vagueness: a problem for ai

HIT Summer School 2008, K.v.Deemter

Vagueness: a problem for AI

Kees van Deemter

University of Aberdeen

Scotland, UK


Overview

1. Meaning: the received view

2. Vague expressions in NL

3. Why vagueness?a. vague objects

b. vague classes

c. vague properties

4. Vagueness and context


Overview (ctd.)

5. Models of vaguenessa. Classical modelsb. Partial Logic c. Context-based modelsd. Fuzzy Logice. Probabilistic models

6. Generating vague descriptions• Natural Language Generation

7. Project: advertising real estate


The plan

• The lectures will give you a broad overview of some of the key concepts and issues to do with vagueness– Lots of theory (very briefly of course)– Some simple/amusing examples

• The project will let you apply some of these concepts practically

• The project gives you a lot of freedom: you can do it in your own way!


Not Exactly:In Praise of Vagueness

Oxford University Press To appear, 2009



• Linguistic Theory (e.g., Kripke, Montague) : in a given “world”, each constituent denotes a set-theoretic object.

• For example, Domain = {a,b,c,d,e,f,g,h,i,j,k}

[[cat]] = {a,b,f,g,h,i,j} [[dog]] = {c,d,e,k}

• There are no unclear cases. Everything is crisp. (Nothing is vague.)Sentences are always true or false.



• Linguistic Theory (e.g., Kripke, Montague) : in a given “world”, each constituent denotes a set-theoretic object.

• Example: Domain of animals {a,..,k} in a village[[black]] = {a,b,c,d,e}[[white]] = {f,g,h,i,j,k}[[cat]] = {a,b,f,g,h,i,j} [[healthy]] = {d,e,f}

[[dog]] = {c,d,e,k} [[scruffy]] = {a,b,h,i} [[black cat]] = {a,b,c,d,e} {a,b,f,g,h,i,j} = {a,b} [[All black cats are scruffy]]= {a,b} {a,b,h,i} = 1


Complications

• This was only a rough sketch ...– One needs syntax to know how to combine

the meanings of constituents– The Example neglects intensionality:

• [[fake cats]] = [[fake]][[cat]] ??

• These and other issues are tackled in tradition started by Richard Montague– (Not the focus of these lectures)


Complications

• There’s more to communication.– Why was the sentence uttered?

• warn a child not to approach a particular cat?

– The same information could be expressed differently• ``Cats are always scruffy``• ``Don’t touch that cat`` • What motivates the choice?

• This is a key question when you try to let a computer generate sentences: Natural Language Generation (NLG)


Modern Computational Semantics

• ... is still broadly in line with this ``crisp`` picture• Statistical methods do not really change matters• Example: recent work on logical entailment

– uses stats to learn entailment relations and test theories by comparing with human judgments

– Entailment itself is typically assumed to be crisp • But what if the extension of a word is vague?

– clearly scruffy: a,b,h,i– clearly not scruffy: f– unclear cases: d,g,j : maybe scruffy? a little scruffy?

• Does x is scruffy entail x is not healthy? (Perhaps just a bit?) A subtler semantic theory is needed, which takes vagueness into account


2. Vague expressions in NL

Vagueness in every linguistic category. E.g.,• adjectives: large, small, young, old, ...• adverbs: quickly, slowly, well, badly, ...• determiners/articles: many, few, ...• nouns: girl, boy, castle, fortune, poison (?)...• verbs: run, improve (significantly), cause (?) ...• prepositions: (right) after, (right) above, ...• intensifiers: very, quite, somewhat, a little


In some categories, vagueness is normal:

• Adjectives: hard to find crisp ones

• Adverbs: hard to find crisp ones

British National Corpus (BNC): 7 of the 10 most frequent adjectives are vague; the others are ... borderline cases of adjectives ( `last`, `other`, `different` ):


British National Corpus (BNC)

last (140.063 occurrences) Adj?other (135.185) Adj?new (115.523)good (100.652) old (66.999)great (64.369) high (52.703)small (51.626) different (48.373) Adj?large (47.185)


Non-lexical vagueness

• Generics: ``Cats are scruffy``. All cats?? ``Dutchmen are tall``. All Dutchmen??

• Temporal succession: ``He came, he saw, he conquered’’. In quick succession!

• Irony: ``George is not the brightest spark`. George is probably quite stupid.

