The Utility of Hedged Assertions in the Emergence of Shared Categorical Labels

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by  Martha Lewis

The Utility of Hedged Assertions in the Emergence of
Shared Categorical Labels
MarthaLewis and JonathanLawry1
Abstract. We investigate the emergence of shared concepts in a Conceptualspacestheoryrendersconceptsasconvexregionsofa
community of language users using a multi-agent simulation. We conceptualspace-ageometricalstructurewithqualitydimensions
extend results showing that negated assertions are of use in devel- andadistancemetric.Examplesare:theRGBcolourcube,pictured
opingsharedcategories,toincludeassertionsmodifiedbylinguistic infigure1;physicaldimensionsofheight,breadthanddepth;orthe
hedges.Resultsshowthatusinghedgedassertionspositivelyaffects tastetetrahedron.Sinceconceptsareconvexregionsofsuchspaces,
theemergenceofsharedcategoriesintwodistinctways.Firstly,us- thecentroidofsucharegioncannaturallybeviewedastheprototype
ingcontractionhedgeslike‘very’givesbetterconvergenceovertime. oftheconcept.
Secondly, using expansion hedges such as ‘quite’ reduces concept
overlap.However,boththeseimprovementscomeatacostofslower
6
speedofdevelopment.
1
0
2 1 INTRODUCTION

n
Anevolutionaryapproachtosemanticsenablesthedevelopmentin
a
J robots and autonomous agents of flexible, mutable concepts that
could be learnt through interaction and can change over time [13].
5
2 This approach is investigated by Eyre and Lawry in [2], in which
they develop a model of language evolution based in the label se-

] manticsframework.Theyshowthatusingamixtureofpositiveand
I
A negated assertions enables the development of languages that are Figure1. TheRGBcuberepresentscoloursinthreedimensionsofRed,
both shared, and discriminate effectively between elements within GreenandBlue.Acolourconceptsuchas‘purple’canberepresentedasa
.
s theenvironment.Weextendthisworktoincludeassertionsmodified regionofthisconceptualspace.
c by the words ‘very’ and ‘quite’, and show that doing so improves
[
performanceintwoways.Useofthehedge‘very’improveslevelsof
Labelsemantics[7]isarandomsetapproachtoconceptswhich
1 convergenceattained.Usingthehedge‘quite’reducestheamountof
quantifies an agent’s uncertainty about the extent of application of
v overlapwithinanagent’slabelset.Wedescribeindetailthetheoreti-
a concept. We refer to this as subjective uncertainty [8] to empha-
5 calapproachtoconceptstakenandlinguistichedgesintheremainder
sisethatitconcernsthedefinitionofconceptsandcategories,incon-
5 ofthissection.Section2givesdetailsofthemathematicalandcom-
7 putationalmodelusedinthesimulations.Section3givesresultsof trasttostochasticuncertaintywhichconcernsthestateoftheworld.
6 LawryandTang[8]combinethelabelsemanticsapproachwithcon-
the simulations which are discussed in section 4. Lastly, section 5
0 ceptualspacesandprototypetheory,togiveaformalisationofcon-
givesconclusionsandfurtheravenuesofresearch.
. ceptsasbasedonaprototypeandathreshold,locatedinaconceptual
1
space.
0
6 1.1 Arepresentationmodelforconcepts Within this framework, agents use sets of labels LA =
1 {L1,L2,...,Ln}todescribeanunderlyingconceptualspaceΩwith
We model concepts within the label semantics framework [7, 8],
: distancemetricd(x,y)betweenpoints.Theconceptualspacecould
v combined with prototype theory [11] and the conceptual spaces
be,asmentioned,theRGBcolourspace.LabelsL wouldthenbe
i modelofconcepts[3].Prototypetheoryoffersanalternativetothe i
X concepts such as ‘red’, ‘blue’, ‘purple’, ‘orange’ and so on. These
classical theory of concepts, basing categorization on proximity to
labelsareviewedasregionsoftheconceptualspace.Sotheconcept
r a prototype. This approach is based on experimental results where
a ‘blue’ is represented by the blue region in the colour cube. Within
human subjects were found to view membership in a concept as a
label semantics, these regions are specified by prototypes P and
i
matterofdegree,withsomeobjectshavinghighermembershipthan
thresholds ε . This is in contrast to Ga¨rdenfors’ original approach
others[11].Fuzzysettheory[15],inwhichanobjectxhasagraded i
whichistoviewthespaceaspartitionedbyaVoronoitessellation.If
membership µ (x) in a concept L, was proposed as a formalism
L thislatterapproachistaken,eachindividualpointintheconceptual
forprototypetheory.However,numerousobjectionstoitssuitability
space is allocated to exactly one label. With a prototype-threshold
havebeenmade[10,12,6,5,4].
approach,itiseasytoaccommodatetheideaofanobjectbeingac-
1 University of Bristol, England, email: [email protected], curately described by more than one concept, or conversely, some
[email protected] pointswithinthespacenotbeingassignedtoanyconcept.Thisdif-
ferenceisillustratedinfigures2and3. labelL todescribeanelementxisthengivenbytheprobabilitythat
i
xlieswithintheneighbourhoodNεi,i.e.thatthedistanced(x,P )
Li i
0.5 islessthanε .So:
i
0.4 (cid:90) ∞
µ (x)=P(d(x,P )≤ε )= δ (ε )dε
0.3 Li i i i i i
d(x,Pi)
0.2
Figure5showshowthisappropriatenessmeasureworksinasetup
0.1 similartothatinfigure4.
