Spring 2012
Juyeon Kang
qkang@fi.muni.cz
B410, Faculty of Informatics, Masaryk University,
Brno, Czech Rep.
IA165
Combinatory Logic for
Computational Semantics
➢ Combinators allow to introduce and define new operators which mark the
aspecto-temporal relation. ”→ aspecto-temporal operators”
We show the aspecto-temporal relation of the given text in
the SDRT and define the aspecto-temporal operators by
means of the combinators
→ propose a formal semantic analysis by taking into account of
the aspecto-temporal relation in the text
→ establish the temporal relations between sentences
We show the aspecto-temporal relation of the given text in
the SDRT and define the aspecto-temporal operators by
means of the combinators
→ propose a formal semantic analysis by taking into account of
the aspecto-temporal relation in the text
→ establish the temporal relations between sentences
● Hypothesis for the computational and semantic
representation of the temporality
The temporality of language can not be described without taking account
of the aspectuality. All aspectual notions imply an underlying temporality;
→ most of situations require topological relations between open closed
boundaries of intervals compounded by instants.
(show the examples of the topological relations on the board)
(i) STATEO
( ) is developed on the topological open interval ‘O’ and is true for eachΛ
instant of ‘O’;
(ii) EVENF
( ) is developed on the closed interval ‘F’ and is true at the right closedΛ
boundary ‘ (F)’;δ
(iii) PROCJ
( ) is developed on the interval ‘J’ with a left-closed boundary ‘ (J)’ andΛ γ
right-open boundary ‘ (J)’ and is true at each instant ‘t’ of ‘J’ before the right openδ
boundary of ‘ (J)’ (t < (J))δ δ
(i) STATEO
( ) is developed on the topological open interval ‘O’ and is true for eachΛ
instant of ‘O’;
(ii) EVENF
( ) is developed on the closed interval ‘F’ and is true at the right closedΛ
boundary ‘ (F)’;δ
(iii) PROCJ
( ) is developed on the interval ‘J’ with a left-closed boundary ‘ (J)’ andΛ γ
right-open boundary ‘ (J)’ and is true at each instant ‘t’ of ‘J’ before the right openδ
boundary of ‘ (J)’ (t < (J))δ δ
π1.1. Last night (reform: All that follows occurred last night): Temporal Framework, STATEO1
 
(state)
π1.2. Fred had a great evening : EVENF1
 (event)
π2. He had a great meal: EVENF2
 (event)
π3. He ate salmon: EVENF3
 (event)
π4. He devoured lots of cheese: EVENF4 
(event)
π5. He then won a dancing competition: EVENF5
 (event)
π1.1. Last night (reform: All that follows occurred last night): Temporal Framework, STATEO1
 
(state)
π1.2. Fred had a great evening : EVENF1
 (event)
π2. He had a great meal: EVENF2
 (event)
π3. He ate salmon: EVENF3
 (event)
π4. He devoured lots of cheese: EVENF4 
(event)
π5. He then won a dancing competition: EVENF5
 (event)
(1) Fred had a great evening last night (π1). He had a
great meal (π2). He ate salmon (π3). He devoured lots of
cheese (π4). He then won a dancing competition (π5).
(Asher and Lascarides 2003)
(1) Fred had a great evening last night (π1). He had a
great meal (π2). He ate salmon (π3). He devoured lots of
cheese (π4). He then won a dancing competition (π5).
(Asher and Lascarides 2003)
PROCJ0
 ((I­SAY) (& (ASPI
 ( )) [I REP JΛ 0
]))
comment:
the aspectual process PROCJ0
 is applied on the result of the application of (I­
SAY) on a conjunction of an aspectualized predicative relation ASPI
 ( )  and a Λ
temporal relation [I REP J0
] between the interval I related to the predicative 
relation and an interval J0
 related to enunciative process.
PROCJ0
 ((I­SAY) (& (ASPI
 ( )) [I REP JΛ 0
]))
comment:
the aspectual process PROCJ0
 is applied on the result of the application of (I­
SAY) on a conjunction of an aspectualized predicative relation ASPI
 ( )  and a Λ
temporal relation [I REP J0
] between the interval I related to the predicative 
relation and an interval J0
 related to enunciative process.
