Concept Generation in Language Evolution

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by  Martha Lewis

Concept Generation in Language Evolution
MarthaLewis,JonathanLawry
DepartmentofEngineeringMathematics,UniversityofBristol,BS81TR,UK
[email protected],[email protected]
Abstract upofqualitydimensionsandequippedwithadistancemetric,
6
1 forexampletheRGBcolourspace.
Thisthesisinvestigatesthegenerationofnewcon-
0 Label semantics proposes that agents use a set of labels
cepts from combinations of existing concepts as a
2 LA = {L ,...,L } to describe a conceptual space Ω with
1 n
language evolves. We give a method for combin- distancemetricd(x,y). LabelsL areassociatedwithproto-
n i
ing concepts, and will be investigating the utility
a typesPi ⊆ Ωanduncertainthresholdsεi,drawnfromprob-
of composite concepts in language evolution and
5 J thencetheutilityofconceptgeneration. athbailtitayndeisletrmibeunttioxns∈δεiΩ. TishesuthffirecsiehnotlldyεcilocsaepttuorePsithteonboetiloan-
2 belledLi. TheappropriatenessofalabelLi todescribexis
1 Introduction quantifiedbyµLi(x),givenby
]
I Humans are skilled at making sense of novel combinations
A (cid:90) ∞
of concepts, so to create artifical languages for implemen-
µ (x)=P(d(x,P )≤ε )= δ (ε )dε
. tation in AI systems, we must model this ability. Stan- Li i i εi i i
s d(x,Pi)
c dard approaches to combining concepts, e.g. fuzzy set
[ theory, have been shown to be inadequate [Osherson and LabelscanthenbedescribedasLi =<Pi,d(x,y),δεi>.
Smith, 1981]. Composite labels frequently have ‘emergent
1
v attributes’ [Hampton, 1987] which cannot be explicated by 3 ANewModelofConceptComposition
2 decomposing the label into its constituent parts. We argue
Experiments in [Hampton, 1987] propose that human con-
3 thatinthiscaseanewconceptisgenerated.Thisprojectaims
7 to determine conditions for such concept generation, using cept combination can (roughly) be modelled as a weighted
6 multi-agentmodelsoflanguageevolution. sum of attributes such as ‘has feathers’, ‘talks’ (for the con-
0 cept‘Bird’). Theseattributesdifferfromqualitydimensions
. 1.1 ThesisOutline in conceptual spaces: they tend to be binary, complex, and
1
multidimensional. Wethereforevieweachattributeasalabel
0 The project divides into three parts. Firstly, we have devel-
inaconceptualspaceΩ andcombinetheselabelsinabinary
6 oped a model of concept combination within the label se- i
v:1 mmaondteilcsisfirnasmpeirweodrbkyasangdivreenfleinct[sLraewsuryltsanind[THaanmg,p2to0n0,91].98T7h]e, sspuacchela{b0e,l1s}α˜n=ill(cid:86)usnit=ra1te±dLiinmfiagpusreto1a, wbihnearreyaveccotonrju(cid:126)xnαcttiaokninogf
value 1 for positive labels L and 0 for negated labels ¬L .
i inwhichmembershipinacompositeconceptcanberendered i i
X Wetreatmembershipinα˜inthebinaryspacewithinthelabel
astheweightedsumofmembershipsinindividualconcepts.
semanticsframework. Soα˜ isdescribedinthebinaryspace
r Secondly, wemustshowthatcompositionalitycanevolve
a byα˜ =<(cid:126)x ,d((cid:126)x,(cid:126)x(cid:48)),δ>asbefore.
within a population of interacting agents. Preliminary work α
inthisareaexaminestheabilityofapopulationofagentsto
convergetoasharedsetofdimensionweights.
{0,1}n
Thirdly, we will investigate the generation of new unitary
concepts from existing composite concepts, building further
uponthemulti-agentmodel.
2 Background Ω1 Ω2 Ωn
Thisworkisbasedonthelabelsemanticsframework[Lawry,
2004;LawryandTang,2009],togetherwithprototypetheory Figure1: Combininglabelsinabinaryspace
[Rosch, 1975], where membership in a concept is based on
proximitytoaprototype,andconceptualspaces[Ga¨rdenfors, Wedefineadistancemetricinthebinaryspace{0,1}nas:
2004]. Thelatterviewsconceptsasregionsofaspacemade
Definition1 WeightedHammingDistance 1 1
SD SD
For (cid:126)λ ∈ (R+)n, ∀(cid:126)x,(cid:126)x(cid:48) ∈ {0,1}n, where (·) is the scalar 0.8 λ 0.8 λ
product,
H ((cid:126)x,(cid:126)x(cid:48))=(cid:126)λ·|(cid:126)x−(cid:126)x(cid:48)| 0.6 0.6
(cid:126)λ
Theorem2 Let α = (cid:86)n ±L and λ = (cid:80)n λ . Let 0.4 0.4
i=1 i T i=1 i
ε∼U(0,λ ),d=H . Then: 0.2 0.2
T (cid:126)λ
µα˜(Y(cid:126))=(cid:88)n λλi µ±Li(Yi) 0 0.6 w0.8 1 0 0.6 w0.8 1
T
i=1 (a) x ∼ U[0,1], x ∼ (b) x ∼ U[0.25,0.75],
1 2 1
Compound concepts θ˜,ϕ˜ may be combined in a higher U[0,0.5]. λ converges to x2 ∼ U[0,0.5]. λ con-
levelbinaryspace. Thenθ˜•ϕ˜canbeexpressedinthecontin- 0.5forallvaluesofw vergestovaryingvalues.
