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
3
Introduction and elimination rule of
combinators
Elim. Rules Intro. Rules
If f
--- [e-I] ---[i-I]
f If
Kfx f
----- [e-K] ----[i-K]
f Kfx
Elim. Rules Intro. Rules
If f
--- [e-I] ---[i-I]
f If
Kfx f
----- [e-K] ----[i-K]
f Kfx
Elim. Rules Intro. Rules
C*fx xf
--- [e-C*] ---[i-C*]
xf C*fx
Bfxy f(xy)
----- [e-B] ----[i-B]
f(xy) Bfxy
fxyzΦ f(xz)(yz)
----- [e- ]Φ ----[i- ]Φ
f(xz)(yz) fxyzΦ
Elim. Rules Intro. Rules
C*fx xf
--- [e-C*] ---[i-C*]
xf C*fx
Bfxy f(xy)
----- [e-B] ----[i-B]
f(xy) Bfxy
fxyzΦ f(xz)(yz)
----- [e- ]Φ ----[i- ]Φ
f(xz)(yz) fxyzΦ
4
● Bracketing: associativity
((X·Y)·Z)x
≥ B(BXY)Zx
≥ BXY(Zx)
≥ X(Y(Zx))
≤ X(BYZx)
≤ BX(BYZx)
= X(Y·Z)
Bxyz=(Bxy)z
xyz=x(yz)
Bxyz=(Bxy)z
xyz=x(yz)
5
Equivalence to Church's λ- expressions
● The most important difference between the CL and -calculus is theλ
use of the bounded variables.
● Every combinator is an -expression.λ
K xy.x≡ λ
T x.x≡ λ
S xyz.xz(yz)≡ λ
● I =def x.x (identificator)λ
● K =def x. y.x (cancellator)λ λ
● W =def x. y.xyy (duplicator)λ λ
● C =def x. y. z.xzy (permutator)λ λ λ
● B =def x. y. z.x(yz)(compositor)λ λ λ
● S =def x. y. z.xz(yz) (substitution)λ λ λ
● =def x. y. z. u.x(yu)(zu) (distribution)Ф λ λ λ λ
● =def x. y. z. u.x(yz)(yu) (distribution)ψ λ λ λ λ
Application of the CL to NLP
● S. k. Shaumyan (University of Yale, USA)
→ first application of combinators to natural language analysis
● M. Steedman (University of Endinburgh, of Pennsylvania, UK, USA)
→ relation with the extended Categorial Grammar
● J.-P. Desclés (Paris-Sorbonne University, FR)
→ application of combinators to complex phenomena
CCG and CL
● several of the combinators which Curry and Feys (1958) use to
define the -calculusλ and applicative systems in general are of
considerable syntactic interest (Steedman, 1988)
● The relationships of these combinators to terms of the -λ
calculus are defined by the following equivalences (Steedman,
2000):
a. Bfg   ≡ x.f(g x)λ b. Tx   ≡ f.f xλ c. Sfg   ≡ x.fx(g x)λ d. Фfgq    x.fq(g q)≡ λ
Main idea in CCG and application operation
● All natural language consists of operators and of
operands.
 Operator (functor) and operand (argument)
 Application: (operator(operand))
 Categorial type: typed operator and operand
● Application of the combinators using syntactic tools: typed
combinators
→ CCG (Combinatory Categorial Grammar) offers CCG types and rules
→ β -reduction rules of combinators are integrated into the CCG rules
”→ systematic”
● Application of the combinators directly to the natural language
sentences
: need to introduce the brackets and decide yourself when you apply the
combinators
”→ heuristic”
Example: use the combinators as a logical tool of
semantic analysis
a. apply directly the β -reduction rules of combinators
John is brilliant
● The predicate is brilliant is an operator which operate on the operand
John to construct the final proposition.
● The applicative representation associated to this analysis is the following:
(is­brillant)John
● We define the operator John* as being constructed from the lexicon John by
[John* = C* John].
