vagueness facilitates search

Kees van Deemter (AC, Dec 09)

Vagueness Facilitates Search

Kees van Deemter

Computing Science

University of Aberdeen


Two “big” problems with vagueness

A predicate is vague (V) if it hasborderline cases or degrees

1. The semantic problem: How to model the meaning of V expressions?

– Classical models: 2-valued– Partial models: 3-valued– Degree models: many-valued

(e.g. Fuzzy Logic, Probabilistic Logic)

No agreement how to answer this question (Keefe & Smith 1997, Van Rooij (to appear))


The pragmatic problem:

2. Why is language vague? (Variant: When & Why use vague expressions?)

B.Lipman (2000, 2006): “Why have we tolerated a world-wide several-thousand-year efficiency loss?”


The pragmatic problem:

Why is language vague? Speaker ...

– hides information – only has uncertain or vague information– lack an objective metric– reduces processing cost– expresses an opinion– aids understanding or gist memory

(Lipman 2000, 2006, Veltman 2000, de Jaegher 2003, Jäger 2003, van Rooij 2003, Peters et al. 2009)


This talk

• No verdict on these earlier answers van Deemter 2009 [JPL survey]

• Explore tentative new answer: V can “oil the wheels” of communication

• Starting point: it’s almost inconceivable that all speakers arrive at exactly the same concepts


Causes of semantic mismatches

• Perception varies per individual– Hilbert 1987 on colour terms: density of pigment

on lens & retina; sensitivity of photo receptors

• Cultural issues. – Reiter et al. 2005 on temporal expressions.

Example: “evening”: Are the times of dinner and sunset relevant?

• R.Parikh (1994) recognised that mismatches exist …


• Parikh proposed utility-oriented perspective on meaning– Utility as reduction in search effort

• showed communication doesn’t always break down when words are understood (slightly) differently by different speakers


Blue books (Bob)

Blue books (Ann)

25

75

225

Ann: “Bring the blue book on topology”

Bob: Search [[blue]]Bob, then, if necessary, all other books

(only10% probability!)

675


What Parikh did not do: show utility of V – Ann and Bob used crisp concepts!

This talk:

• “tall” instead of “blue”. 2-dimensional [[tall1]] [[tall2]] or [[tall2]] [[tall1]]

• Focus on V


The story of the stolen diamond

“A diamond has been stolen from the Emperor and (…) the thief must have been one of the Emperor’s 1000 eunuchs. A witness sees a suspicious character sneaking away. He tries to catch him but fails, getting fatally injured (...). The scoundrel escapes. (…) The witness reports “The thief is tall” , then gives up the ghost. How can the Emperor capitalize on these momentous last words?”

(book, to appear)


The problem with dichotomies

• Suppose Emperor uses a dichotomy, e.g. Model A:

[[tall]]Emperor = [[>180cm]] (e.g., 500 people)

• What if [[tall]]Witness = [[>175cm]]?

If thief [[tall]]Witness - [[tall]]Emperor

then Predicted search effort: 500+ ½(500)=750Without witness’ utterance: ½(1000)=500

• The witness’ utterance “misled” the Emperor

180cm

thief

175cm

A


• Model A uses a crisp dichotomy between [[tall]]A and [[tall]]A

• Contrast this with a partial model B, which has a truth value gap [[?tall?]]B


Model A Model B

tallA

tallA

tallB

?tall?B

tallB

2-valued 3-valued

180cm 180cm

165cm


How does the Emperor classify the thief?

Three types of situations

s(X) =def expected search time given model X

Type 1: thief [[tall]]A = [[tall]]B

In this case, s(A)=s(B)


How does the emperor classify the thief?

Type 2: thief [[?tall?]]B

model A: search all of [[tall]]A in vain, then (on average) half of [[tall]]A

model B: search all of [[tall]]B in vain, then (on average) half of [[?tall?]]B

B searches ½(card([[tall]]B)) less!

So, s(B)<s(A)


How does the emperor classify the thief?

Type 3: thief [[tall]]B

model A: search all of [[tall]]A in vain, then (on average) half of [[tall]]A

model B: search all of [[tall]]B in vain, then all of [[?tall?]]B

then (on average) half of [[tall]]B

Now A searches ½(card([[?tall?]]B)) less.So, s(A)<s(B)


“Normally” model B wins

• “Type 2 is more probable than Type 3”

• Let S = xy((x[[?tall?]]B & y[[tall]]B) p(“tall(x)”) > p(“tall(y)”)

• S is highly plausible! (Taller individuals are more likely to be called tall)



S: xy ((x[[?tall?]]B & y[[tall]]B) p(“tall(x)”) > p(“tall(y)” )

S implies

xy ((x[[?tall?]]B & y[[tall]]B) p(thief(x)) > p(thief(y) )

• It pays to search [[?tall?]]B before [[tall]]B

• A priori, s(B) < s(A)


Why does model B win?

• 3-valued models beat 2-valued models because they distinguish more finely

• Therefore, many-valued models should be even better!– All models that allow degrees or ranking

(including e.g. Kennedy 2001)


Consider degree model C

• v(tall(x)) [0,1] (e.g., Fuzzy or Probabilistic Logic)

xy ((v(tall(x)) > v((tall(y)) p(“tall(x)”) > p(“tall(y)”)

height(x)

probability of x being called “tall”



• Analogous to the Partial model

• C allows the Emperor to– rank the eunuchs, and – start searching the tallest ones, etc.