uousspaceasaweightedsumofθ˜andϕ˜. Figure 2: Mean SD and λ at time t = 2000 for different
Theorem3 Letθ˜•ϕ˜=<{(1,1)},H ,δ>.Thenµ (Y(cid:126))= valuesofw. Eachpointaverages25simulationsrunwith10
w(cid:126) θ˜•ϕ˜ agents.
(cid:80)ni=1(w1λϕTwλTθλiθ+Twλ2ϕλTθTλϕi)µ±Li(Y(cid:126)).
We have therefore shown that combining labels in a When w = 1 we can predict the value to which λ will
weightedbinaryspaceleadsnaturallytothecreationofcom- converge. Consider the quantity A − λt which determines
positeandcompoundconceptsasweightedsumsofindivid- whethertheupdateispositiveornegativeateachstep.
ual labels, reflecting results in [Hampton, 1987]. We have Definition4 A positive region R+ ⊂ Ω is a set of points
furthercharacterisednotionsofnecessaryandimpossibleat- R+ ={(cid:126)x∈Ω:A−λ ≥0}
t
tributesusingideasfrompossibilitytheory.
Theorem5 Let p+ denote the probability of a point(cid:126)x ∈ Ω
fallinginapositiveregionandletw = 1acrossthepopula-
4 ConvergenceofDimensionWeightsAcross tion. Thentheexpectedvalueofλconvergestop+.
aPopulation
5 FurtherWork
We investigate how a population of agents in a multi-agent
simulationplayingaseriesoflanguagegamesmightconverge Wearecurrentlyworkingonanalyticalresultstopredictthe
to a shared set of dimension weights. Agents with equal la- value of λ to which agents converge. Under certain circum-
belsL =L =<1,d,U[0,1]>∈Ω =Ω =[0,1](disEu- stances, such as the case where w = 1, or with an altered
1 2 1 2
clidean distance), and randomly initiated weights λ ∈ [0,1] updatingmodel,analyticresultsarepossible. Wewillextend
engage in a series of dialogues about elements in the con- thisworktolookattheutilityofusingconjunctiveassertions
ceptual space, adjusting their weights after each dialogue is withinthesesimulations.
completed. Ateachtimestep,speakeragentsmakeassertions Work in the third year will focus on examining how new
α = ±L ∧±L aboutelements(cid:126)x ∈ Ω ×Ω whichmax- concepts might be generated from the combination of exist-
i 1 2 1 2
imiseµ ((cid:126)x)=λµ (x )+(1−λ)µ (x ). ingones. Wewillbuildonthelanguageevolutionmodelcur-
αi L1 1 L2 2
Thelisteneragentassessesα againstitsownlabelset. If rentlyindevelopment.
i
µ (x) ≤ w,thereliabilityofthespeakeragent,thelistener
αi
agentupdatesitslabelset. References
Theupdateconsistsinincrementingthedimensionweight [Ga¨rdenfors,2004] P. Ga¨rdenfors. Conceptual spaces: The
λtowardsavalueA,sothatλ = λ +h(A−λ )where
t+1 t t geometryofthought. TheMITPress,2004.
h=10−3and
[Hampton,1987] J.A. Hampton. Inheritance of attributes
A= w−µL2(x2) in natural concept conjunctions. Memory & Cognition,
µ (x )−µ (x ) 15(1):55–71,1987.
±L1 1 ±L2 2
[LawryandTang,2009] J. Lawry and Y. Tang. Uncertainty
Thisisthequantitythatsatisfiesµ (x)=w. IfA<0(or
αi modelling for vague concepts: A prototype theory ap-
A>1)wesetA=0(orA=1).
proach. ArtificialIntelligence,173(18):1539–1558,2009.
Theconvergenceacrossthepopulationismeasuredbythe
standarddeviation(SD)oftheλacrossthepopulation. [Lawry,2004] J. Lawry. A framework for linguistic mod-
Figure2showstheresultsoftwosetsofsimulationsacross elling. ArtificialIntelligence,155(1-2):1–39,2004.
varyingvaluesofw.Thetwosetsofsimulationshavedistinct [OshersonandSmith,1981] D.N.OshersonandE.E.Smith.
distributionsofelementsencounteredwithinthespace.When On the adequacy of prototype theory as a theory of con-
wis0.5orbelow,theagentsdonotconvergetoshareddimen- cepts. Cognition,9(1):35–58,1981.
sionweights(notshown).Whenw >0.5,agentsdoconverge
[Rosch,1975] E. Rosch. Cognitive representations of se-
to shared dimension weights: SD is low. The weights con-
mantic categories. Journal of experimental psychology:
vergedtodependbothonthereliability,w,ofeachagent,and
General,104(3):192,1975.
thedistributionofelementsintheconceptualspace.