1. John* (is­brillant)
2. [John* = C* John]
3. C*John (is­brillant)   [e­C*]
4. is­brillant (John)
● Combinator B and C*:
John loves Mary = (loves Mary)John 
1. C*John loves Mary [i­C*]
2. C*John (loves Mary) bracketing
3. B(C*John) loves Mary [i­B]
4. (C*John) (loves Mary) [e­B]
5. (loves Mary) John [e­C*]
John reads a book at home = (at home(reads (a book)))John
1. (C*John) reads a book at home [i­C*]
2. (B(C*John)reads) a book at home [i­B]
3. (B(B(C*John)reads) a) book at home [i­B]
4. (B(B(B(C*John)reads) a) book at home [i­B]
5. (B(B(B(B(C*John)reads) a) book) at home [i­B]
?6. (B(B(B(B(B(C*John)reads) a) book) at) home [i­B]
?7. (B(B(B(B(B(B(C*John)reads) a) book) at) home) [i­B]
.....
?. (C*John)(reads(a book((at home)))) [e­B]
?. (reads(a book((at home))))John   [e­C*]
Example: use the combinators as a logical tool of
semantic analysis
b. use the CCG types and rules by integrating the β
-reduction rules of combinators into the CCG rules
Introduction to CCG types and rules
● 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
● CCG rules: basic CG rules + combinators
Functional application (>,<); Functional composition (>B, <B); Typeraising
(<C*, >C*); Distribution (<S, >S); Coordination (< , >Φ Φ)
Forward(>) and backward (<) functional application rules
a. X/Y Y X⇒ (>)
b. Y X\Y X⇒ (<)
x/y y x→ ≈ 2/4 * 2 = 4
x/y y x→ ≈ 2/4 * 2 = 4
x/y means that an expression e1 of type x/y waits an expression e2 of 
type y to its left in order to obtain an expression (e1 e2)of type x.
Example
x is an expression e1 of type (S\NP)/NP: loves
y is an expression e2 of type NP: Anna
(S\NP)/NP:loves NP:Anna (S\NP)→
loves (Anna)
x/y y x→ ≈ 2/4 * 2 = 4
x/y y x→ ≈ 2/4 * 2 = 4
Function composition (FC) rules with the combinator B
X/Y:f Y/Z:g ⇒B
X/Z:λx.f(gx) (>B)
Y\Z:f X\Y:g ⇒B
X\Z:λx.f(gx) (<B)
e1: (x/y) e2:(y/z)
-------------------->(>B)
(x/z): B e1 e2
An expression e1 of type x/y waits an expression e2 of type 
y/z to obtain an expression Be1 e2 of type x/z by the 
composition rule.
Example: e1 of type (S\NP)/NP: likes  and e2 of type (NP/N): 
a
(S\NP)/NP:likes  (NP/N):a 
               ­­­­­­­­­­­­­­­­­­­­­­> (>B)
               ((S\NP)/N): (B likes a)
An expression e1 of type x/y waits an expression e2 of type 
y/z to obtain an expression Be1 e2 of type x/z by the 
composition rule.
Example: e1 of type (S\NP)/NP: likes  and e2 of type (NP/N): 
a
(S\NP)/NP:likes  (NP/N):a 
               ­­­­­­­­­­­­­­­­­­­­­­> (>B)
               ((S\NP)/N): (B likes a)
Type-raising rules with the combinator C*
X:a ⇒ S/(S\X):λf.fa (>C*)
X:a ⇒ S\(S/X):λf.fa (<C*)
e1:x
---------------> (>C*)
S/(S\x): C*x
An expression e1 of type x are type­raised to transform e1 
into C*e1 by the type­raising rule.
Example: e1 of type N*:John
e1:John
       ­­­­­­­­> (>C*)
S/(S\NP):C*John
An expression e1 of type x are type­raised to transform e1 
into C*e1 by the type­raising rule.
Example: e1 of type N*:John
e1:John
       ­­­­­­­­> (>C*)
S/(S\NP):C*John
● Syntax-semantic interface using CCG and typed combinators
→ apply the elimination rules of the combinators to the result obtained
from the syntactic analysis
→ get the semantic representation in terms of operator and operand
→ make clear two steps at the syntactic and semantic level
● Combinator B and C* in the CCG rules
John likes the puzzle
John      likes      the   puzzle
NP        (S\NP)/NP  NP/N   N
S/(S\NP): (C*John)                   [>C*]
     S/NP: B(C*John) likes    [>B]
                     NP: (the puzzle)   [>]
S: B(C*John)likes (the puzzle)  [>]
S: (C*John) (likes (the puzzle))   [e­B]
S: (likes (the puzzle))John   [e­C*]
23
Next week...
● Continue about the application of the combinators to natural
language analysis: more complex phenomena