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure2. ConceptualspacedividedintoconceptsaccordingtoaVoronoi
tessellationaroundprototypes.Eachpartofthespacecorrespondstoexactly 1
oneconcept. 0.8
0.6
(x)µL0.4
0.5
0.2
0.4
0
0.3 1
0.8 1
0.2 0.6 0.4 0.6 0.8
0.1 x2 0.2 0 0 0.2 0x.41
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5. Membership in a concept. The prototype of the label is at
Figure3. Conceptualspacedividedintoconceptsaccordingtoaprototype- [0.5,0.5]andthethresholdεhasdistributionU[0,0.3].Whenx=[0.5,0.5],
thresholdapproach.Somepointsinthespacecorrespondtomorethanone µL(x) = 1.Aswemovefurtherawayfromtheprototype,membershipin
concept,andsomecorrespondtonone. theconceptdecreases,andis0whend(x,Pi)>0.3
Inthismodel,however,agentsareuncertainastoexactlywhere ThisappropriatenessmeasureissimilartoZadeh’sdescriptionof
thethresholdslie.Toillustratethis,considertheconcept‘tall’.Itis fuzzymembershipinaconcept[15].
easytopointoutatallperson,andtopointoutapersonwhoisnot
tall,butitisdifficulttospecifytheexactthresholdbetween‘tall’and
1.2 Linguistichedges
‘nottall’.Thisuncertaintyconcerningwherethethresholdliesisrep-
resentedinthelabelsemanticsframeworkbysayingthatathreshold Hedgesarewordsorphrasessuchas‘very’,‘quite’,‘strictlyspeak-
ε isdrawnfromaprobabilitydistributionδ .LabelsL areassoci- ing’whichmodifythedomainofapplicationofaconcept.Inpartic-
i i i
atedwithneighbourhoodsNεi ={(cid:126)x∈Ω:d((cid:126)x,P )≤ε },i.e.the ular,‘very’,and‘quite’respectivelycontractorexpandthedomain
Li i i
regionwithinthethreshold.Theseideasarerepresentedinfigure4. ofapplicationofaconcept,sothat,forexample,‘verytall’appliesto
fewer people than does ‘tall’, whereas ‘quite tall’ applies to more.
Within fuzzy set theory, we expect that µ (x) ≤ µ (x) and
b veryL L
µ (x) ≥ µ (x).Applyingthistotheconcept‘tall’,again,this
quiteL L
meansthatmembershipintheconcept‘verytall’shouldalwaysbe
a
lessthanmembershipin‘tall’.Soanyonewhocanbedescribedas
ε
i ‘verytall’canalsobedescribedas‘tall’.Zadeh[16]usesoperations
x
2 ofconcentrationanddilationtorendertheseideas.Concentrationis
Pi describedasCON(µ (x))=(µ (x))2anddilationisoftenren-
Li Li
deredasDIL(µ (x)) = (µ (x))1/2.However,weargue,asdo
Li Li
[1],thatZadeh’sformulaeare,toanextent,arbitrary,sincetheno-
tionoftakingapowerofamembershipvaluedoesnotcorrespondto
anythingthatlanguageusersmightdo.Rather,itsimplyhassomeof
x
1 therighteffects.Aswith[1],wetakeasemanticapproach.
In[9],weproposethataconcept‘veryL’or‘quiteL’berendered
Figure4. Prototype-thresholdrepresentationofaconceptLi.Theconcep- by considering that the prototype of ‘very/quite L’ is equal to that
tual space has dimensions x1 and x2. The concept has prototype Pi and ofthebaseconceptL,butthatthethresholdofthehedgedconcept
thresholdεi.Theuncertaintyaboutthethresholdisrepresentedbythedotted ‘very/quiteL’isrespectivelysmallerorlargerthanthatofthebase
line.TheneighbourhoodNεi istheareawithinthedottedline.Elementain concept. This approach is grounded in the idea that ‘very/quite L’
Li
theconceptualspaceiswithinthethreshold,soitisappropriatetoassert‘ais shouldapplytorespectivelyfewerormoreobjectsthanL.Narrow-
Li’.Elementbisoutsidethethreshold,soitisnotappropriatetoassert‘bis ingorwideningthethresholdachievesthisinanaturalway.Thisis
Li’ illustratedinfigure6.
Ourmodelofthehedges‘very’and‘quite’thereforerequiressim-
Thethresholdεiisuncertain,however,sothereissomeprobability plythatvεi ≤ εi andthatqεi ≥ εi.Weimplementthismodelin
thatεiinfigure4isactuallywideenoughtoincludetheobjectb,i.e. aversionofthemulti-agentsimulationcreatedin[2]inordertoin-
thatLiisappropriatetodescribeb.TheappropriatenessµLi(x)ofa vestigatehowtheuseofthesehedgesinamodeloflanguagehelps
2.3 Assertionmodel
Ateachtimestep,halftheagentsaredesignatedspeakeragentsand
b
makeassertions,determinedbytheassertionmodelused.Theasser-
a εi tionmodelisbasedontheprobabilityofmakingaparticularasser-
x2 vε tionθ,giventhattheobjectbeingdescribedisx.Followingmethods
i Pi in[2,8],wecalculatetheposteriorprobabilityofeachθ∈AS,given
anelementx∈Ω.Theassertionmadebyaspeakeragentistheas-
sertionwiththehighestprobability.Theposteriorprobabilityofeach
qε θ,givenx,isdeterminedbytheappropriatenessoftheassertionθto
i
describex,i.e.µ (x),andthepriorprobabilityP(θ)ofassertingθ.
