CONSEQUENCE RELATIONS IN DRT 
Seiki Akama 
Computational Logic Laboratory, Department of Information Systems, 
Teikyo University of Technology, 2289 Uruido, Ichihara-shi, 
Chiba, 290-01, Japan. 
(TEL) ÷81-436-74-5511, (FAX) +81-436-74-2592. 
Yotaro Nakayama 
Nihon Unisys Ltd., Tokyo Bay Development Center, 
1-10-9, Shinonome, Koto-ku, Tokyo, 135, Japan. 
Abstract 
We discuss some consequence relations in DRT 
useful to discourse semantics. We incorporate 
some consequence relations into DRT using se- 
quent calculi. We also show some connections 
of these consequence relations and existing par- 
tial logics. Our attempt enables us to display 
several versions of DRT by employing different 
consequence relations. 
1. Introduction 
Discourse Representation Theory (DRT) due 
to Kamp (1981) provides a promising frame- 
work for discourse semantics. DRT is in fact 
successul in formalizing several challenging top- 
ics, e.g. anaphora, tense and belief; see Asher 
(1986, 1993) and Helm (1982). Due to its clear 
semantics with the construction algorithm, DRT 
is also used as a background theory in compu- 
tational lhtguistics; see Wada and Asher (1986) 
and Ishikawa and Akama (1992). 
However, DRT lacks a deductive (proof- 
theoretic) formulation to serve as an abstract 
interpreter for discourse understanding, since 
it is formalized by means of the notion of par- 
tial models. This prevents us from utilizing 
DRT in various ways for natural language un- 
derstanding systems. To make DR'\]' more flex- 
ible for computational linguistics, we need to 
generalize a formal basis in a proof-theoretic 
fashion. If this is successful, computational lin- 
guists will be able to reconstruct DRT for their 
own systems using programming languages like 
Prolog and LISP. There are perhaps several 
ways to give an operational semantics of DRT. 
One of the attractive approaches is to investi- 
gate consequence relations associated with DR'I?. 
It is thus very useful to study some conse- 
quence relations in DRT to develop different 
versions of DRT. 
The purpose of this paper is to explore con- 
sequence relations in DRT, one of which ex- 
actly produces Kamp's original semantics. We 
incorporate some consequence relations defined 
by partial semantics into DRT using sequent 
calculi. Our attempt enables us to display sev- 
eral versions of DRT by employing different 
consequence relations. We also show some con- 
nections of the proposed consequence relations 
and partial logics in the literature. 
2. Overview of DRT 
In this section, we give a brief introduction 
to DRT. For a detailed exposition, the reader 
should consult Asher (1993). The basic idea 
of DRT is to formalize a dynamic represen- 
tation of partial interpretations by means of 
classical models using a construction algorithm 
of discourse representation structures (DRSs). 
Observe that DRSs can be regarded as such 
abstract objects as partial models, mental rep- 
resentations, or (partial) possible worlds. But, 
1114 
such identifications do not seem essential to the 
significance of DRT. 
'\]'he language of DRT is called Discourse 
Representation Language (DRL), which is like 
a standard quantifier-free first-order language 
except discourse referents and conditions. The 
logical symbols of DRL include =: (identity), 
--~ (comlitional), V (disjunction) and ~ (nega- 
tion). A discourse representation (DR) K is 
expressed as a pair (UK, ConE), where UE is a 
set of discourse re\]erents, and Conic is a set of 
conditions. Each condition is either atomic or 
complex. Complex conditions are of the form: 
K1 :~ K2, KI V K2 or ~K1, where both K1 
and K2 are Dl~s. 
A discourse representation structure (DRS) 
is a partially ordered set of DRs, which can 
be constructed by means of DRS construction 
rules whose application reflects the syntactic 
composition of the sentences in the discourse. 
When each DR of a DRS is maximal, the DRS 
is called a complete DRS. Intuitively speaking, 
each stage in the construction algorithm can 
be viewed as a partial possible worlds, in which 
more information resulting from the processing 
of a further bit of the discourse changes it into 
a more precise description of the world. 
A model for DRL is an ordered pair (DM, 
FM), where DM is the domain of M and FM 
is an interpretation function of constants and 
predicates. An embedding \]'unction for a DR 
K in a model M is a mapping from discourse 
referents in UK into the domain of M. An ex- 
tension of an embedding flmction f for K in M 
to an embedding function g for K' in M is de- 
fined as g: (Dora(f) U UE, ) --~ DM. We write 
f C K g to mean that g extends an embedding 
function f to an embedding of K'. The notaion 
M ~-t,K C abbreviates that M satisfies C un- 
der ffor K. A proper embedding of K in M is an 
embeddhtg flmetion such that f ~K g and for 
any condition C in K, M ~g,E C. The notions 
of proper embedding and satisfaction can be 
extended to general cases by slmnltaneous re- 
cursion; see Asher (1993). A DR K is shown to 
be true in a model M iff there is a proper em- 
bedding of K in M. A DR K implies a DR K' iff 
every model in which K is true is also a model 
in which K' is true. This definition induces a 
consequence relation in DRT, but we have no 
reason to consider it as the only plausible for 
DRT. In fact, it is our job in tMs paper to seek 
alternate definitions. 
