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## Vagueness Facilitates Search

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**Vagueness Facilitates Search**Kees van Deemter Computing Science University of Aberdeen Kees van Deemter (AC, Dec 09)**Two “big” problems with vagueness**A predicate is vague (V) if it has borderline cases or degrees • 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)) Kees van Deemter (AC, Dec 09)**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?” Kees van Deemter (AC, Dec 09)**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) Kees van Deemter (AC, Dec 09)**This talk**• No verdict on these earlier answersvan 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 Kees van Deemter (AC, Dec 09)**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 … Kees van Deemter (AC, Dec 09)**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 Kees van Deemter (AC, Dec 09)**Blue books (Bob)**75 225 25 Blue books (Ann) 675 Ann: “Bring the blue book on topology” Bob: Search [[blue]]Bob, then, if necessary, all other books (only10% probability!) Kees van Deemter (AC, Dec 09)**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 Kees van Deemter (AC, Dec 09)**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) Kees van Deemter (AC, Dec 09)**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)=750 Without witness’ utterance: ½(1000)=500 • The witness’ utterance “misled” the Emperor A 180cm thief 175cm Kees van Deemter (AC, Dec 09)**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 Kees van Deemter (AC, Dec 09)**2-valued**3-valued Model A Model B tallA tallB 180cm 180cm ?tall?B tallA 165cm tallB Kees van Deemter (AC, Dec 09)**How does the Emperor classify the thief?**Three types of situations s(X) =defexpected search time given model X Type 1: thief [[tall]]A = [[tall]]B In this case, s(A)=s(B) Kees van Deemter (AC, Dec 09)**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) Kees van Deemter (AC, Dec 09)**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?]]Bthen (on average) half of [[tall]]B Now A searches ½(card([[?tall?]]B)) less. So, s(A)<s(B) Kees van Deemter (AC, Dec 09)**“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) Kees van Deemter (AC, Dec 09)**“Normally” model B wins**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) Kees van Deemter (AC, Dec 09)**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) Kees van Deemter (AC, Dec 09)**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)”) probability of x being called “tall” height(x) Kees van Deemter (AC, Dec 09)**Consider degree model C**• 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) Kees van Deemter (AC, Dec 09)**An example of model C**v(tall(a1)=v(tall(a2))=0.9 v(tall(b1)=v(tall(b2))=0.7 v(tall(c1)=v(tall(c2))=0.5 v(tall(d1)=v(tall(d2))=0.3 v(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) Kees van Deemter (AC, Dec 09)**This analysis suggests …**Kees van Deemter (AC, Dec 09)**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 Kees van Deemter (AC, Dec 09)**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 Kees van Deemter (AC, Dec 09)**Objections …**Kees van Deemter (AC, Dec 09)**Objections …**• Objection 1: “Smart search would have been equally possible based on a dichotomous (i.e., classical) model”. Kees van Deemter (AC, Dec 09)**Objections …**• Objection 1: “Smart search would have been equally possible based on a dichotomous (i.e., classical) model”. • 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. Kees van Deemter (AC, Dec 09)**Objections …**• Objection 1: “Smart search would have been equally possible based on a dichotomous (i.e., classical) model”. • 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. Kees van Deemter (AC, Dec 09)**Objections …**• Objection 2: “Truth value gaps (or degrees) do not imply vagueness”. Kees van Deemter (AC, Dec 09)**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. Kees van Deemter (AC, Dec 09)**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 Kees van Deemter (AC, Dec 09)**Objections …**• Objection 3: “Why was the witness not more precise?” Kees van Deemter (AC, Dec 09)**Objections …**• Objection 3: “Why was the witness not more precise?” • The idea: Witness should have said “the thief was 185cm tall”, giving an estimate Kees van Deemter (AC, Dec 09)**Objections …**• Objection 3: “Why was the witness not more precise?” • 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. Kees van Deemter (AC, Dec 09)**Objections …**• Objection 4: “This contradicts Lipman’s theorem” Kees van Deemter (AC, Dec 09)**Objections …**• Objection 4: “This contradicts Lipman’s theorem” • The idea: Lipman (2006) proved that every V predicate can be replaced by a crisp one that has higher utility. Kees van Deemter (AC, Dec 09)**Objections …**• Objection 4: “Thiscontradicts Lipman’s theorem” • 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”). Kees van Deemter (AC, Dec 09)**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 !) Kees van Deemter (AC, Dec 09)**“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) Kees van Deemter (AC, Dec 09)**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 Psychology15 (3): 213-227 Kees van Deemter (AC, Dec 09)**Appendix**• Extra slides Kees van Deemter (AC, Dec 09)**“Normally” model B wins**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) Kees van Deemter (AC, Dec 09)**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 then s(<+,?,->)= 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 Kees van Deemter (AC, Dec 09)**Consider degree model C**• 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)) probability of x being called “tall” height(x) Kees van Deemter (AC, Dec 09)**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)) ) Kees van Deemter (AC, Dec 09)**End**Kees van Deemter (AC, Dec 09)