θ
x Wefirstconsiderwhichsetsoflabelsthatareappropriatetode-
1
scribe x ∈ Ω. The probability that any particular set of labels
F ⊆LAareappropriatetodescribexisgivenbyaprobabilitymass
Figure6. Representationof‘veryLi’and‘quiteLi’.‘VeryLi’hasproto- functionm :2LA →[0,1].Onewayofdeterminingm isviathe
typePiandthresholdvεi ≤ εi.‘QuiteLi’hasprototypePiandthreshold x x
consonantselectionfunctionintroducedin[8].Thisstates:
qεi ≥ εi.NoticethatalthoughLiisappropriatetodescribea,vLiisnot.
Also,althoughLiisnotappropriatetodescribeb,qLiis.
Definition1(Consonantselectionfunction) Given non-zero ap-
propriateness measures on basic labels µ (x) : i = 1,...,n or-
theemergenceofsharedcategoriesacrossacommunityoflanguage Li
dered such that µ (x) ≥ µ for i = 1,...,n, the consonant
users. Li Li+1
selectionfunctionidentifiesthemassfunction
m ({L ,...,L })=µ (x)
2 METHODS x 1 n Ln
m ({L ,...,L })=µ (x)−µ (x)fori=1,..n−1
x 1 i Li Li+1
m (∅)=1−µ (x)
2.1 Overview x L1
m (F)=0ifF (cid:54)={L ,L ,...,L }forsomek≤n
x 1 2 k
Toinvestigatetheutilityofhedgedassertionsweimplementamulti-
Because we have ordered the labels by µ (x) ≥ µ , if the
agentsimulationofaversionofthecategorygame[14],following Li Li+1
labelL isappropriatetodescribex,alllabelsL :j <imustalso
[2],inwhichsharedcategoriesdevelopovertimeasaresultofthe i j
beappropriatetodescribex.Thequantityµ (x)−µ (x)cor-
interactionsofthecategoryusers.Anoverviewofthegameisasfol- Li Li+1
respondstotheideathatxinsomesenseliesbetweenthethresholds
lows. Agents use labels to describe a conceptual space Ω. At each
ε andε ,sothatL isappropriatetodescribex,butL isnot.
timestep,agentsarerandomlypairedintospeakersandlisteners,and i+1 i i i+1
Weextendthisdefinitiontothecaseofhedgedlabelssimplybycon-
eachpairisshownadistinctelementx ∈ Ω.Thespeakermakesan
sideringallhedgedlabelsasbasiclabels,explainedintheexample
assertionθabouttheelementbasedonitslabelset.Thelistenerthen
below.
updates its own label set to be more similar to that of the speaker,
basedonthisassertionandaparameterw whichcanbethoughtof
Example2(Determiningthemassfunction) Suppose we are de-
astheageofthespeaker.Theupdatemadebythespeakerisacom-
termining the mass function for subsets F ⊆ {kL ,kL : k =
binationofshiftingtheprototypeoftherelevantlabelandchanging 1 2
very, quite, basic},giventhepointa ∈ x ×x ,asillustratedin
thesizeofthethreshold.Theaimisthatafteranumberoftimesteps, 1 2
figure7.
labelsetsacrossthepopulationhaveconvergedtoacommonsetof
sharedcategories.
2.2 Conceptualmodels
EachagentisequippedwiththesamenumbernoflabelsLi,with x2 P2
pointprototypesP ∈Ω,whereΩ=[0,1]3.Atthestartofthesimu- a
i
lationsthePiareuniformlydistributedaroundthespace.Thresholds P1
ε arealsorandomlyinitiated,andconsideredtobesomemultipleof
i
abasethresholdε.Eachthresholdε ∼ U(0,b ),whereagain,the
i i
b canbeconsideredtobeamultipleofsomecommonb,andtheb
i i
aretakenfromU[0.5,2].ThedistancemetricisEuclidean.
EachagentthereforehasalabelsetLA={L1,L2,...Ln}.These x1
labelscanbehedgedtoformasetLA+ =LA∪{veryL ,quiteL :
i i
i = 1,...,n}.HedgedconceptshavethesameprototypePi asba- Figure7. Determiningthemassfunctiononsubsetsof{kL1,kL2 : k =
sic labels, but a scaled threshold vεi or qεi where v < 1 and very, quite, basic}.P1 andP2 representprototypesforeachlableL1 and
q > 1. Agents can assert positive or negated, hedged or basic la- L2,andthedottedlinesgivethedifferentthresholdsaccordingtothehedges,
bels,givinganassertionsetAS = {kLi,¬kLi : i = 1,...,n;k = asinfigure6.Noticethat‘quiteL2’,L2,‘veryL2’and‘quiteL1’areall
very, quite, basic}, where k = basic means that the label is not appropriatetodescribea,althoughwithdifferentappropriatenessmeasures
hedged. (notshown),butL1and‘veryL1’arenot.