3. Consequence Relations and Sequent 
C alcull 
A partial semantics for classical logic is implicit 
in the so-called Beth tableaux. This insight can 
be generalized to study consequence relations 
in terms of Gentzen calculi. The first impor- 
tant work in this direction has been done by 
van Benthem (1986, 1988). We here try to ap- 
ply this technique to DRT. Since our approach 
can replace the base logic of DRT by other in- 
teresting logics, we obtain alternative versions 
of DttT. 
Recall the basic tenet of Beth tableaux. 
Namely, Beth tableaux (also semantic tableaux) 
prove X --~ Y by constructing a counterexam- 
pie of X K: ~Y. In fact, Beth tableaux induce 
partial semantics in the sense that there may 
be counterexamples even if a branch remains 
open. Let X and Y be sets of formulas, and 
A and B be formulas. And we write X b Y 
to mean that Y is provable from X. Van Ben- 
there's partial semantics for classical logic can 
be axiomatized by the Gentzen calculus, which 
has the axiom of the form: 
X, A P A, Y 
and the following sequent rules: 
(Weakening) X bY ~ X, A F A, Y. 
(Cut) X, Ab Y and XF A,Y 
=--~ X F Y. 
(~R) X, AbY ~ Xb~A,Y. 
(~L) XPA, Y ~=~ X,~At-Y. 
(&R.) X P Y, h and X F Y,B 
=-.~ Xt-Y, A & B. 
(&L) X,A,B bY ---.s X,A &B P Y. 
(vR) XPA, B,Y ~ XPAvB, Y. 
(vL) X,A F Y and X, B t- Y 
==> X, AVBbY. 
Van Benthem's formulation can be extended 
for partial logics. Because such an extension 
1115 
uses the notion of partial valuations, it is not 
difficult to recast the tzeatment for DRT. 
Let V be a partial valuation assigning 0, 1 
to some atomic formula p. Now, we set V(p) 
= 1 for p on the left-hand side and V(p) = 0 
for p on the right-hand side in an open branch 
of Beth tableaux. This construction can be 
easily accommodated to sequent calculi. Then, 
we can define the following two consequence 
relations: 
(C1) for all V, if V(Pre) = 1 
then V(Cons) = 1, 
(C2) for all V, if V(Pre) = 1 
then V(Cons) # 0, 
where Pre and Cons stand for premises (an- 
tecedent) and conclusion (succedent) of a se- 
qnent, respectively. In a classical setting, (C1) 
and (C2) coincide. It is not, however, the case 
for partial logics. 
The Gentzen calculus G1 for C1 is obtain- 
able from the above system without right (~)- 
rule by introducing the following rules: 
(~R) X~-A,Y ~ XP~A,Y. 
(~L) X, AF-Y ~ X,~-A~-Y. 
aR) x ~A, Y 
X ~- ~(A & B), Y. 
(~ &L) X,,-~A F- Y and X, ~B ~- Y 
X, ~(A & B) ~- Y. 
(,-~VR) XF- ~A,Y and X~- NB, Y 
X P --~(A V B), Y. 
(~ VL) X, ,-~A, ~B t- Y 
x, ~(A v B) t- Y. 
Van Benthem (1986) showed that G1 is a Gentzen 
type axiomatization of C1. To guarantee a cut- 
.free formulation, we need to modify van Ben- 
them's original system. We denote by GC1 the 
sequent calculus for GC1, which contains the 
axioms of the form: (A1) A }- A and (A2) A, 
--~A ~-, with the right and left rules for (&), 
(V), (~), (~ &) and (~ V) together with 
(Weakening) and (Cut). It is shown that GC1 
is equivalent to G1 without any difficulty. As 
a consequence, we have: 
Theorem 1 
C1 can be axiomatized by GC1. 
The Gentzen system GC2 for C2 can be ob- 
tained from (GC1) by adding the next axiom: 
(A3) A, ~A. 
Theorem 2 
C2 can be axiomatized by GC2. 
There are alternative ways to define con- 
sequence relations by means of sequent calculi. 
For example, it is possible to give the following 
alternate definitions. 
(C3) for aH V, if V(Pre) -- 1 
then V(Cons) = 1 
and if V(Cons) = 0 
then V(Pre) = 0. 
The new definition obviously induces inconsis- 
tent valuations. The Gentzen system GC3 is 
obtainable from GC1 by replacing (A2) by the 
following new axiom: 
(A4) A, -A ~ B, ~B. 
Theorem $ 
C3 can be axiomatized by GC3. 