• Aside: Same mechanisms in Chinese? Many possible linguistic research projects!


3. Why do we use vagueness?

1. Vague objects

Example: Old Number One. (A court case made famous by a recent article by the philosopher Graeme Forbes.)


Old Number One

[Details omitted here, but see the following animation]


Other objects are vague too ...

• Rowing boats are repaired by replacing planks. When no plank is the same, is it the same boat?

• Think of a book or PhD thesis. How is it written? Doesn’t it change all the time? What if it’s translated? What if it’s translated badly?

• How many languages are spoken in China? Every linguists gives a different answer.

• Are you the same person as 3 weeks after conception? After you’ve lost your memory?

• The cat is loosing hair. Is it still part of the cat?


Why vagueness?

2. Vague classes

Biology example made famous by Richard Dawkins: the ring species called ensatina.

Standard principle: a and b belong to the same species iff a and b can produce fertile offspring.


Why vagueness?

2. Vague classes

Biology example made famous by Richard Dawkins: the ring species called ensatina.

Standard principle: a and b belong to the same species iff a and b can produce fertile offspring.

It follows that species overlap, and that there are many more of them than is usually assumed!


The ensatina salamander

[Details omitted here]


Why vagueness?

3. Vague predicates

Why do we use words like `large/small`, `dark/light`, `red/pink/...`, `tall/short` ?

Question: would you believe me if I told you that Dutch has no vague word meaning `tall`, but only the word `lang` which means `height 1.85cm`? For example,

Kees is lang means height(Kees) 1.85cm


Why vagueness?

`Kees is taller than 1.85cm`

1.85cmThe threshold of 185cm makes a crisp distinction between

A : height >185cm

B: height 185cm

A

B


Why vagueness?

`Kees is taller than 185cm`

184.999cm185.001cm

A

B

xySome elements of A and B are

too close to be distinguished by anyone!

Example: x in A y in B


Vagueness can have other reasons, including

• We may not have an objective measurement scale (e.g., `John is nice`)

• You may not share a scale with your audience (Celcius/Fahrenheit)

• You may want to add your own interpretation (e.g., `Blood pressure is too high`)

• You’re passing on words verbatim (e.g., I tell you what I heard in the weather forecast)

Many of these reasons are relevant for NLP (including NLG)!


4. Vagueness and context


4. Vagueness and Context

• Vague expressions can often not be interpreted without context

• We know (roughly) how to apply `tall` to a person. Roughly: Tall(x) Height(x) >> 170cm

• If we used the same standards for buildings as for people, than no building would ever be tall!

• Context-dependence allows us to associate one word with very different standards. Very efficient! (Cf. Barwise & Perry, “Situations and Attitudes”)


Variant of old example

[[animal]] = {a,...,k}

[[black]] = {a,b,c,d,e}[[cat]] = {a,b,f,g,h,i,j}

[[elephant]] = {c,d,e,k}

[[black cat]] = {a,b,c,d,e} {a,b,f,g,h,i,j} = {a,b}

[[small elephant]] = ?

[[small]] = ? Any answer implies that

x is a small elephant x is a small animal


Other types of vagueness are also context dependent

• [[many]] = how many?• [[(increase) rapidly]] = how rapidly?• temporal succession. Compare the time lags

between the events reported:

1. ``John entered the room. [..] He looked around.less than a second

2. ``Caesar came, [..] he saw, [..] he conquered``weeks or months

3. ``The ice retreated. [..] Plants started growing. [..] New species appeared.``many years


5. Models of vagueness


5. Models of Vagueness

A problem that we would like our models to shed light on: The sorites paradox

Oldest known version (Euboulides): 1 stone makes no heap; if x stones don’t make a heap then x+1 stones don’t make a heap. So, no finite heaps exist (consisting of stones only).

Henceforth: `~` = `indistinguishable` `<<` = observably smaller than


Sorites paradox (version 2)

Short(150.000cm) Short(200.000cm)

150.000cm ~150.001cm

therefore Short(150.001cm)

150.001cm ~ 150.002cm


...



Sorites paradox

• We derive a contradiction

• Of course there is something wrong here

But what exactly is the error?

• One bit that we would like to ``honour``:

Principle of Tolerance: things that resemble each other so closely that people cannot notice the difference should not be assigned (very) different semantic values



1. Partial Logic.

A small modification of Classical Logic.