Suppose that µ (a) = 0.9, µ (a) = 0.7, µ (a) =
quiteL2 L2 quiteL1 Table1. Priorprobabilitiesofeachtypeofassertion±kLi
0.3, µ (a) = 0.1, µ (a) = 0, µ (a) = 0, giving us
veryL2 L1 veryL1 ∗ pv pb pq
the order µ (a) ≥ µ (a) ≥ µ (a) ≥ µ (a) ≥
quiteL2 L2 quiteL1 veryL2 pp P(vL) P(L) P(qL)
µL1(a) ≥ µveryL1(a).Wemaythenassignprobabilitiestosubsets pn P(¬vL) P(¬L) P(¬qL)
oflabelsaccordingtotheconsonantselectionfunction:
ThepriorprobabilityofassertinganyparticularlabelL ∈LAis
i
m (F )=m ({quiteL ,L ,quiteL ,veryL ,L ,veryL })
x 6 x 2 2 1 2 1 1 uniformacrossLA.HencethevalueofP(θ)calculatedaboveshould
=µ (a)=0 bedividedbyn,giving,forexample,
veryL1
mx(F5)=mx({quiteL2,L2,quiteL1,veryL2,L1}) pn∗pv
P(¬vL )=
=µL1(a)−µveryL1(a)=0 2 n
m (F )=m ({quiteL ,L ,quiteL ,veryL })
x 4 x 2 2 1 2 Example5(Determiningtheposteriorprobabilityofassertion)
=µveryL2(a)−µL1(a)=0.1 Suppose,foraneasyexample,wewanttocalculatetheprobability
mx(F3)=mx({quiteL2,L2,quiteL1}) ofasserting‘veryL1’,givenobjecta,asinexample2.Weneedto
calculate
=µ (a)−µ (a)=0.2
quiteL1 veryL2
m (F )=m ({quiteL ,L })=µ (a)−µ (a)=0.4
x 2 x 2 2 L2 quiteL1
mx(F1)=mx({quiteL2})=µquiteL2(a)−µL2(a)=0.2 P(A =‘veryL1’|a)=P(θ) (cid:88) mPa(xG(G))
mx(∅)=1−µquiteL2(a)=0.1 G⊆AS:‘veryL1’∈G
Havingdeterminedtheprobabilitymassfunctiononsetsoflabels, whereGi =Fi∪{¬kLj :kLj ∈LA+\Fi}.However,theonly
amassassignmentonsetsofassertionsisthendefined. subsetGi (cid:51)‘veryL1’isG6,so
Definition3(Massassignmentonassertions) ma : 2AS →
x ma (G )
[0,1]isdefinedsuchthat: P(A =‘veryL ’|x)=P(‘veryL ’) x 6
1 1 P(G )
6
(cid:88)
ma (G)= m (F) =0
x x
F⊆LA+:C(F)=G
Suppose,foramoreinvolvedexample,thelabelsetLA+isasin
whereC(F) = {θ ∈ AS : F ∈ λ(θ)},andλ(θ)isdefinedrecur- example2,withpp=0.7,pv =0.7,pq =0.2,andwewanttode-
sivelyby termineP(A = quiteL |a).ThepriorprobabilityP(quiteL ) =
1 1
0.7∗0.2 =0.07.Sowehave:
2
λ(kL )={F ⊆LA+ :kLi∈F}
i
λ(¬θ)=(λ(θ))c P(A =quiteL1|a)
λ(θ∧φ)=λ(θ)∩λ(φ) =0.07 (cid:88) max(Gi)
P(G )
λ(θ∨φ)=λ(θ)∪λ(φ) Gi:quiteL1∈Gi i
ma (G ) ma (G ) ma (G ) ma (G )
=0.07( x 6 + x 5 + x 4 + x 3 )
P(G ) P(G ) P(G ) P(G )
6 5 4 3
ThisdefinitionhastheimplicationthatforGi = Fi ∪{¬kLj : 0.1 0.2
kLj ∈LA+\Fi},max(Gi)=mx(Fi). =0.07(0+0+ (cid:80) P(ϕ) + (cid:80) P(ϕ))
Thentheprobabilityofanassertionθ beingmade,giventhatan ϕ∈G4 ϕ∈G3
0.1 0.2
objectxisbeingdescribed,canbecalculatedbysummingoverG⊆ =0.07( + )
0.54 0.4
ASthatcontainθ.
=0.048
Definition4 GivenapriordistributiononAS,aposteriordistribu-
tiongivenanobjectxcanbecalculatedby:
Having calculated the probability of each assertion, the speaker
(cid:88)
P(A =θ|x)= max(G)P(A =θ|A ∈G) agentmakesthemostprobableassertionθ∈AS.
