Spring 2012
Juyeon Kang
qkang@fi.muni.cz
B410, Faculty of Informatics, Masaryk University,
Brno, Czech Rep.
IA165
Combinatory Logic for
Computational Semantics
Summing up: last lecture
● How to apply the combinators to natural language analysis
1) using introduction and elimination rules by beta-reduction of1) using introduction and elimination rules by beta-reduction of
combinators: control heurstic of combinatorial application andcombinators: control heurstic of combinatorial application and
bracketingbracketing
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
Remind...1
● The combinator C
: expresses the conversion, that is, the permutation of two arguments of an
binary operator.
: takes one functor f and two arguments x and y. The elimination of the
combinator C by β-reduction allows to converse the position of the
arguement x with y.
The introduction and elimination rule of
the combiantor C
Remind...2
Associativity of the combinatory logic
x(yz)=xyz
Bxyz= (Bxy)z
Associativity of the combinatory logic
x(yz)=xyz
Bxyz= (Bxy)z
Proof
((X·Y)·Z)x ≥ B(BXY)Zx ≥ BXY(Zx) ≥ X(Y(Zx)) ≤ X(BYZx)
≤ BX(BYZx) = X(Y·Z)
Short introduction to ”Passivization”
● Consider the following sentences
a. The man has been killed.
b. One has killed him.
→ Invariant of meaning
→ Relation between two sentences
a’. unary passive predicate (has-been-killed)
b’. active transitive predicate (have-killed)
● active agent corresponds optional by phrase in passive; passivization is a
form of ‘intransitivation’;
– same semantic role (e.g. agent) – (1)
– some differences (e.g. case) – (2)
• other complements are unaffected;
• changes to morphosyntax of verb (aux bebe plus passive participle)
(1) Kim stole the most expensive picture.
(1’) The most expensive picture was stolen by Kim
(2) He saw her.
(2’) She was seen by him.
● Definition of the operator of passivisation 'PASS'
where B and C are the combinator of composition and of conversion and
where is the existential quantificator which, by applying to a binaryΣ
predicate, transforms it into the unary predicate.
[PASS = B   C ]Σ[PASS = B   C ]Σ
(EΣ 1
 E2
)   (E→ 1
 x E2
)(EΣ 1
 E2
)   (E→ 1
 x E2
)
Formal semantic analysis of the ”Passivization”
1/ has­been­killed (the­man)   hypothesis
2/ [has­been­killed=PASS(has killed)]  passive lexical predicate
3/ PASS (has­killed)(the­man)  repl.2.,1.
4/ [PASS = B   C ] Σ definition of 'PASS'
5/ B   CΣ  (has­killed)(the­man)  repl.4.,3.
6/   (CΣ (has­killed))(the­man)  [e­B]
7/ (C(has­killed)) x (the­man)  [e­ ]Σ
8/ (has­killed)(the­main) x  [e­C]
9/ [x in the agentive subject position = one]  definition of 'one'
10/ (has­killed)(the­man)one  repl.9.,8., normal form
We establish the paraphrastic relation between the passive sentence with
expressed agent and its active counterpart:
The man has been killed by the enemy
↓
The enemy has killed the man
The man has been killed by the enemy
↓
The enemy has killed the man
Relation between give-to and receive-from
z gives y to x
↕
x receives y from z
The lexical predicate “give­to” has a predicate converse
associated to “receive­from”;
[receive­from z y x = give­to x y z]
The lexical predicate “give­to” has a predicate converse
associated to “receive­from”;
[receive­from z y x = give­to x y z]
Anna gave a DVD to Nancy
Nancy received a DVD from Anna 
Anna gave a DVD to Nancy
Nancy received a DVD from Anna 
z gave y to x   → (gave_to x y z)
x received y from z   → (received_from z y x) 
z gave y to x   → (gave_to x y z)
x received y from z   → (received_from z y x) 
1/ ((receive­from) z) y x
2/ C((receive­from) z) x y
3/ BC(receive­from) z x y
4/ C(BC(receive­from))  x z  y
5/ C(C(BC(receive­from)) x) y z
6/ BC(C(BC(receive­from))) x y z
7/ [give­to=BC(C(BC(receive­from)))]
8/ give­to x y z
x: Anna
y: a DVD
z: Nancy
x: Anna
y: a DVD
z: Nancy
Prove the following relation
[((receive­from) z) y x = give­to x y z]
x lead y   → (lead y) x
y follow x   → (follow x) y
x lead y   → (lead y) x
y follow x   → (follow x) y
x chase y   → (chase y) x
y flee x   → (flee x) y
x chase y   → (chase y) x
y flee x   → (flee x) y
Combinator C of conversion
(f(x)) y   (→ Cf) y x
What is the semantic relations between these 
couples of sentences? (Show in the classwork)
Combinator C of conversion
(f(x)) y   (→ Cf) y x
What is the semantic relations between these 
couples of sentences? (Show in the classwork)
Multilingual examples of Passives-1
● Passive transformation In Czech
Transformation T1
(a) [NP1
]NOM
 – [V1
lex
]fin
 – [NP2
]ACC
 ⇒ (b)
(b) [NP2
]NOM
 – [V2
be]fin
 – [V1
lex]part
 – [NP1
]INSTR
Transformation T1
(a) [NP1
]NOM
 – [V1
lex
]fin
 – [NP2
]ACC
 ⇒ (b)
(b) [NP2
]NOM
 – [V2
be]fin
 – [V1
lex]part
 – [NP1
]INSTR
Examples
(a) Petr sliboval Pavlovi, že přijde ⇒ (b)
PeterNOM promised PaulDAT that arrives3SM ⇒ (b)
‘Peter promised to Paul that he arrives.’
(b) Pavlovi bylo slibováno (Petrem), že přijde
PaulDAT was promised (PeterINSTR) that arrives3SN
‘It was promised to Peter (by Paul) that he arrives.’
Examples
(a) Petr sliboval Pavlovi, že přijde ⇒ (b)
PeterNOM promised PaulDAT that arrives3SM ⇒ (b)
‘Peter promised to Paul that he arrives.’
(b) Pavlovi bylo slibováno (Petrem), že přijde
PaulDAT was promised (PeterINSTR) that arrives3SN
‘It was promised to Peter (by Paul) that he arrives.’
Multilingual examples of Passive-2
We need to define the operator of the passivisation
[PASS=B   C =   Σ Σ • C]
Consider the following relation:
bylo­slibováno and sliboval
x sliboval y
 → ((sliboval y) x)
⇓
y bylo­slibovano (by x)
 → (bylo­slibovano) y
● Formal analysis of the passivisation in Czech
1/ bylo­slibováno y (by x=Petrem)
2/ [bylo­slibováno=PASS (sliboval)]
3/ PASS (sliboval) y
4/ B   C (sliboval) yΣ
5/   (C sliboval) yΣ
6/ (C sliboval)   x’   y
7/ [x’= agentive subject]
8/ sliboval y x’
[bylo­slibováno Pavlovi] = 
Passivisation of [sliboval Pavlovi Petr]
[bylo­slibováno Pavlovi] = 
Passivisation of [sliboval Pavlovi Petr]
17
Next week...
● Continue about the application of the combinators to natural
language analysis: aspecto-temporal operatorsaspecto-temporal operators