4. Relation to Partial Logics 
In this section, we compare the proposed Gentzen 
systems with some existing partial logics, in 
particular, three-valued and four-valued log- 
ics in the literature; see Urquhart (1986). To 
make connections to partial logics clear, we ex- 
tend DRL with weak negation "--" to express 
the lack of truth rather than verification of fal- 
sity in discourses. We denote the extended lan- 
guage by EDRL. In the presence of two kinds 
of negation, we can also define two kinds of 
implication as material implications. We need 
the next rules for weak negation: 
(-R) X, A ~- V ~ X ~- -~A, Y. 
X A, X, Y. 
In fact, these rules provide a new consequence 
reation of EDRL denoted by ~EDRL. Our 
first result is concerned with the relationship of 
GC1 and Kleene's (1952) strong three-valued 
logic KL, namely 
Theorem 4 
The consequence relations of GC1 and KL are 
equivalent. 
From this theorem, EDRL can be identified 
with the extended Kleene logic EKL. Let A 
-~,, B be an abbreviation of ~A V B. Then, we 
can also interpret Lukasiewicz's three-valued 
1116 
logic L3. In fact, the Lukasiewicz huplication 
D can be defined as follows: 
A D B =a~t (A -~0 B) & (~B-*,~ ~A) 
which implies 
t=EKL h D B iff A ~:EKL B and ~B ~EKL 
~i. 
This is closely related to the consequence rela- 
tion C3. 
Theorem 5 
AFc, a B iff ~EKL A D B. 
If we drop (A2) from GC1, we have the sequent 
calculus GCI-, which is shown to be equiva- 
lent to Belnap's (1977) four-valued logic BEL. 
Theorem 6 
~-BEL = ~GC1-" 
The four-valued logic BEL can handle both 
incomplete and inconsistent information. We 
believe that four-vaNed semantics is plausible 
as a basis for representational semantics like 
DRT, which should torelate inconsistent infor- 
mation in discourses. In view of these results, 
we can develop some versions of DRT which 
may correspond to current three-valued and 
four-vahed logics; see Akama (1994). 
5. Conclusions 
We have studied a proof-theoretic foundation 
for DRT based on consequence relations de- 
fined by partial semantics. These consequence 
relations yield alternative versions of DRT to 
be used for different applications. We have 
noted some connections between these relations 
and partial logics, in particular three-valued 
and four-valued logics. We believe that the 
significance of our work lies in reformulating 
DRT in sequent calculi to be easily applied to 
computational linguistics. 
There are several topics that can be further 
developed. First, we should give a more de- 
tailed discussion of what sort of completeness 
proof is involved, although we have established 
some correspondence results. Second, it is very 
interesting to show how the proposed conse- 
quence relations affect DRT in mo~e detailed 
ways. Third, we need to extend the present 
work for the predicate case to take care to cap- 
ture the dynamic effect of the quantificational 
structure of DRT. 
References 
Akama, S. (1994): A proof system for useful 
three-valued logics, to appear in Proc. 
of Japan-CIS Symposium on Knowledge 
Based So\]tware Engineering. 
Asher, N. (1986): Beliefiu discourse represen- 
tation theory, Journal o\] Philosophical 
Logic 15, 127-189. 
Asher, N. (1993): Re\]erence to Abstract Objects 
in Discourse, Kinwer, Dordrecht. 
Belnap, N. D. (1977): A useful four-valued logic, 
J. M. Dunn and G. Epstein (eds.), Mod- 
ern Uses of Multiple- Valued Logic, 8-37, 
Reidel, Dordrecht. 
Heim, i. (1982): The Semantics o\] Indefinite 
and Definite Noun Phrases, Ph.D. dis- 
sertation, University of Massachussetts 
at Amherst. 
Ishikawa, A. and Akama, S. (1991): A seman- 
tic interface for logic granmlars and its 
application to DRT, C. Brown and G. 
Koch (eds.), Natural Language Under- 
standing and Logic Programming III, 281- 
292, North-Holland, Amsterdam. 
Kamp, H. (1981): A theory of truth and se- 
mantic representation, J. Groenendijk, 
T. Janssen and M. Stokhof (eds.), For- 
mal Methods in the Study of Language, 
277-322, Mathematisch Centrum Tracts, 
Amsterdam. 
Kleene, S. C. (1952): Introduction to Metamath- 
ematics, North-Holland, Amsterdam. 
\["1'(111\]1~1"1. m. (19~6): l~\[any-valued logic, I). (~-nb- 
hily ;|l|,fl |¢. (~111,111,\[1\[1(,i; (('ds.). lhz,dt~o(,/,: 
o.\[ l~/~.ih).~ophic,.l Logic vol. IV. 71 liB, 
llcidcl, l)ordrm'hl. 
Wada, H. and Asher, N. (1986): BUILDRS: An 
implementation of DR Theory and LFG, 
Proc. of COLING-86, 540-545. 
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