G⊆AS:θ∈G
=P(θ) (cid:88) max(G) 2.4 Updatingalgorithms
P(G)
G⊆AS:θ∈G
Oncethespeakeragenthasmadeassertionθ,thelisteneragentcom-
Here,P(G)=(cid:80)ϕ∈GP(ϕ) putesµθ(x)basedonitscurrentlabelset.Ifµθ(x) < w,wherew
ThevalueofP(θ)foroneparticularlabelLisaproductoftwo isaparameterthatcanbethoughtofastheageofthespeakeragent,
elements:thepriorprobabilityppofmakingapositiveassertion(or thelisteneragentupdatesitslabelsetLAbymovingtheprototype
1−ppforanegatedassertion);andthepriorprobabilityofmaking and/orchangingthethresholdoftheconcept,untilµθ(x)=w.For-
ahedgedassertion,givenbypvformakinganassertionhedgedwith mulaefortheseupdatesareagainbasedon[2].AlabeldefinedbyPi
theword‘very’,pq formakinganassertionhedgedwith‘quite’or andεiisupdatedtoPi(cid:48) =Pi−λ(x−Pi)andε(cid:48)i =αεi.Valuesfor
1−pv−pqformakingabasicassertion,summarisedintable1. λandαaresought,suchthatµ(cid:48)(x)=w.
θ
2.4.1 Case1:θ =kL and
i
(cid:40)
Recallthatεi ∼U(0,bi),sothatforx∈Ω, E(λ)= (1−wk)(1− ||xw−kPbii||) ifwq<1
||x−P || 0 otherwise
µ (x)=1− i <wbyassumption.
kLi kb
i So at each timestep, each listener agent, for whom µ (x) < w,
θ
ThelabelL isupdatedtoL(cid:48),whereP(cid:48) = P −λ(x−P )and updatestherelevantlabelusingthethequantitiesE(α),E(λ).
i i i i i
ε(cid:48)i = αεi, such that µkL(cid:48)(x) ≥ w, and minimising the distance
i
betweentheinterpretationsasmeasuredbytheHaussdorffdistance 2.5 Performancemetrics
betweenthetwoneighbourhoods,
Performancemetricsfrom[2]areused,measuringtheAveragePair-
H(NLi,NL(cid:48)i)=||Pi−Pi(cid:48)||+|εi−ε(cid:48)i| (1) wiseDistancebetweenlabelsets(APD)andtheAverageLabelOver-
εb lap (ALO). APD measures the difference in label sets in the com-
=|λ|||x−Pi||+ bi|1−α|(*) munity,andALOindicatestheextenttowhichanagent’sconcepts
overlap.Weseeklowvaluesforeachmetric.
Tominimisetheupdate,wesetµkL(cid:48)(x)=w,so: APD is calculated using the Haussdorff distance between two
i
neighbourhoodsasgiveninequation1.Thedifferencebetweenthe
||x−P(cid:48)|| |1−λ|||x−P || labelsetsofanyonepairofagentsisgivenby
w=µkL(cid:48)i(x)=1− kb(cid:48)i i =1− αkbi i n
IPD=(cid:88)H(Nj ,Nk )
whichgives Li Li
i=1
α= |1−λ|||x−Pi|| wherenisthenumberoflabelseachagenthasandj andkrefer
(1−w)kbi todistinctagents.
(1−λ)||x−P || Thisisaveragedoverpairsofagents.ThereareN agents,there-
= (1−w)kb i sinceλ=1→Pi(cid:48) =x fore(cid:0)N(cid:1)pairs,giving:
i 2
ToupdateLiwewillalwayswantλ≥0,α≥1,aswearedealing 2 (cid:88)N (cid:88)N
withapositivelabel. APD= N(N −1) IPDjk
Substitutingαintoequation(*),weobtain k=j+1j=1
ALO is the extent to which labels overlap. To calculate this, we
εb (1−|λ|)||x−P ||
H(N ,N(cid:48) )=|λ|||x−P ||+ i( i −1) take the maximum value of the intersection of a pair of labels, as
Li Li i b (1−w)kb
i measuredbyaminrule.Weaveragethisvalueoverpairsoflabels.
ε ε||x−P || εb
=|λ|||x−P ||(1− )+ i − i (2) Theoverlapwithinanindividual’slabelsetistherefore
i b(1−w)k b(1−w)k b
Thenif1− b(1−εw)k > 0,i.e.ε < b(1−w)k,thequantity(2) ILO= 2 (cid:88)n (cid:88)n max{min{µ (x),µ (x):x∈Ω}}
canbeminimisedbysettingλ=0soα= ||x−Pi|| .Otherwise,we n(n−1) Li Lj
(1−w)kbi j=i+1i=1
haveα=1,λ=1− (1−w)kbi.
||x−Pi|| Averagedacrossthepopulationthisis:
Sinceεisarandomvariable,soisthechoicebetweenλandα.We
thereforeneedaconcreteupdatingrule.WeupdatePiandbiwiththe 1 (cid:88)N
expectedvaluesofλandαrespectively.ε∼Uniform[0,b],so ALO= ILO
N k
k=1
(cid:40)
(1−w)k if(1−w)k<1 whereILOksiginifiesagentk’slabeloverlap.
P(ε<b(1−w)k)=
1 otherwise
2.6 Simulationprocess
Wecanthereforecalculate
Simulationswithn = 100agentswererunforT = 104 timesteps.
(cid:40)
||x−Pi|| +1−(1−w)k if(1−w)k<1 Agentweightsw ∈ [0.2,0.8]wereupdatedateachtimestepinin-
E(α)= bi
||x−Pi|| otherwise crementsof1/T.Whenw≥0.8,agentsarerebornwithrandomised
(1−w)kbi labelsandw=0.2.20simulationsarerunforeachreportedcombi-
and nationofparameters.
