Spring 2012
Juyeon Kang
qkang@fi.muni.cz
B410, Faculty of Informatics, Masaryk University,
Brno, Czech Rep.
IA165
Combinatory Logic for
Computational Semantics
Formal semantic analysis of aspecto-temporal relation
➢ Many works based on the analysis of temporal relations (Asher and Vieu
2005; Grosz and Sidner 1986; Lascarides and Asher 1993; Mann and
Thompson 1987) assume that a text (or discourse) has hierarchical
structures
➢ Representing this structure is quite different in various theories (SDRT,
CAG, )
➢ The Segmented Discourse Representation Theory (SDRT) (Asher 1993) is
anchored in the formal semantics to study the complex interplay between
the semantic contribution of propositions with their components and the
segmentation of discourse. dynamical and theoretical tool for the→
analysis of discourse
(Asher, 1993; Asher and Vieu, 2005; Descles et al., 2011)
➢ 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
● Preliminary works
1) Introduce the SDRT (Asher, 1993; variant of the DRT by Kamp)
2) Introduce the derived combinators: the powers of a combinator and
the deferred combinators
General introduction to SDRT
● Method of modelling dialogue
● SDRT (Asher 1993, Asher & Lascarides 2003)
– task:
compute pragmatically preferred interpretation of discourse / model
pragmatic competence more than what grammar outputs, less than full→
belief revision
– for dialogue
→ SDRT: logic(s) for representing & reasoning about cognitive states
● The SDRT defines a set of speech-act labels, π1
, … πn
, related by
discourse relations R
→ each speech-act label is associated with a ’discourse constituent’,
which is either simple—the logical formula representing a simple clause— or
complex —a SDRS representing a discourse segment.
● Discourse relations used in SDRT for modelling dialogue
: Background, Continuation, Parallel, Contrast, Topic, Precondition,
Commentary, etc.
: relations used for temporal structuring Narration, Result,→
Elaboration, Explanatation
relations used for temporal structuring
→ Narration, Result, Elaboration, Explanatation
− Narration( , ): The event described in is a consequenceα β β
of the event described in ;α
− Result( , ) : the event described in caused the eventα β α
or state described in ;β
− Elaboration( , ) : ’s event is part of ’s;α β β α
− Explanation ( , ): the event describe in explains whyα β β
’s event happened.α
− Narration( , ): The event described in is a consequenceα β β
of the event described in ;α
− Result( , ) : the event described in caused the eventα β α
or state described in ;β
− Elaboration( , ) : ’s event is part of ’s;α β β α
− Explanation ( , ): the event describe in explains whyα β β
’s event happened.α
 These relations are appeared where the clause   α  presents in the 
text before β .
Discourse relations
− Elaboration links the first clause (π1) to the rest of the discourse
(π2-π5);
− Narration links the message of great meal to the dancing competition,
i.e. (π2) and (π5);
− Elaboration links the message of great meal to two following clause,
i.e. (π3) and (π4).
(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)
(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)
Clauses (π3­π4) elaborate the meal (π2), which in turn elaborates the evening (π1). (π5) 
also elaborates the evening, but unlike (π3­π4) it doesn’t elaborate the meal. Rather, 
it forms a narrative with (π2).
● 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)
Definition of the aspecto-temporal operators
● To analyze semantically the expressions of linguistic temporality, a
predicative relation, noted ‘ (lexis)’ (see Culioli 1999), is aspectualizedΛ
as a state, or an event, or a process (Comrie 1976; Desclés 1980; 1990b;
Mourelatos 1981) in using aspectual operators STATEO
, EVENF
and PROCJ
which are actualized on topological intervals of instants:
(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
● Example of event, state, process
STATE:  The air smells of jasmine.
PROC: It’s snowing.   
EVENT: He crossed the street upon seeing her. 
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
”
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
)])
Derived combinators
● The powers of a combinator: Xn
Given a combinator X,
X0
=def
I
X1
=def
X
Xn+1
=def
X•Xn
Application:
B2
f x y z ≥ B(Bf)xyz ≥ Bf(xy)z ≥ f(xyz)
C2
f x y ≥ C(Cf)xy ≥ Cfyx ≥ fxy
W2
f x ≥ W(Wf)x ≥ Wfxx ≥ fxxx
K2
f x y ≥ K(Kf)xy ≥ Kfy ≥ f
B2
=B•B=BBB
B3
=B•B2
=B•(BBB)=BBBBB
B4
=B•B3
=B•(BBBBB)=BBBBBBB
B2
=B•B=BBB
B3
=B•B2
=B•(BBB)=BBBBB
B4
=B•B3
=B•(BBBBB)=BBBBBBB
● The deferred combinators: Xn
X(k) defers the action of X by K steps
X(k)
fX1
...Xm+k
≥ fx1
...xk
Χ1
..Χ’ n
’
Application:
C(k)
interchanges xk+1
and xk+2
; W(k)
causes a repetition of xk+1
; and K(k)
causes the cancellation of xk+1
B(k)
fx1
x2
...xk
xk+1
xk+2
≥β
fx1
x2
...xk
(xk+1
xk+2
)
C(k)
fx1
x2
...xk
xk+1
xk+2
≥β
fx1
x2
...xk
xk+2
xk+1
W(k)
fx1
x2
...xk
xk+1
≥β
fx1
x2
...xk
xk+1
xk+1
K(k)
fx1
x2
...xk
xk+1
≥β
fx1
x2
...xk
I(k)
fx1
x2
...xk
≥β
fx1
x2
...xk
19
Next week...
● Continue about the application of the combinators to natural
language analysis: aspecto-temporal analysisaspecto-temporal analysis