the literature [2, 3, 6, 16]. However, this has been pro- 0 posed in an ad hoc fashion, and many concepts do not adhere to k=1 thisformulation.Further,nomechanismfordeterminingtheweights h =Var(λ )(1−h)2t+ Var(A)(1−(1−h)2t) hasbeenproposed.Wehavedevelopedahierarchicalmodelofcon- 0 2−h cept combination from which the characterisation of concepts as a Again,consider weightedsumofattributesarisesnaturally,summarisedinsection2. Wehaveimplementedasimpleversionofthismodelwithinthelan- h guagegameframeworkinwhichagentsmakeassertionsthatconsist (cid:15)≥|Var(λ )− Var(A)| t 2−h ofaweightedsumoftwoconstituentconcepts.Wehaveshownthat h within a multi-agent simulation of a community of such language =|Var(λ )− Var(A)|(1−h)2t 0 2−h users agents can converge to shared weightings (section 4.2). The t≥ (log((cid:15))−log(|Var(λ0)− 2−hhVar(A)|)) wtoetihgehtdsisλtriibtoutwiohnicohfethleemceonmtsmxuniintythoefcaognecnetsptcuoanlvsepragcee,artheeremlaetmed- 2log(1−h) j bershipfunctionsµ (x )usedfortheconstituentconcepts,andthe Lj j reliabilitywoftheagents.Wehavederivedexplicitexpressionsre- [4] J.Hampton,‘Conceptualcombinationsandfuzzylogic’,inConcepts h = 10−2 h = 10−3 andFuzzyLogic,eds.,R.BelohlavekandG.J.Klir,TheMITPress, 0.1 0.1 Simulations (2011). Predicted [5] H.KampandB.Partee,‘Prototypetheoryandcompositionality’,Cog- 0.08 Resting State 0.08 nition,57(2),129–191,(1995). 0.06 0.06 [6] G.Lakoff,‘Hedges:Astudyinmeaningcriteriaandthelogicoffuzzy λVar()0.04 0.04 [7] cJ.oLnacweprtys,’‘,AJofurarmnaelwoofrpkhfiolorsloinpghuicisatliclomgoicd,e2ll(i4n)g,’4,5A8r–ti5fi0c8ia,l(I1n9t7e3ll)i.gence, 155(1-2),1–39,(2004). 0.02 0.02 [8] J. Lawry and Y. Tang, ‘Uncertainty modelling for vague concepts: Aprototypetheoryapproach’,ArtificialIntelligence,173(18),1539– 0 0 1558,(2009). 0 10 20 30 40 50 0 10 20 30 40 50 [9] MarthaLewisandJonathanLawry,‘Theutilityofhedgedassertionsin theemergenceofsharedcategoricallabels’,inProceedingsoftheAISB h = 10−4 h = 10−5 Convention2013.SocietyfortheStudyofArtificialIntelligenceand 0.1 0.1 theSimulationofBehaviour,(2013). [10] MarthaLewisandJonathanLawry,‘Ahierarchicalconceptualspaces 0.08 0.08 modelofconceptcomposition’,ArtificialIntelligence,(toappear). [11] D.N.OshersonandE.E.Smith,‘Ontheadequacyofprototypetheory 0.06 0.06 asatheoryofconcepts’,Cognition,9(1),35–58,(1981). λar() [12] E.Rosch,‘Cognitiverepresentationsofsemanticcategories.’,Journal V0.04 0.04 ofExperimentalPsychology:General,104(3),192,(1975). [13] E.E.SmithandD.N.Osherson,‘Conceptualcombinationwithproto- 0.02 0.02 typeconcepts’,CognitiveScience,8(4),337–361,(1984). [14] LucSteels,TonyBelpaeme,etal.,‘Coordinatingperceptuallygrounded 0 0 categoriesthroughlanguage:Acasestudyforcolour’,Behavioraland 0 10 20 30 40 50 0 10 20 30 40 50 Timesteps Timesteps brainsciences,28(4),469–488,(2005). [15] L.A. Zadeh, ‘Fuzzy sets’, Information and Control, 8(3), 338–353, Figure12: Predicted value of Var(λ) and actual value of Var(λ) (1965). over time for different values of h. The rate at which Var(λ) ap- [16] L.A.Zadeh,‘Afuzzy-set-theoreticinterpretationoflinguistichedges’, proachesitsrestingstateisslowerforsmallerh JournalofCybernetics,(1972). latingthemeanofthefinalweightsλ acrossallagentstothedis- i tributionofthex ,µ (x ),andw.Inamodifiedupdatingmodel j Lj j thatismoreamenabletoanalysis,wehavederivedexpressionsfor thevarianceoftheλ intermsofx ,µ (x ),wandalsoh,therate i j Lj j atwhichagentsadoptotheragents’concepts.Whenhissmall,i.e. agents are slower to adopt others’ concepts, the variance of the λ i issmaller,perhapsindicatingthatmorerobustconceptsareformed. Wehavealsoderivedexpressionsforthespeedatwhichtheresting distribution of the λ are approached, dependent on the value of h i (section4.3).Theresultsfromthismodelindicatehowweightingsin aweightedsumofconceptscanberelatedtotheelementsencoun- teredinaconceptualspace. This model is of course extremely simple, and numerous exten- sionsareongoing.TheuseofmorecomplexlabelsL isunderin- i vestigation,withlabelsbeingextendedtomultipledimensions.We arealsodevelopingtheupdatingalgorithminordertocombinemore than two labels and to include noise in agents’ labels. In previous work[1,9],theagentsinthesimulationshavehadvaryingreliability w,whichcouldbeincorporatedintothesesimulations.Futurework will also incorporate the theoretical aspects alluded to in section 2 thataccountfornon-compositionalpropertiessuchasemergentat- tributes. ACKNOWLEDGEMENTS MarthaLewisgratefullyacknowledgessupportfromEPSRCGrant No.EP/E501214/1 REFERENCES [1] HenriettaEyreandJonathanLawry,‘Languagegameswithvaguecate- goriesandnegations’,AdaptiveBehavior,1059712314547318,(2014). [2] P.Ga¨rdenfors,Conceptualspaces:Thegeometryofthought,TheMIT Press,2004. [3] J.Hampton,‘Inheritanceofattributesinnaturalconceptconjunctions’, Memory&Cognition,15(1),55–71,(1987).
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