(cid:40)(1−(1−w)k)(1− (1−w)kbi) if(1−w)k<1 [2] show that if pp ∈ [0.5,0.6] then performance of the system
E(λ)= ||x−Pi|| changesfromlowALOandhighAPDtoviceversaatapproximately
0 otherwise pp=0.56.Weransimulationsinaslightlyextendedrangeforcom-
parison, varying the prior probabilities pv, pb and pq of asserting
2.4.2 Case2:θ =¬kL the different hedges ‘very’, ‘basic’, and ‘quite’. We present results
i
fromthreesetsofparameters.Asabaselinewerunsimulationswith
Byanentirelysimilarargument,weobtain
no hedges, i.e. pv = 0, pb = 1, pq = 0. To investigate the ef-
(cid:40) fectsofusingcontractionhedges,werunsimulationswithparame-
||x−Pi|| +1−wk ifwk<1
E(α)= bi terspv = 0.7,pb = 0.2,pq = 0.1.Forexpansionhedges,weuse
||xw−kPbii|| otherwise parameterspv=0.1,pb=0.2,pq=0.7.
3 RESULTS
0.4
Theresultspresentedshowperformanceagainstthetwometricsafter
104simulationtimesteps.Bythispoint,thepopulationhasgenerally
reachedasteadystateinwhichperformancedoesnotgreatlychange.
Figure 8 shows the steady state of APD achieved after 104
0.35
timestepsforarangeofvaluespp ∈ [0.4,0.6].Threesetsofresults
arepresented:resultsusingunhedgedassertions;resultswithahigh D
P
A
priorprobabilityofusingcontractionhedges;andresultsfromsim-
ulationswithahighpriorprobabilityofassertingexpansionhedges,
0.3
wherethesepriorprobabilitiesareasstatedin2.6.
AhighpriorofassertingcontractedlabelsreducesminimumAPD
achieved from 0.38 when pp = 0.56 or pp = 0.6 to 0.29 when
pp = 0.57(figure8).Performingapairedt-testacrossthe20sim-
ulations gives the mean difference between these values as 0.097. 0.25
Thisdifferenceisstatisticallysignificantwithp < 0.001andwith Unhedged, pp = 0.56Contraction hedged, pp = 0.57
95% confidence interval [0.084,0.110]. The median and range of
Figure 9. Using contraction hedges reduces minimum APD. Box and
results are given in figure 9. At pp = 0.57, ALO decreases, from whiskerplotofAPDafter104timestepsfor20simulations,forpp = 0.56
0.92to0.89(figure11).Themeanvalueofthisdifferenceacrossthe
unhedged,pp = 0.57withahighprobabilityofcontractionhedges(values
20 simulations is 0.032. Again, this is statistically significant with
ofppatwhichminimumAPDisachieved).Themiddlelineshowsmedian
p < 0.001 and 95% confidence interval of [0.029,0.035], further
value,theboxshowsthe25thand75thpercentile.Whiskersshowtherange
illustratedinfigure10.Theseresultsimplythatahighpriorproba-
ofdataexcludingoutliers,andcrossesshowoutliers.
bilityofassertingcontractionhedgesenablesustoimproveconver-
gencebetweenagents’labelsetsaswellasreducingoverlapwithin
labelsetsslightly.
0.925
0.92
1
0.915
0.9
0.91
0.8
0.905
O
0.7 L
A 0.9
0.6 0.895
D
P0.5 0.89
A
0.4 0.885
0.88
0.3
0.875
0.2
Unhedged Contraction hedged
pp = 0.57
Unhedged
0.1 Contraction Hedges
Expansion Hedges Figure 10. Using contraction hedges slightly reduces ALO. Box and
0
0.45 0.5 0.55 whiskerplotofALOafter104timestepsfor20simulations,forpp=0.57.
pp
Themiddlelineshowsmedianvalue,theboxshowsthe25thand75thper-
Figure8. APDafter104timesteps.Usingcontractionhedgesreducesthe centile.Whiskersshowtherangeofdataexcludingoutliers,andcrossesshow
minimumAPDachievedfrom0.38atpp = 0.56to0.29atpp = 0.57. outliers.
ExpansionhedgesreduceAPDfrom0.85to0.73atpp=0.45
steadystate.Figure14showsthatatshorttimescales(t<2000),bet-
terconvergencemaybeachievedallowingonlyunhedgedassertions.
Withahighpriorprobabilityofassertingexpandedlabels,lower
Inamoreextremecase,figure15showsthatforpp=0.5,betterper-
valuesofALOcanbeachievedwhentheprobabilityofassertingpos-
formanceontheALOmetricisonlyachievedafter7500timesteps.
itivelabelsis0.45,decreasingto0.02comparedto0.1,figure11.