Definition of the speech act operator ”I-am-saying”
: a result of a functional composition of the two operators:
”I­SAY” and ”PROCJ0
”
Definition of the speech act operator ”I-am-saying”
: a result of a functional composition of the two operators:
”I­SAY” and ”PROCJ0
”
p1.1. PROCJ0
 (I­SAY (& (STATEO1
 (All that follows occurred last night)) [ (Oδ 1
) < 
(Jδ 0
)])
p1.2. PROCJ0
 (I­SAY (& (EVENF1
 ((have (a great evening))(Fred))) [ (Fδ 1
) <  (Jδ 0
)])
p2. PROCJ0
 (I­SAY (& (EVENF2
 ((have (a great meal))(Fred))) [ (Fδ 2
) <  (Jδ 0
)])
p3. PROCJ0
 (I­SAY (& (EVENF3
 ((eat (salmon)) (x)))[ (Fδ 3
) <  (Jδ 0
)])
p4. PROCJ0
 (I­SAY (& (EVENF4
 ((devour (lots of cheese))(x))) [ (Fδ 4
) <  (Jδ 0
)])
p5. PROCJ0
 (I­SAY (& (EVENF5
 ((win (a dancing competition)) (x))) [ (Fδ 5
) <  (Jδ 0
)])
p1.1. PROCJ0
 (I­SAY (& (STATEO1
 (All that follows occurred last night)) [ (Oδ 1
) < 
(Jδ 0
)])
p1.2. PROCJ0
 (I­SAY (& (EVENF1
 ((have (a great evening))(Fred))) [ (Fδ 1
) <  (Jδ 0
)])
p2. PROCJ0
 (I­SAY (& (EVENF2
 ((have (a great meal))(Fred))) [ (Fδ 2
) <  (Jδ 0
)])
p3. PROCJ0
 (I­SAY (& (EVENF3
 ((eat (salmon)) (x)))[ (Fδ 3
) <  (Jδ 0
)])
p4. PROCJ0
 (I­SAY (& (EVENF4
 ((devour (lots of cheese))(x))) [ (Fδ 4
) <  (Jδ 0
)])
p5. PROCJ0
 (I­SAY (& (EVENF5
 ((win (a dancing competition)) (x))) [ (Fδ 5
) <  (Jδ 0
)])
Fred devoured lots of cheese
p4. PROCJ0
 (I­SAY (& (EVENF4
 ((devour (lots of cheese))(x))) [ (Fδ 4
) <  (Jδ 0
)])
Fred devoured lots of cheese
p4. PROCJ0
 (I­SAY (& (EVENF4
 ((devour (lots of cheese))(x))) [ (Fδ 4
) <  (Jδ 0
)])
Definition the ascpecto­temporal marker in term of 
the combinators
­ed
COMPLETE­EVENT­PAST
Definition the ascpecto­temporal marker in term of 
the combinators
­ed
COMPLETE­EVENT­PAST
1/ Fred devoured lots of cheese
2/ ((devour­ed (lots of cheese))(Fred))
3/ past­suffix P2
A2
A1
4/ COMPLETE­EVENT­PAST (P2
A2
A1
)
5/ [COMPLETE­EVENT­PAST=X & ([ (Fδ 4
) <  (Jδ 0
)]) I­am­saying EVENF4
]
6/ [X= B6
C3
C3
CB2
]
7/ I­am­saying (& (EVENF4
(P2
A2
A1
))([ (Fδ 4
) <  (Jδ 0
)]))
8/ [I­am­saying =B PROCJ0
 (I­SAY)]
9/ PROCJ0
 (I­SAY (& (EVENF4
 (P2
A2
A1
)) ([ (Fδ 4
) <  (Jδ 0
)])))
10/ PROCJ0
 (I­SAY (& (EVENF4
 ((devour (lots of cheese))(x))) [ (Fδ 4
) <  (Jδ 0
)])
1/ Fred devoured lots of cheese
2/ ((devour­ed (lots of cheese))(Fred))
3/ past­suffix P2
A2
A1
4/ COMPLETE­EVENT­PAST (P2
A2
A1
)
5/ [COMPLETE­EVENT­PAST=X & ([ (Fδ 4
) <  (Jδ 0
)]) I­am­saying EVENF4
]
6/ [X= B6
C3
C3
CB2
]
7/ I­am­saying (& (EVENF4
(P2
A2
A1
))([ (Fδ 4
) <  (Jδ 0
)]))
8/ [I­am­saying =B PROCJ0
 (I­SAY)]
9/ PROCJ0
 (I­SAY (& (EVENF4
 (P2
A2
A1
)) ([ (Fδ 4
) <  (Jδ 0
)])))
10/ PROCJ0
 (I­SAY (& (EVENF4
 ((devour (lots of cheese))(x))) [ (Fδ 4
) <  (Jδ 0
)])
­ed= past­suffix
Past­suffix=COMPLETE­EVENT­PAST 
­ed= past­suffix
Past­suffix=COMPLETE­EVENT­PAST 
Discursive structure
Temporal relation of discourse in the enunciative
referential framework
Incomplete present-process
● The police chase the red car (at this moment)
Definition of the aspect incomplete present
Aspectual operators
INC_PRST: grammaticalized aspectual operator
prst_process
: pre­morphologic asepctual operator
Definition of the aspect incomplete present
Aspectual operators
INC_PRST: grammaticalized aspectual operator
prst_process
: pre­morphologic asepctual operator
INC_PRST: 
Not associated to the lexical predicate but concern the whole predicative 
relation
The linguistic trace can be expressed at the morpho­syntactic level in the 
form of the verbal morphemes (pre­verb, suffix, affix..)
prst_process
: 
Takes an unique lexical predicate as operande and expresses the verbal 
aspect being attached to the verb
INC_PRST: grammaticalized aspectual operator
prst_process
: pre­morphologic asepctual operator
INC_PRST: grammaticalized aspectual operator
prst_process
: pre­morphologic asepctual operator
● Hypothesis
(­e (chas­)) (the red car)(the police)
 (prst–process
 (P2
) ) A2  
A1
 INC_PRST (P2
 A2
 A1
) 
I­am­sayingJ0
 (& (PROCJ1
 (P2
 A2
 A1
)) ([ (Jδ 1
)= (Jδ 0
)]) ) 
The police chase the red car (at this moment)
4/ I­am­sayingJ0
 (& (PROCJ1
 (P2
 A2
 A1
)) ([ (Jδ 1
)= (Jδ 0
)]) )  hyp.
