1. CSA4050: Advanced Topics in NLP
Semantics III
Quantified Sentences
November 2008
HLT Semantics III

2. Outline Language Sentences Determiners Noun Phrases
Syntactic Structure
Logic
Generalised Quantifiers
Higher order functions
Translation into Prolog
Syntax-Semantics Interface
November 2008
HLT Semantics III

3. Determiners and Quantifiers in Language and Logic
A dog barked
x dog(x) & bark(x)
Every dog barked
x dog(x)  bark(x)
Fido chased a cat
x cat(x) & chase(fido,x)
Every dog chased
a cat
x dog(x)  (y cat(x) & chase(x,y)))
November 2008
HLT Semantics III

4. Syntactic Shape vs. Semantic Shape
John walks semantics: walk(suzie).
Every man talks semantics: all(X, man(X)  talk(X)) S
NP VP
Suzie walks
Det N talks
Every man
November 2008
HLT Semantics III

5. Problem Similar syntactic shape Dissimilar semantic shape
How is this possible if the syntax drives the combination of semantic fragments as per rule-to-rule hypothesis?
Answer: be creative about logical forms and semantic combination rules
November 2008
HLT Semantics III

6. Montague Solution
Reorganising the semantic combination rules operating between VP and NP in rules such as s(S) --> np(NP), vp(VP).
We will be considering [NP]([VP]) versus [VP]([NP]).
NPs as higher order functions
Analyse LF of quantified sentences
November 2008
HLT Semantics III

7. LF of Quantified Sentences
LF of quantified sentences has a general shape involving
a restrictor predicate R
a scope predicate S
R restricts the set of things we are talking about
S says something further about set element(s)
a logical quantifier Q
a bound variable V
a logical operator O connecting R and S
November 2008
HLT Semantics III

8. x lecturer(x)  lazy(x)
Examples
All lecturers are lazy
x lecturer(x)  lazy(x)
Restrictor = lecturers
Scope = lazy
Quantifier = All
Operator = implies
Bound Variable = x
November 2008
HLT Semantics III

9. Examples There is a lazy lecturer x lecturer(x) & lazy(x)
Restrictor = lecturers
Scope = lazy
Quantifier = exist
Operator = and
Bound Variable = x
November 2008
HLT Semantics III

10. Anatomy of Quantified Sentences
Logic Q V R O S
x m(x)  w(x)  x
m(x) 
w(x)
x d(x) & b(x) 
d(x)
&
b(x)
x d(x)  (h(x) & b(x))
h(x) & b(x)
November 2008
HLT Semantics III

11. Generalized Quantifiers
We adopt the following generalized quantifier representation for LF in which quantifier is a 3-place predicate: Q(<variable>,<restrictor>,<scope>)
Operator is omitted.
Examples all(X,man(X),walk(X)) exist(X,man(X),walk(X)) the(X,man(X),climbed(X,everest)) most(X,lecturer(X),poor(X))
November 2008
HLT Semantics III

12. NP as higher order function
Q^all(X,man(X),Q)
every man VP
Y^walk(Y)
walks S
all(X,man(X),walk(X))
November 2008
HLT Semantics III

13. Encoding in Prolog The VP remains as before, ie X^walks(X)
The quantified NP every man will be of the form Q^all(X,man(X),Q)
The semantic rule for S now ensures that the NP function is applied to the VP function. s(S)--> np(NP),vp(VP), {reduce(NP,VP,S)}
November 2008
HLT Semantics III

14. DCG with Quantification Program 1
% grammar
s(S) --> np(NP), vp(VP), {reduce(NP,VP,S)}
vp(VP) --> v(V).
% lexicon
v(X^walk(X)) > [walks].
np(Q^all(X,man(X),Q)) --> [every,man].
November 2008
HLT Semantics III

15. Result ?- s(X,[every,man,walks],[]).
X = all(_G397, man(_G397), _G405^walk(_G405))
all(x, man(x), y^walk(y))
What is wrong with this?
How can we fix it?
We need to force the variables to be identical using reduce
November 2008
HLT Semantics III

