Spring 2012
Juyeon Kang
qkang@fi.muni.cz
B410, Faculty of Informatics, Masaryk University,
Brno, Czech Rep.
IA165
Combinatory Logic for
Computational Semantics
Application of the combinators to natural language
analysis: basic
example of the formal semantic analysis
some structures: cooridination
Summing up: last lecture
● How to apply the combinators to natural language analysis
1) using introduction and elimination rules by beta-reduction of
combinators: control heurstic of combinatorial application and bracketing
2) using a syntactic tool for controlling the application of combinators
: CCG assumes the preliminary steps to find a well-structured normal
form, that is, a formal semantic structure
Example: use the combinators as a logical tool of
semantic analysis
a. apply directly the β -reduction rules of combinators
b. use the CCG types and rules by integrating the β -reduction
rules of combinators into the CCG rules
Remind 1....METHOD
● Goal: Formal semantic structure in term of operator and of operand
: bracketed expression written by convention: (operator(operand))
● Two methods of application of the combinators: both can be useful
a. useful for defining some specific operators: passivisation,
reflexivisation, quantification..
b. useful for general formal semantic analysis; can handle the analysis at
the syntactic level and semantic level in one representation; informatic
implementation more systemactic see the nexe slides ”Remind 2 & 3”→
Remind 2...TYPES
● CCG types
primitive types: S for sentence, NP for noun phrase, N for noun
derived types: S/NP, N/N, N\N, (S/NP)/NP, NP/N...
● Directionality: / (over) and \ (under)
a/b: a applies to b, a\b: b is applied to a
→ direction of application of operator to operand
x/y:e1    y:e2  
­­­­­­­­­­­­­­>(>)
     x:(e1(e2))
Remind 3...RULES
(S\NP)/NP:loves   NP:Anna 
­­­­­­­­­­­­­­­­­­­­­­­­­> (>)
(S\NP):(loves (Anna))
(S\NP)/NP:loves   NP:Anna 
­­­­­­­­­­­­­­­­­­­­­­­­­> (>)
(S\NP):(loves (Anna))
1. Forward(>) and backward (<) functional application rules
e1: (x/y) e2:(y/z)
-------------------->(>B)
(x/z): B e1 e2
 (S\NP)/NP:likes  (NP/N):a 
               ­­­­­­­­­­­­­­­­­­­­­­> (>B)
               (S\NP)/N: (B likes a)
 (S\NP)/NP:likes  (NP/N):a 
               ­­­­­­­­­­­­­­­­­­­­­­> (>B)
               (S\NP)/N: (B likes a)
2. Function composition (FC) rules with the combinator B
e1:x
---------------> (>C*)
S/(S\x): C*x
For NP subj
e1:John
       ­­­­­­­­> (>C*)
S/(S\NP):C*John
For NP subj
e1:John
       ­­­­­­­­> (>C*)
S/(S\NP):C*John
3. Type-raising rules with the combinator C*
Some preliminary exercises to introduce the
combinators:
Finding a ccg type adequate to the given lexique
11
Adverbs
Adverb of verb
(S\NP)/(S\NP)
(S\NP)/NP/
(S\NP)/NP
Adverb of verb
(S\NP)/(S\NP)
(S\NP)/NP/
(S\NP)/NP
Adverb of adverb
(S\NP)/(S\NP)/(S\NP)/(S\NP)
(S\NP)/NP/(S\NP)/NP/(S\NP)/NP/
(S\NP)/NP
Adverb of adverb
(S\NP)/(S\NP)/(S\NP)/(S\NP)
(S\NP)/NP/(S\NP)/NP/(S\NP)/NP/
(S\NP)/NP
Adverb of
adjective
(N/N)/(N/N)
(N\N)/(N\N)
Adverb of
adjective
(N/N)/(N/N)
(N\N)/(N\N)
Adverb of
proposition
S/S
Adverb of
proposition
S/S
12
Preposition
Prep. 1:
constructor of adverbial phrase
(S\NP)\(S\NP)/NP
(S/S)/NP
(S/S)/N
Prep. 1:
constructor of adverbial phrase
(S\NP)\(S\NP)/NP
(S/S)/NP
(S/S)/N
Prep. 2:
constructor of adjectival phrase
(N\N)/NP
(N\N)/N
Prep. 2:
constructor of adjectival phrase
(N\N)/NP
(N\N)/N
13
Example: Dictionary of typed words
Syntactic categories Syntactic types Lexical entries
Nom. N Olivia, apple...