5/ B2
 I­am­sayingJ0
 & (PROCJ1
 (P2
 A2
 A1
)) ([ (Jδ 1
)= (Jδ 0
)])  int. B2
6/ B (B2
 I­am­sayingJ0
 &) PROCJ1
 (P2
 A2
 A1
) ([ (Jδ 1
)= ()]) δ int. B
7/ C2
 B (B2
 I­am­sayingJ0
 &) PROCJ1
 ([ (Jδ 1
)= (Jδ 0
)]) (P2
 A2
 A1
)  int. C2
8/ B2
 (C2
 B) B2
 I­am­sayingJ0
 & PROCJ1
 ([ (Jδ 1
)= (Jδ 0
)]) (P2
 A2
 A1
)  int. B2
9/ [INC_PRST J1
 J0
 =déf B2
 (C2
 B) B2
 I­am­sayingJ0
 & PROCJ1
 ([ (Jδ 1
)= (Jδ y
)])] 
def.
10/ [INC_PRST=déf   ∃ J0
 J1
{B2
(C2
 B) B2
 I­am­sayingJ0
 & PROCJ1
 ([ (Jδ 1
)= (Jδ 0
)])}] 
int. ∃
11/ INC_PRST (P2
 A2
 A1
)  rempl. 6, 5
12/ B2
 INC_PRST P2
 A2
 A1
  int. B2
13/ [prst–process
 = déf B2
INC_PRST]  def.
14/ (prst–process
 (P2
) ) A2 
A1
  rempl. 10, 9
A. Definition of the INC_PRST (Incomplete process)
9/ [INC_PRST J1
 J0
 =déf B2
 (C2
 B) B2
 I­am­sayingJ0
 & PROCJ1
 ([ (Jδ 1
)= (Jδ y
)])] 
B. Introduction of the existential quantificator ∃
10/ [INC_PRST=déf   ∃ J0
 J1
{B2
(C2
 B) B2
 I­am­sayingJ0
 & PROCJ1
 ([ (Jδ 1
)= (Jδ 0
)])}] 
A. Definition of the INC_PRST (Incomplete process)
9/ [INC_PRST J1
 J0
 =déf B2
 (C2
 B) B2
 I­am­sayingJ0
 & PROCJ1
 ([ (Jδ 1
)= (Jδ y
)])] 
B. Introduction of the existential quantificator ∃
10/ [INC_PRST=déf   ∃ J0
 J1
{B2
(C2
 B) B2
 I­am­sayingJ0
 & PROCJ1
 ([ (Jδ 1
)= (Jδ 0
)])}] 
Comment:
The operator INC_PRST depends on intervals 'J1
' and 'J0
'. It
is thus necessary to abstract from the operator INC_PRST by
supposing simply the existence of such intervals which respect a
temporal condition. To do this, we introduce the operator of
the existential quantification at the step 10.
Temporal scheme of the incomplete present process
J0
 I am saying....
J1
 The police chase the car at this moment....
T0
 The real present moment...now...
Summing up-1
● Operators of the aspectuality
a. COMPLETE-EVENT-PAST: e.g. Verbal ending -ed
 [COMPLETE­EVENT­PAST=X & ([ (Fδ 4
) <  (Jδ 0
)]) I­am­saying EVENF4
]
where X is  B6
C3
C3
CB2
b. INC-PRST: verbal ending -e
[INC_PRST J1
 J0
 =déf B2
 (C2
 B) B2
 I­am­sayingJ0
 & PROCJ1
 ([ (Jδ 1
)= (Jδ y
)])] 
Summing up-2
Computaional semantic representation of the asepcto­temporality
1. morpho­syntactic representation
Peter devour ­ed lots of cheese
­ed is a verbal morpheme which mark the aspecto­temporal relation
2. logico­grammatical representation
(COMPLETE­EVENT­PAST devour) (lots­of­cheese)(Peter)
3. discursive representation
PROCJ0
 (I­SAY (& (EVENF4
 ((devour (lots of cheese))(x))) [ (Fδ 4
) <  (Jδ 0
)])
Computaional semantic representation of the asepcto­temporality
1. morpho­syntactic representation
Peter devour ­ed lots of cheese
­ed is a verbal morpheme which mark the aspecto­temporal relation
2. logico­grammatical representation
(COMPLETE­EVENT­PAST devour) (lots­of­cheese)(Peter)
3. discursive representation
PROCJ0
 (I­SAY (& (EVENF4
 ((devour (lots of cheese))(x))) [ (Fδ 4
) <  (Jδ 0
)])
20
Next week...
● Continue about the application of the combinators to natural
language analysis: QuantificationQuantification