relating label semantics and prototype theory · relating label semantics and prototype theory –...

Relating Label Semantics andPrototype Theory

Jonathan Lawry and Yongchuan Tang

University of Bristol and Zhejiang University

Relating Label Semantics and Prototype Theory – p. 1/24

Outline

Prototype theory of concepts.

Prototype interpretation of fuzziness.

Random set interpretation of fuzziness.

Background to label semantics.

A prototype based interpretation of appropriatenessmeasures.

Conclusions.


Concept Representation

In logic (e.g. Kripke Semantics) concepts arerepresented as mappings from a set of possible worldsinto sets of element, each corresponding to a possibleextension of the concept.

This fails to take account of the role similarity plays inestablishing the meaning of concept labels.

In Prototype theory (Rosch 1975) concepts are definedin terms of similarity to prototypical points (in aconceptual space).


Prototype Interpretation of Fuzzy Sets

Prototype similarity has been proposed as aninterpretation of fuzzy membership functions.

A similarity measure S is defined on the underlyingdomain, taking values in [0, 1].

Ruspini (1991): Membership of x in Li corresponds tosupS(x, y) : y ∈ Pi, where Pi is the set of prototypes ofLi.

Resulting calculus is not truth-functional.


Random Set Interpretation of Fuzzy Sets

Seminal work of Nguyen and Goodman proposed arandom set interpretation of fuzzy sets.

The extensions of vague concepts are represented byrandom sets of the underlying domain characterisinguncertain boundaries.

Membership functions then correspond to single pointcoverage functions of random sets; i.e. the probabilitythat an element is a member of the random set.

Resulting calculus is not truth-functional.

Labels semantics is also a random set model but wherethe random sets are on labels rather than values.

This new work links random set and prototype theory.


The Label Semantics Framework

Label Semantics provides an alternative model of theuncertainty associated with the use of vague linguisticlabels to describe objects/values.

The basis of the model is in judging theappropriateness or assertibility of vague expressions todescribe an object or an instance.

Appropriateness in this context is governed by theconventions of language use emergent from apopulation of communicating agents.

An agent’s knowledge of appropriateness is based onpartial evidence concerning the previous use of labelsand is consequently uncertain.


The Epistemic Stance

Label semantics (and other random set models)assumes that communicating agents adopt anepistemic stance regarding labels.

Within a population of communicating agents, individuals assumethe existence of a set of labelling conventions for the populationgoverning what linguistic labels and expression can be appropriatelyused to describe particular instances.

Making an assertion to describe an object or instance x

involves making a decision as to what labels can beappropriately used to describe x.


Measures of Appropriateness

Assume that there is a finite set of labelsLA = L1, . . . , Ln for describing elements of theuniverse Ω.

LE is the set of expressions generated from LA throughrecursive application of the connectives ∧,∨ and ¬.

For θ ∈ LE, x ∈ Ω, µθ (x) = the subjective probabilitythat θ is appropriate to describe x.


Mass Functions

Dx is the complete set of labels appropriate to describex.

Dx is a random set into 2LA.

mx : 2LA → [0, 1] is a probability mass function onsubsets of labels.

For S ⊆ LA mx(S) is the subjective probability thatDx = S.

The mass function mx and the appropriatenessmeasure µ are strongly related...

µθ(x) is the sum of mx over those values for Dx

consistent with θ


General Relationships

‘x is θ’ requires that Dx ∈ λ (θ) where ∀θ, ϕ ∈ LE

∀L ∈ LA λ(L) = S ⊆ LA : L ∈ S (= S ∈ F : L ∈ S)

λ(θ ∧ ϕ) = λ(θ) ∩ λ(ϕ), λ(θ ∨ ϕ) = λ(θ) ∪ λ(ϕ)

λ(¬θ) = λ(θ)c

This results in the following equation relating mx as µ:

µθ(x) = P (Dx ∈ λ(θ)) =∑

S∈λ(θ) mx(S)

Functionality:

By functionality we mean that for any θ there is afunction fθ : [0, 1]n → [0, 1] such that ∀x

µθ(x) = fθ(µL1(x), . . . µLn

(x)).

Appropriateness measures are not in general functionalexcept under certain conditions . . .


Ordering on Labels

Agents rank the labels LA in terms of theirappropriateness according to the ordering x, so thatL x L′ means that L′ is judged at least as appropriateas L for describing x.

If L x L′ then L ∈ Dx implies that L′ ∈ Dx.

If x is a total ordering then possible values of Dx forma nested hierarchy i.e. Dx is a consonant random set.

If x is a partial ordering then...


The Consonance Assumption

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

small medium large

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

µL1(x), . . . , µLn(x) ordered so that

µLi(x) ≥ µLi+1

(x) for i = 1, . . . , n − 1

mx (L1, . . . , Ln) = µLn(x)

mx (L1, . . . , Li) = µLi(x) − µLi+1

(x)

and mx (∅) = 1 − µL1(x)

mx (s)

mx (s, m)

mx (m)

mx (m, l)

mx (l)

µm∧¬l (x)

min (µm (x) , 1 − µl (x))


Properties under Consonance

If |= θ then ∀x, µθ(x) = 1

If θ ≡ ϕ then ∀x, µθ(x) = µϕ(x)

∀θ,∀x, µ¬θ(x) = 1 − µθ(x)

∀θ, ϕ ∈ LE∧,∨, ∀x ∈ Ω it holds thatµθ∧ϕ(x) = min(µθ(x), µϕ(x)) andµθ∨ϕ(x) = max(µθ(x), µϕ(x))

Law of Excluded Middle: λ(θ) ∪ λ(θ)c = 22LA

therefore∀x µθ∨¬θ(x) = 1

Law of Non-contradiction: λ(θ) ∪ λ(θ)c = ∅ therefore∀x µθ∧¬θ(x) = 0


A Prototype Interpretation

Let d : Ω2 → [0,∞) be a distance metric satisfyingd(x, x) = 0 and d(x, y) = d(y, x).