Completed nom. NP an apple, the school
Pron. NP She, he...
Adj. (N/N), (N\N) pretty woman,...
Adv. (N/N)/(N/N),
(S\NP)\(S\NP)...
very delicious,...
Vb (S\NP), (S\NP)/NP... run, give...
Prep. (S\NP)\(S\NP)/NP
(NP\NP)/NP...
run in the park,
book of John,...
Relative (S\NP)/S... I believe that...
Some structures of natural language:
coordination
➢ Cecila picks the pear.
➢ Cecilia eats the pear.
➢ Cecilia picks and eats the pear.
● Any logical form of this kind must of course express that the arguments
appear in two predications.
((and (picks (the pears)) (eats (the pears))) Cecilia)((and (picks (the pears)) (eats (the pears))) Cecilia)
15
Coordination (Φ)
X CONJ X⇒Φ
X (Coordination )Φ
x:e1  CONJ  x:e2
­­­­­­­­­­­­­­­­­> (> )Φ
x:   CONJ e1 e2 Φ
e1 of type x coordinated with e2 of type x by the conjonction 'and'
 
(S\NP)/NP:pick    CONJ:'and'    (S\NP)/NP:eat
       ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­> (>Φ)
(S\NP)/NP:   Φ and pick eat
e1 of type x coordinated with e2 of type x by the conjonction 'and'
 
(S\NP)/NP:pick    CONJ:'and'    (S\NP)/NP:eat
       ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­> (>Φ)
(S\NP)/NP:   Φ and pick eat
16
Follow the next steps.
(1) Attribute first the CCG types to each linguistic expression
(2) calculate theses types to obtain the syntactic analysis by applying 
the CCG rules. 
(3) eliminate the applied combinators with respect to each  b­reduction 
of combinators
(4)  check  if  your  semantic  representation  is  well­structured  normal 
form.
17
● Example : Cecilia picks and eats the pear.
1/[NP:Cecilia]­[(S\NP)/NP: picks]­[CONJ:and]­[(S\NP)/NP: eats]­[NP: the pear]
2/[(S/(S\NP):C*Cecilia]­[(S\NP)/NP: picks]­[CONJ:and]­[(S\NP)/NP: eats]­[NP: the 
pear]   (>C*)
3/[(S/(S\NP):C*Cecilia]­[(S\NP)/NP: Φ and picks eats]­[NP: the pear]   (>Φ)
4/[(S/(S\NP):C*Cecilia]­[(S\NP)/NP: Φ and picks eats]­[NP: the pear]   (>B)
5/[(S/NP:B(C*Cecilia) (Φ and picks eats)]­[NP: the pear]   (>B)
6/[(S:(B(C*Cecilia) (Φ and picks eats))(the pear)]   (>)
18
7/(B(C*Cecilia) (Φ and picks eats))(the pear)
8/(C*Cecilia) ((Φ and picks eats))(the pear))  (elimination of B)
9/((Φ and picks eats))(the pear))(Cecilia)  (elimination of C*)
10/((and (picks(the pear)) (eats(the pear)))(Cecilia) ) (elimination of Φ)
●We get the well­formed semantic structure of the 
sentence Cecila picks and eats the pear.
●
●
((and (picks(the pear)) (eats(the pear)))(Cecilia) )
●We get the well­formed semantic structure of the 
sentence Cecila picks and eats the pear.
●
●
((and (picks(the pear)) (eats(the pear)))(Cecilia) )
19
Next week...
● Continue about the application of the combinators to natural
language analysis: extraction asymmetries, subordinative structure
and relative