Three truth values: True, False, Undecided (in the “gap” between True and False).


Partial Logic

185cm

Partial Logic makes some formulas true, others false, yet others undecided.

In the partial model on the right, if Height(John)=170cm then [[Tall(John)]] = undecided

Tall

Not tall

Gap

165cm


Some connectives defined

[[p q]] = True iff [[p]]=True or [[q]]=True

[[p q]] = False iff [[p]]=[[q]]=False

[[p q]] = Undecided otherwise

[[p]] = True iff [[p]] = False

[[p]] = False iff [[p]] = True

[[p]] = Undecided otherwise


Consider ...

[[Tall(John) Tall(John)]] (an instance of the Law of Excluded Middle)

Is this true, false, or undecided?


Consider ...

[[Tall(John) Tall(John)]]

[[Tall(John)]] = undecided, therefore

[[Tall(John)]] = undecided, therefore

[[Tall(John) Tall(John)]] = undecided


Repair by means of supervaluations

Suppose I am uncertain about something(e.g., the exact threshold for ``Tall``)

Suppose p is true regardless of how my uncertainty is resolved ...

Then I can conclude that p


Consider [[Tall(John) Tall(John)]]

The yes/no threshold for ``Tall`` can be anywhere between 165 and 185cm.

Where-ever the threshold is, there are only two possibilities:

1. [[Tall(John)]] = True. In this case[[Tall(John) Tall(John)]] = True.

2. [[Tall(John)]] = False. In this case [[Tall(John)]] = True. Therefore[[Tall(John) Tall(John)]] = True.

The formula must therefore be True.


Partial Logic + supervaluations

• Supervaluations enable Partial Logic to be ``almost Classical`` in its behaviour.

• How good is this as a model of vagueness?

• Like the Classical model that put the threshold at 185cm, the partial model makes a distinction that people could never make:


Partial Logic

This time, we even have

2 such artificial

boundaries:

This still contradicts the

Principle of Tolerance

185.001cm

Tall

Not tall

Gap

165.001cm

184.999cm

164.999cm



2. Context-based models (Version by N.Goodman, M.Dummett, F.Veltman, R.Muskens)

Suppose x~y, but h << x h~y

The observer can deduce that y<x, so y and x become distinguishable. Short(y) nolonger implies Short(x)

184.999

A

B

185.001xy

184.000h



h is called a help element

Sorites introduces more and more things into the argument that can serve as help elements.

184.999

A

B

185.001xy

184.000h



New Principle of Tolerance: things that resemble each other so closely that people cannot notice the difference even whentaking help elements into accountshould not be assigned different semantic values

Theorem: this new toleranceprinciple does not lead to sorites

184.999

A

B

185.001xy

184.000h



Problem:

New Tolerance Principle assumes that `~` is crisp

184.999

A

B

185.001xy

184.000h



If you ask a subject to compare x,y and h 1000 times, you get different responses, e.g.

1. x~y, h<<x, h~y. 2. x~y, h~x, h~y.3. y<<x, h<<x, h<<y.4. y<<x, h<<x, h~y.5. x<<y, h<<x, h~y.6. x~y, h<<x, h~y. 7. ... (etc) ...



If you ask a subject to compare x,y and h 1000 times, you get different responses

The idea of a fixed “just noticeable difference” is a simplification

This is a serious problem for context-based approaches.

Other context-based models exist, e.g. H.Kamp 1981, but these are further removed from classical logic


Models of vagueness

• Book: agent-based model which takes the probabilistic nature of `~` into account. Inspired by J.Halpern’s work. Model revolves around Margin of Error (MoE):

• x~y def |MeasuredHeight(x)-MeassuredHeight(y)| MoE

• Clearly(Short(x)) def MeasuredHeight(x) (thresholdshort – MoE)

Tolerance: Clearly(Short(x)) & x~y Short(y)


Models of vagueness

ClearlyShort(150.000cm) MoE=5cm

ClearlyShort(150.001cm)

...

ClearlyShort(160.000cm)

160.000cm ~ 165.000cm

therefore, Short(165.000cm)

ClearlyShort(165.000cm), because the individual measured as 165.000cm might by as tall as 170.000cm.