For Li ∈ Ω let Pi ⊆ be a set of prototypical elements.

For x ∈ Ω let d(x, Pi) = supd(x, y) : y ∈ Pi

Let ǫ be a random variable into [0,∞) with densityfunction δ.

Li is appropriate to describe x iff d(x, Pi) ≤ ǫ.

Dǫx = Li : d(x, Pi) ≤ ǫ.

∀F ⊆ Ω mx(F ) = δ(ǫ : Dǫx = F)

Li x Lj iff d(x, Pi) ≥ d(x, Pj) (a total ordering)


Appropriateness of Basic Labels

µLi(x) = δ(ǫ : d(x, Pi) ≤ ǫ) = δ([d(x, Pi),∞)).

Example Li = about 5 and Pi = 5

0 0.5 1.0 1.5

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

k r ǫ

1r−k

δ

µLi(x)

2rx

2k


GeneratingDǫx

x

ǫ3ǫ2

ǫ1p1

p2

p3

p4

p5p6

p7

Pi = pi : i = 1, . . . , n: Identifying Dǫx; Dǫ1

x = ∅, Dǫ2x = L1, L2, Dǫ3

x = L1, L2, L3, L4


Appropriateness as Intervals ofǫ

∀x ∈ Ω and ∀θ ∈ LE, I(θ, x) ⊆ [0,∞) is definedrecursively by:

∀Li ∈ LA I(Li, x) = [d(x, Pi),∞)

∀θ ∈ LE I(¬θ, x) = I(θ, x)c

∀θ, ϕ ∈ LE I(θ ∨ ϕ, x) = I(θ, x) ∪ I(ϕ, x)

∀θ, ϕ ∈ LE I(θ ∧ ϕ, x) = I(θ, x) ∩ I(ϕ, x)

Theorem: ∀θ ∈ LE,∀x ∈ Ω I(θ, x) = ǫ : Dǫx ∈ λ(θ)

Corollary: ∀θ ∈ LE,∀x ∈ Ω µθ(x) = δ(I(θ, x))


Area under δ: Appropriateness

I(Li ∧ ¬Lj , x) = [d(x, Pi), d(x, Pj))

ǫd(x, Pi) d(x, Pj)

µLi∧¬Lj(x)

δ(ǫ)


Area under δ: Mass

mx(F ) = δ([maxd(x, Pi) : Li ∈ F, mind(x, Pi) : Li 6∈ F))

ǫ

δ(ǫ)

d(x, P1) d(x, P2) d(x, P3) d(x, P4)

mx(∅)

mx(L1)

mx(L1, L2)

mx(L1, L2, L3)

mx(L1, L2, L3, L4)


Another Perspective

Consider a neighbourhood of label prototypes Pi:N ǫ

Li= x : d(x, Pi) ≤ ǫ = x : Li ∈ Dǫ

x

= x : Li is appropriate to describe x.

The ǫ-extension of the concept Li is the set of pointsǫ-similar to the prototype(s) of Li.

Extended to any expression θ:N ǫ

θ = x : Dǫx ∈ λ(θ) = x : ǫ ∈ I(θ, x)

Appropriateness measures: (Theorem)µθ(x) = P (x ∈ N ǫ

θ ) = δ(ǫ : x ∈ N ǫθ)

Viewing N ǫθ as a random set then µθ(x) is the single

point coverage function for N ǫθ .

If θ ∈ LE∧,∨ then ∀ǫ ≤ ǫ′ N ǫθ ⊆ N ǫ′

θ .


Prototype Neighbourhoods

Pi

N ǫLi

pi

pj

N ǫLi∧¬Lj


Recursive Definition ofN ǫθ

The neighbourhoods N ǫθ can also be defined directly in

a recursive manner:

For Li ∈ LA N ǫLi

= x ∈ Ω : d(x, Pi) ≤ ǫ

For θ, ϕ ∈ LE N ǫθ∧ϕ = N ǫ

θ ∩N ǫϕ

For θ, ϕ ∈ LE N ǫθ∨ϕ = N ǫ

θ ∪N ǫϕ

For θ ∈ LE N ǫ¬θ = (N ǫ

θ )c

Hence, appropriateness measures can be viewed assingle point coverage functions of random sets into 2Ω,generated recursively from neighbourhoods ofprototypes according to standard boolean combinationrules.


A General Model

For each label Li we define a distinct metricdi : Ω → [0,∞) and threshold ǫi.

δ(ǫ1, . . . , ǫn) is then the joint density on all thresholdvariables.

D~ǫx = Li : di(x, Pi) ≤ ǫi and µθ(x) = δ(~ǫ : D~ǫ

x ∈ λ(θ))

Independence: δ(ǫ1, . . . , ǫn) =∏n

i=1 δi(ǫi) andmx(F ) =

∏Li∈F µLi

(x) ×∏

Li 6∈F (1 − µLi(x))

Consonance: ǫi = fi(ǫ) where fi : [0,∞) → [0,∞) is anincreasing function.


Conclusion

A prototype interpretation of label semantics has beenintroduced.

Appropriateness measures are interpreted as theprobability that a distance threshold ǫ lies with aparticular interval of [0,∞) as determined by therelevant expression.

ǫ is a random variable representing the upper bound ond(x, Pi) at which Li can still be deemed appropriate todescribe x.

Appropriateness measures can also be defined interms of random set neighbourhoods of prototypes. i.e.Li = approximately Pi


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