Although this improvement takes a longer time to achieve, it goes
Themeandifferencebetweenthesevaluesacrossthe20simulations
togetherwithimprovedperformanceonAPDwhichisachievedina
is0.083,whichisstatisticallysignificantwithp<0.001anda95%
similartimescaletotheunhedgedmodel16.
confidenceintervalof[0.078,0.089].Thedataisrepresentedinfig-
ure13.Atthisvalueofpp,APDachievedis0.73comparedto0.85
forunhedgedassertions,figure8.Themeanvalueofthisdifference 4 DISCUSSION
acrossthe20simulationsis0.12.Thisfigureisstatisticallysignif-
icant with p < 0.001 and 95% confidence interval [0.116,0.126]. Theseresultsshowthat,inamodeloflanguagedevelopmentacross
Thedataisagainrepresentedinfigure12.Ahighpriorprobabilityof apopulation,hedgedassertionscanimproveboththelevelofcon-
assertingexpansionhedgesthereforeenablesminimaloverlaptobe vergence to shared language as measured by average pairwise dif-
maintainedatlowppwhilstimprovingconvergence. ferencebetweenlabelsets(APD)and,toanextent,thediscrimina-
Wecanalsoexaminehowfastthecommunityofagentsarrivesata torypowerofindividuals’labelsets,asmeasuredbyaveragelabel
1 0.11
0.9 0.1
0.8 0.09
0.7 0.08
0.07
0.6
O
LO0.5 AL0.06
A
0.05
0.4
0.04
0.3
0.03
0.2
0.02
Unhedged
0.1 Contraction Hedges 0.01
Expansion Hedges
0 Unhedged Expansion hedged
0.45 0.5 0.55 pp = 0.45
pp
Figure13. ExpansionhedgesreduceminimumALO.Boxandwhiskerplot
Figure 11. Contraction hedges slightly reduce high levels of ALO. At
ofALOafter104 timestepsfor20simulations,forpp = 0.45.Themid-
pp = 0.57, ALO is reduced from 0.92 to 0.89. Expansion hedges re-
dle line shows median value, the box shows the 25th and 75th percentile.
duceminimumALOfrom0.1forunhedgedassertionsto0.02forexpansion
Whiskersshowtherangeofdataexcludingoutliers,andcrossesshowout-
hedgedassertions,atpp=0.45.
liers.
0.86
1
0.84 0.9
0.82 0.8
0.7
0.8
D 0.6
P
A0.78 D
P0.5
A
0.76 0.4
0.3
0.74
0.2
0.72 Unhedged
0.1 Contraction Hedges
Expansion Hedges
Unhedged pp = 0.45Expansion hedged 00 2000 4000 6000 8000 10000
timestep
Figure12. ExpansionhedgesreducemaximumAPD.Boxandwhiskerplot
ofAPDafter104 timestepsfor20simulations,forpp = 0.45.Themid- Figure14. APDvstimeformodelswithnohedgedassertions,contracted
assertionsandexpandedassertions,forapriorprobabilityofpositiveasser-
dle line shows median value, the box shows the 25th and 75th percentile.
tionspp = 0.56.Althoughthefinalvaluereachedislowerwhenthereisa
Whiskersshowtherangeofdataexcludingoutliers,andcrossesshowout-
highprobabilityofmakingcontractedassertions,thecommunityofagents
liers.
takeslongertoreachthatvalue.
overlap (ALO). The two different types of hedges improve perfor-
manceindistinctways.Ifoverallconvergenceisimportant,ahigh manceonALOgoestogetherwithimprovedperformanceonAPD
prior probability of asserting contraction hedges should be used to whichisattainedatthesamespeedastheintheunhedgedmodel.
improveperformanceontheAPDmetric.Conversely,iftheabilityof Ifthespeedofdevelopmentofsharedcategoriesisnotimportant,
theagentstodiscriminatepreciselybetweenobjectsintheenviron- the two types of hedges would be useful in different types of sit-
mentismoreimportant,thenexpansionhedges,togetherwithlower uation, depending whether convergence or discriminatory power is
probabilitiesofassertingpositivelabels,shouldbeusedtomaintain moreimportant.Thismightbedependenton,forexample,thestruc-
lowlevelsofALOwhilststillimprovingperformanceonAPD. ture of the underlying environment. In the current simulation, ob-
Theimprovedperformanceagainstthetwometricsistemperedby jectsarepresenteduniformlyacrossthespace.Ifobjectsweredis-
thefactthatthespeedatwhichthesteadystateisachievedissome- tributednon-uniformly,perhapsclumpinginvariousregionsofthe
whatslowerthanwhenusingsimplyunhedgedassertions.However, space,thenperhapstheabilitytodiscriminatepreciselybetweendif-
the improvement in APD is seen relatively quickly at pp = 0.56, ferentlabelswouldbelessimportant,sincetheenvironmentprovides
soonaftertheunhedgedmodelhasreacheditssteadystate.Theim- thatdistinctionnaturally.Convergencetosharedlabelswouldthenbe
provementinALOwhenusingexpansionhedges,forpp=0.5,does moreimportant.
notoccuruntilafter7,500timesteps,wellaftertheunhedgedmodel Ifspeedisimportant,usingcontractionhedgescanstillimprove
has reached its steady state. However, the improvement in perfor- levelsofconvergenceinarelativelyshorttimeframe.
in two distinct ways. Firstly, using contraction hedges, i.e. words
1
Unhedged like ‘very’, allows improved levels of convergence to shared cate-
0.9 CExopnatrnascitoionn H Heeddggeess gories,whilstslightlyimprovingtheextenttowhichlabelsoverlap.
0.8 Secondly,usingexpansionhedges,orwordslike‘quite’,enablesthe
developmentoflabelsetsthataremorediscriminatoryoftheenvi-
0.7
ronmentandalsohavebetterlevelsofconvergence.However,both
0.6 these improvements come with a slower speed of development of
O sharedlabels.Itmaybepossibletoimprovethesespeedsbytuning
L0.5
A other parameters such as the age range of agents or the values of
0.4 hedgesused.