16. Result ?- s(X,[every,man,walks],[]).
X = all(_G397, man(_G397), _G405^walk(_G405))
all(x, man(x), y^walk(y))
What is wrong with this?
The variables _G397 and _G405 are distinct. They should be identical.
The consequent of the implication is a λ expression
How can we fix it?
We need to force the variables to be identical using reduce
November 2008
HLT Semantics III

17. DCG with Quantification Program 2
% grammar
s(S) --> np(NP), vp(VP), {reduce(NP,VP,S)}
vp(VP) --> v(V).
% lexicon
v(X^walk(X)) > [walks].
np(Q^all(X,man(X),P)) --> [every,man],
{reduce(Q,X,P)}.
November 2008
HLT Semantics III

18. Result The effect of the reduce clause is
?- s(X,[every,man,walks],[]).
X = all(_G397, man(_G397),walk(_G397))
The effect of the reduce clause is
to identify the appropriate variables
to remove the λ variable
November 2008
HLT Semantics III

19. Handling Quantified NPs
Before we cheated by having every man as a lexical item. np(Q^all(X,man(X),P)) --> [every,man], { reduce(Q,X,P)}.
Now we see what is involved in analysing the NP from its parts.
Step 1 is to write a new syntactic rule np(NP) --> d(D), n(N).
How does the semantics work?
November 2008
HLT Semantics III

20. LF of determiners
Key idea is determiner has LF of a 2-argument function corresponding to R and S which become bound during processing.
λR.λS.Q(V,R,S)
where Q is associated with the particular determiner
When we apply this function to the adjacent noun, we obtain the LF of the NP.
November 2008
HLT Semantics III

21. How NP is created D R^S^all(X,R,S) every N Y^man(Y) man NP
S^all(X,man(X),S)
November 2008
HLT Semantics III

22. Fitting the Semantics Together
Handle the quantified NP np(NP) --> d(D), n(N), {reduce(D,N,NP)}.
Add lexical entry for every
d(RL^SL^all(X,R,S)) >[every], {reduce(RL,X,R), reduce(SL,X,S) }.
November 2008
HLT Semantics III

23. DCG with Quantification Program 3
% grammar
s(S) --> np(NP), vp(VP), {reduce(NP,VP,S)}.
np(NP) --> d(D), n(N), {reduce(D,N,NP) }.
vp(VP) --> v(VP).
% lexicon
v(X^walk(X)) > [walks].
n(X^man(X)) > [man].
d(RL^SL^all(X,R,S) --> [every], {reduce(RL,X,R), reduce(SL,X,S) }.
November 2008
HLT Semantics III

24. Trace >: (7) s(_G510, [every, man, walks], [])
>: (8) np(_L183, [every, man, walks], _L184)
>: (9) d(_L205, [every, man, walks], _L206)
<: (9) d((X^R)^ (X^S)^all(X, R, S), [every, man, walks], [man, walks])
>: (9) n(_L207, [man, walks], _L208)
<: (9) n(Z^man(Z), [man, walks], [walks])
>: (9) reduce((X^R)^ (X^S)^all(X, R, S), Z^man(Z), _L183)
<: (9) reduce((X^man(X))^ (X^S)^all(X, man(X), S), X^man(X), (X^S)^all(X, man(X), S))
<: (8) np((X^S)^all(X, man(X), S), [every, man, walks], [walks])
>: (8) vp(_L185, [walks], _L186)
>: (9) v(_L185, [walks], _L186)
<: (9) v(Y^walk(Y), [walks], [])
<: (8) vp(Y^walk(Y), [walks], [])
>: (8) reduce((X^S)^all(X, man(X), S), Y^walk(Y), _G510)
<: (8) reduce((X^walk(X))^all(X, man(X), walk(X)), X^walk(X), all(X, man(X), walk(X)))
<: (7) s(all(X, man(X), walk(X)), [every, man, walks], [])
November 2008
HLT Semantics III

25. Summary Quantification crops up a lot in NL
To handle quantified sentences, there is a mismatch in shape between syntax and semantics
Need to reorganise the semantic rules (NP applies to VP not vice versa).
Representation of quantifier as a higher-order function.
November 2008
HLT Semantics III