• Assuming >3 differences in height (i.e., more than 3 differences in v(tall(x)), this is even quicker than B: s(C) < s(B)


An example of model C

v(tall(a1)=v(tall(a2))=0.9v(tall(b1)=v(tall(b2))=0.7v(tall(c1)=v(tall(c2))=0.5v(tall(d1)=v(tall(d2))=0.3v(tall(e1)=v(tall(e2))=0.1

• Search {a1,a2} first, then {b1,b2}, etc. (5 levels)• This is quicker than a partial model (3 levels)• which is quicker than a classical model (2 levels)


This analysis suggests …


This analysis suggests …

• ... that it helps the Emperor to understand “tall” as having borderline cases or degrees

• But, borderline cases and degrees are the hallmark of V

• It appears to follow that V has benefits for search


This analysis also suggests …

... that degree models offer a better understanding of V than partial models– Compare the first “big” problem with V

• Radical interpretation: V concepts don’t serve to narrow down search space, but to suggest an ordering of it


Objections …


Objections …

• Objection 1: “Smart search would have been equally possible based on a dichotomous (i.e., classical) model”.


Objections …


• The idea: You can use a classical model,yet understand that other speakers use other classical models. Start searching individuals who are tall on most models.


Objections …


• The idea: you can use a classical model yet understand that other speakers use other classical models. Start searching individuals who are tall on most models

• Response: Reasoning about different classical models is not a classical logic but a Partial Logic with supervaluations. Presupposes that “tall” is understood as vague.


Objections …

• Objection 2: “Truth value gaps (or degrees) do not imply vagueness”.


Objections …

• Objection 2: “Truth value gaps (or degrees) do not imply vagueness”.

• The idea: Truth value gaps that are crisp fail to model V. (Similar for degrees.) Higher-order V needs to be modelled.


Objections …

• Objection 2: “A truth value gap (or degrees) does not imply vagueness”.

• The idea: Truth value gaps that are crisp fail to model V. (Similar for degrees.) Higher-order V needs to be modelled.

• Response: few formal accounts of V pass this test


Objections …

• Objection 3: “Why was the witness not more precise?”


Objections …


• The idea: Witness should have said “the thief was 185cm tall”, giving an estimate


Objections …


• The idea: Witness should have said “the thief was 185cm tall”, giving an estimate

• Response: (1) Why are such estimates vague (“approximately 185cm”) rather than crisp (“185cm =/- 0.5cm”)? (2) This can be answered along the lines proposed.


Objections …

• Objection 4: “This contradicts Lipman’s theorem”


Objections …


• The idea: Lipman (2006) proved that every V predicate can be replaced by a crisp one that has higher utility.


Objections …


• The idea: Lipman (2006) proved that every V predicate can be replaced by a crisp one that has higher utility.

• Response: Lipman’s assumptions don’t apply: (1) Theorem models V through probability distribution. (2) Theorem assumes that hearer knows what crisp model the speaker uses (e.g. “>185cm”).


Lipman’s theorem assumes that hearer knows what crisp model the speaker uses

Our starting point: in “continuous” domains, perfect alignment between speakers/hearers would be a miracle

(pace epistemicist approaches to V !)


“Not Exactly: in Praise of Vagueness”.

Oxford University Press, Jan. 2010

Part 1: Vagueness in science and daily life.

Part 2: Linguistic and logical models of vagueness.

Part 3: Working models of V in Artificial Intelligence.

(Related to his talk: pp. 187-189; 269-271)


Empirical tests of various hypotheses concerning the effects of vague expressions (aiding understanding or gist memory):

E.Peters, N.Dieckmann, D.Västfjäll, C.Mertz, P.Slovic, and J.Hibbard (2009). “Bringing meaning to numbers: the impact of evaluative categories on decisions”. J. Experimental Psychology 15 (3): 213-227


Appendix

• Extra slides



Let S abbreviate:

p(t[[?tall?]]B | “tall(t)”) > p(t[[tall]]B | “tall(t)”)

For example, S is plausible if

card([[?tall?]]B)=card([[tall]]B)


Example (draft)

Let [[tall]]={a,b,c,d} p(thief [[tall]]) = 1/2 [[?tall]]={e,f} p(thief [[?tall?]]) = 1/4 [[tall]]={g,h,i,j} p(thief [[tall]]) = 1/4

thens(<+,?,->)= 1/2*(2)+1/4*(4+1)+1/4*(6+2)= 4.25

s(<+,-,?>)= 1/2*(2)+1/4*(4+2)+1/4*(8+1)= 4.75

s(<-,?,+>)= 1/4*(2)+1/4*(4+1)+1/2*(6+2)= 5.75



• In C: v(tall(x)) [0,1] (e.g., Fuzzy or Probabilistic Logic)

• S’= ab: a>b p(“tall(x)” | v(tall(x)=a)) > p(“tall(x)” | v(tall(x)=b))

height(x)

probability of x being called “tall”


rich=earn more than 106 EUR

(1) xy: ( x South & y North p(rich(x)) > p(rich(y)) )

(2) x: (rich(x) oil(x))

Therefore

(3) xy: ( x South & y North p(oil(x)) > p(oil(y)) )


End

vagueness facilitates search

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