Models of vagueness

Intuition behind this agent-based model:• We are aware of our own MoE• This allows us to distinguish between things we

know and things that we are not so sure about• This suggests a solution to sorites that

distinguishes between the two. But:– Still the same problems with Principle of Tolerance– Is MoE really crisp??

• Perhaps we need a non-crisp model!



3. Fuzzy Logic.

A much more drastic deviation from Classical Logic.


• Given that `crisp boundaries’ are a disadvantage of 3-valued logic, how about using real numbers as values?

• Best known version is fuzzy logic(Zadeh 1975): [φ] ε [0,1], where

[φ] ≤ [] is at least as true as φ• That’s all (for atomic formulas).

Fuzzy logic does not say how truth values may be obtained


Truth definition (fuzzy logic)Everything is done truthfunctionally:

1. Negation: [¬φ] = 1- [φ]

2. Disjunction [φv] = max([φ] , [])

3. Conjuntion [φ] = min([φ] , [])

4. Implicaton [φ]. Various options, e.g.:1. [φ] = [¬φ v ]

(b) If [φ] ≤ [] then [φ] = 1, otherwise [φ] = 1- ([φ]-[])


Using option (a)

[small(i) small(i+1)]= [¬ small(i) v small(i+1)]= max([¬ small(i)] , [small(i+1)])= max(1-[small(i)] , [small(i+1)])

For i=150, this is [small(151)]. For i=151, this is [small(152)] …Presumably, these are all close to 1

For some i, [small(i) ]=1/2+ε , [small(i+1)]= 1/2-ε,At which point, we have [small(i) small(i+1)] = max((1-1/2)+ ε, 1/2-ε) = 1/2-ε.

Some very different values, some of which are quite low!


Using option (b)

(b) If [φ] ≤ [] then [φ] = 1 otherwise, [φ] = 1- ([φ]-[])

Assume that [small(i)]- [small(i+1)] = ε,for some small constant ε.

Then [small(i)small(i+1)] =1- [small(i)]-[small(i+1)], regardless of i.

This is better! Let’s look at the paradox again


Conclusions get increasingly low values

small(0) [Some high value, let’s say 1] small(1) [1-ε] small(2) [1-2ε] small(3) [1-3ε], etc.

This is great !

But other drawbacks remain. For example,for most i, we have

[small(i) v ¬small(i)] ≠ 1


The problem with truthfunctionality

Truthfunctionality implies:

If [q] = [¬p] then

[p V ¬p] = [p V q]

Truth definition of `V` cannot take into account whether the disjuncts are related


Other forms of the same problem

Let’s take a vote:Which one is the big expensive car?

a. [car(x)]=1[big(x)]=0.5[expensive(x)]=0.5

b. [car(y)=1[big(x)]=0.5[expensive(x)]=1


Other forms of the same problem

a. [car(x)]=1[big(x)]=0.5[expensive(x)]=0.5

b. [car(y)=1[big(y)]=0.5[expensive(x)]=1

Fuzzy Logic says, counter-intuitively, that [car(x) big(x) exp(x)]=0.5[car(y) big(y) exp(y)]=0.5


• To address these problems, Fuzzy Logic allows different kinds of conjunction, for example

Conjunction 1: [φ] = min([φ] , [])

Conjunction 2: [φ] = [φ] * []


Pros and cons of using fuzzy logic

• Fuzzy logic captures the intuition that sorites leads to increasingly doubtful conclusions.

• By using many truth values, it becomes difficult to determine the truth values of atoms

• Different values must sometimes be attributed to [small(x)] and [small(y)] where x and y are indistinguishable. If not, the sorites paradox returns! Imagine the line-up of people again: a1 ~ a2 ~ … ~ a4001

• Is this not at odds with the Tolerance Principle?


Pros and cons of using fuzzy logic (Ctd)

• Definition of logical operators proves to be a bit arbitrary

• Logical principles tend to be sacrificed(e.g. excluded middle)

• Fuzzy Logic not popular as a solution for the sorites paradox

• But lots of maths and engineering (e.g., fuzzy control)


Fuzzy Expert Systems

• Essentially, the same problems plague fuzzy engineering

• each fuzzy system defines the connectives in its own, somewhat ad hoc way.

• accrual is often an additional problem• Fuzzy Logic is not really one logic, but a

toolkit for making logics

• An example


Fuzzy Expert Systems

R1: (female(x) smoking(x)) at_risk(x)

R2: hypertension(x) at_risk(x)

• What kind of conjunction should we use?

