Emerging Dimension Weights in a Conceptual Spaces Model of Concept Combination

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by  Martha Lewis

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: [e-mail], close to Pi to be labelled Li. The membership of an object x in a
[e-mail] 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
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Press,2004.
[3] J.Hampton,‘Inheritanceofattributesinnaturalconceptconjunctions’,
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