0.3
ACKNOWLEDGEMENTS
0.2
0.1 MarthaLewisgratefullyacknowledgessupportfromEPSRCGrant
No.EP/E501214/1
0
0 2000 4000 6000 8000 10000
timestep
REFERENCES
Figure15. ALOvstimeformodelswithnohedgedassertions,contracted
[1] P.Bosc,D.Dubois,A.HadjAli,O.Pivert,andH.Prade,‘Adjustingthe
assertionsandexpandedassertions,forapriorprobabilityofpositiveasser-
coreand/orthesupportofafuzzyset-anewapproachtofuzzymod-
tionspp = 0.5.Althoughthefinalvaluereachedislowerwhenthereisa
ifiers’, in Fuzzy Systems Conference, 2007. FUZZ-IEEE 2007. IEEE
highprobabilityofusingexpansionhedges,thecommunityofagentstakes International,pp.1–6.IEEE,(2007).
longertoreachthatvalue,andmayevenreachalowervaluestill. [2] HenriettaEyreandJonathanLawry,‘Languagegameswithvaguecate-
goriesandnegations’,AdaptiveBehavior,1059712314547318,(2014).
[3] P.Ga¨rdenfors,Conceptualspaces:Thegeometryofthought,TheMIT
1 Press,2004.
[4] J.Hampton,‘Inheritanceofattributesinnaturalconceptconjunctions’,
0.9
Memory&Cognition,15(1),55–71,(1987).
0.8 [5] J.Hampton,‘Conceptualcombinationsandfuzzylogic’,inConcepts
andFuzzyLogic,eds.,R.BelohlavekandG.J.Klir,TheMITPress,
0.7 (2011).
[6] H.KampandB.Partee,‘Prototypetheoryandcompositionality’,Cog-
0.6
nition,57(2),129–191,(1995).
D [7] J.Lawry,‘Aframeworkforlinguisticmodelling’,ArtificialIntelligence,
P0.5
A 155(1-2),1–39,(2004).
0.4 [8] J. Lawry and Y. Tang, ‘Uncertainty modelling for vague concepts:
Aprototypetheoryapproach’,ArtificialIntelligence,173(18),1539–
0.3 1558,(2009).
[9] Martha Lewis and Jonathan Lawry, ‘A label semantics approach to
0.2
linguistic hedges’, International Journal of Approximate Reasoning,
Unhedged 55(5),1147–1163,(2014).
0.1 Contraction Hedges
[10] D.N.OshersonandE.E.Smith,‘Ontheadequacyofprototypetheory
Expansion Hedges
0 asatheoryofconcepts’,Cognition,9(1),35–58,(1981).
0 2000 4000 6000 8000 10000
timestep [11] E.Rosch,‘Cognitiverepresentationsofsemanticcategories.’,Journal
ofExperimentalPsychology:General,104(3),192,(1975).
Figure16. APDvstimeformodelswithnohedgedassertions,contracted [12] E.E.SmithandD.N.Osherson,‘Conceptualcombinationwithproto-
typeconcepts’,CognitiveScience,8(4),337–361,(1984).
assertionsandexpandedassertions,forapriorprobabilityofpositiveasser-
[13] LucSteels,‘Whyweneedevolutionarysemantics’,KI2011:Advances
tions pp = 0.5. Lower APD is achieved with a high prior probability of inArtificialIntelligence,14–25,(2011).
assertingexpansionhedges,inasimilartimescaletotheunhedgedmodel [14] LucSteels,TonyBelpaeme,etal.,‘Coordinatingperceptuallygrounded
categoriesthroughlanguage:Acasestudyforcolour’,Behavioraland
Therearemanyparametersinthesimulationthatbearfurtherin- brainsciences,28(4),469–488,(2005).
[15] L.A. Zadeh, ‘Fuzzy sets’, Information and Control, 8(3), 338–353,
vestigation.Thedistributionofobjectsintheenvironment,asmen-
(1965).
tioned above, is likely to have an effect on performance again the [16] L.A.Zadeh,‘Afuzzy-set-theoreticinterpretationoflinguistichedges’,
twometrics.Inthecurrentsimulations,hedgevaluesofv=0.5and JournalofCybernetics,(1972).
h=2areused.Increasinganddecreasingthesevaluescouldhavean
impactonperformance,aswould,perhaps,allowingagentstohave
difference values of v and h. The range of w allowed also affects
performance.Whenw=[0.01,0.99],agentsnolongerachievehigh
levelsofconvergenceatpp>0.55(resultsnotshown).Otherweight
rangesmaypositivelyaffectperformance,however.
5 CONCLUSIONS
Wehaveinvestigatedtheutilityofhedgedassertionsinthedevelop-
mentofasharedlanguage,andshownthatallowingagentstomake
hedgedassertionsimprovestheabilitytodevelopcommoncategories