• What if both rules fire; does this increase the value [at_risk(x)]?



4. Probabilistic Logic.

An alternative to Fuzzy Logic:

-- based on probability

-- conditional probabilities are not truth functional

-- the result is a much more classical and principled logic.


Probabilistic Logic

[details omitted]

• See various papers by D.Edgington• See also my book “Not Exactly: in Praise

of Vagueness” (2009).


6. Generating Vague Descriptions

• My own work in this area: generation of phrases like– “the large dog”– “the largest two of the cheapest dogs”– etc.

from a purely numerical database.• Van Deemter 2006 ``Generating Referring Expressions

that Involve Gradable Properties’’. Computational Linguistics 32 (2).


7. Project: vagueness in real estate

• NLP research at the University of Aberdeen focusses on Natural Language Generation

• Project: explore vagueness from an NLG viewpoint• Concretely, I’d like you to study the use of vague

expressions in adverts for real estate• Next slides: a computational project: specify, construct

and document an NLG program that generates adverts.

• You can use a computing platform and programming language of your own choosing: Whatever works best for you!


Hints for a computational project

• Input: a self-made database containing information on real estate (flats, houses, etc.):

• size of each room/garden, using numbers (e.g., the widest extensions of the kitchen may be 2.00cm by 3.23cm)

• something about their quality (e.g., the kitchen may be modern, well-equipped, with nice sea views).

• Use your imagination!



Output: (Parts of) an advert, containing sentences that the estate agent might use

• For example: “We are pleased to offer this magnificent 5-bedroom apartment, which overlooks the sea. It has a spacious, well-equipped kitchen, whose widest dimensions are 4.30cm by 5.13cm. The living room is ...” (This is only an example of the kind of text a program might generate given one input. Your own program might generate much simpler text.)

• Don’t worry about the user interface (e.g. how the user selects an apartment). Keep it simple.



Things to bear in mind:• context-dependence: large for a bathroom is not

the same as large for a living room• perspective: you’re the estate agent. Are you

honest (“a small kitchen”) or just trying to sell (“a cosy kitchen”).

• Whatever you do, please document it by writing a short report, written in clear English. What is not documented does not exist!


Questions:

• Can a house be small if all its rooms are large? Can a house be modern if all its rooms are old-fashioned?

• What’s your answer to the Principle of Tolerance?

• Which of the formal models of vagueness did you find most useful (if any)?


Your report should contain

• An explanation of the kinds of inputs that your program accepts

• An explanation of the mechanisms that you use for modelling the facts

• An explanation of your generation algorithm• Examples of output texts (along with the inputs from

which they were generated)• A brief discussion of the pro’s and con’s of your

approach. What would you do differently the next time? What would you do next (if you had the time)?

• A clear installation manual for your program.


• The previous slides assume a project that is mainly computational.

• A more linguistic alternative:– Study a corpus of real-estate adverts, and

report on your findings.• Example texts can be found on the web, e.g. the

ASPC real-estate web site in Aberdeen: http://www.aspc.co.uk/(Also relevant for computational project)


• The previous slides assume a project that is mainly computational.

• A more linguistic alternative:– Study a corpus of real-estate adverts, and

report on your findings.• Example texts can be found on the web, e.g. the

ASPC real-estate web site in Aberdeen.

– Ideal: something in between (a mix?) of these two types of projects


Concluding

Interested in working on vagueness, or NLG, or both?

Contact me: [email protected]

Kees van Deemter

University of Aberdeen

Scotland, UK


Appendix: vagueness in computational semantics


Survey of 2007 International Workshop on Computational Semantics (IWCS-7)

Semantics used in 45 non-invited papers:• DRT or UDRT: 4• Hole Semantics: 2• MRS: 1• Description Logic: 2• Default Logic: 1• Modal and Temporal Logic: 3• Lambda Calculus: 3• Simple atomic formulas: 6• Informal categories (like Agent/Patient, or Event/Action): 18,

of which modelling gradable dimensions (e.g., word similarity): 3• Partial Logic: 1• Prototype Theory: 1


Computational Semantics

• Everything written in blue was overtly Boolean

• The bits in red:– 2 papers model gradability– 3 papers identify gradable phenomena

(word similarity, document similarity, language change and learning)


End of Appendix

vagueness: a problem for ai

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