'TOWARDS DISCOURSE-0RIENTED NONMONOTONIC SYSTEM 
,Barbara Dunin-K~plicz Witold Lukaszewicz 
,Institute of Informatics 
Warsaw University 
P.O.Box 1210 
00-901 Warszawa, POLAND 
ABSTRACT 
The purpose of this paper is to analyse the 
phenomenon of nonmonotonicity in a natural lan- 
guage and £o formulate a number of general prin- 
ciples which should be taken in%o consideration 
while constructing a discourse oriented nonmo- 
notonic formalism. 
INTRODUCTION 
F'or the purpose of this paper we assume 
that to understand a discourse is to specify all 
the conclusions derivable from the discourse 
itself knowledge about the world and knowledge 
of language use. 
To complete discourse-connected conclusions, 
a number of various linguistic phenomena must 
be resolved. The solution of such problems as 
anaphora, quantification, negation and so on, 
should be regarded as full right conclusions. 
The reason is that this purely linguistic informa- 
tion is necessary for obtaining the essential 
one, describing the external world under conside- 
ration. 
The measure of quality of any discourse 
analysis system is the adequacy of its inference 
capabilities to those of human language users. 
This implies that any such a high quality system 
must provide some mechanism modelling common 
sense reasoning. 
In everyday life we are continously forced 
to accept various conclusions which we are pre- 
pared to reject when our knowledge increases. 
The ability of drawing such cancellableinferences, 
beliefs in AI terminology, makes common sense 
reasoning nonmonotonic in the sense that the set 
of derivable conclusions does not increase mono- 
tonically with the set of premises, as in standard 
logics. 
Earlier experiences have proved that ad hoc 
nonmonotonic tools were ineffective. They have 
seemed to work for the simplest cases only. 
Since they lacked theoretical founds\[ions, their 
behaviour has been unclear in more complex 
situations. 
Recently there have been a number of 
attempts to formalize various nonmonotonic mech- 
anisms (see (iI, 1980), (AAAI, 1984\[)). 
In this paper we analyse the phenomenon of 
nonmonotonicity in a natural language. We also 
formulate a number of general principles which 
should be taken into account while specifying 
a discourse-oriented nonmonotonic formalism. 
It is well recognized that ordering of dis- 
course utterances is essential for its understand- 
ing~ On the other hand, logical systems lack 
mechanisms capturing this property. If follows, 
therefore~ that to model the dynamic nature of 
a discourse, some kind of struciuralization is 
needed. A very natural, both computationally and 
conceptually, discourse structuralization has been 
suggested by Kamp (Kamp, 1981). 
According to Kamp, a discourse is represented 
aS a D (iscourse) R (epresentation) S (tructure). 
Reughly speaking, DRS is a sequence of 
D (iscourse)R (epresentations)~ DR is nothing 
else than a partial modeI~ describing discourse 
objects and their tel\[aliens. A ORS is constructed 
as follows. V~'e start with the empby DR. Each 
discourse utterance extends the actual DR by 
adding appropriate information contained in the 
utterance, qFo construct any DR, some kind of 
reasoning mechanism is needed. V~'e postulate 
the application of the nonmonotonic inference 
system for that purpose. 
NONMONOTONICITY 
IN DISCOURSE UNDERSTANDING 
q~o show the universality of nonmonotonicity 
in a natural language 5 we shall present a number 
of examples concerning various linguistic phe- 
nomena commonly occurring in a discourse. 
One should be aware that analogous treatment 
of such different concepts as, for instance, 
anaphora and presupposition is inappropriate. 
Anaphora, as well as quantification or negation 
are purely surface phenomena. One must resolve 
~them while processing a discourse, but their 
solutions lead to conclusions describing the 
discourse rather, than the external world under 
consideration. For this reason we shall refer £o 
those conclusions as surface conclusions. 
Presupposition, on the other hand, as well aS 
time line COnStruction, conditionals or various 
conversational rules, concerns conclusions 
describing directly the world being represented 
by a discourse. Vge shall refer to those conclu- 
sions as .~eneral conclusions. 
Our purpose is not to review all the linguistic 
phenomena dealing with nonmonotonicity, but to 
demonstrate that 
(i) both kinds of conclusions are subject to 
invalidation, 
(ii) canceliable conclusions of both types can 
be supported either by semantic or pragmatic 
sources. 
This leads to an important observation that 
nonmonotonicity exists on various levels of 
discourse understanding, viewing both processing 
method and data to be processed. 
Surface conclusions 
The following examples of pronoun anaphora 
and quantification show that semantic-based as 
well as progmatic-based surface conclusions 
can be invalidated. 
Example i (pronoun anaphora) 
(i) The Vice-President entered the President's 
office. He was nervous and clutching his 
briefcase. 
504 
"l~his example is from (Ascher, 1984). There 
are semantic reasons supporting the conclusion 
that the Vice-President is the one who is nervous 
given the information that he has a meeting with 
the President. "I'his conclusion can be, however, 
overturned by adding 
(2) After all, he couldn't fire the Vice-President 
without making trouble for himself with the 
chairman of the board. 
Example 2 (pronoun anaphora) 
(3) Peter' WaS sitting in a room. When John 
entered the room he seemed nervous. 
Although there are two possible referents of 
"he" in (3), it is pragmatically well motivated 
that "he" refers to "Peter", This follows from 
Gricels Cooperative Principle. Assuming that the 
speaker is obeying it, he should replace (5) by 
the unmistakable 
(4) Peter was sitting in a room. Entering the 
room, John seemed nervous. 
if it were the case that John had been nervous. 
lqeverthelesm, this preferred coreference is inval- 
idated if (4) is extended to 
(5) Peter was sitting in a room. When John 
entered the room he seemed nervous. He 
was afraid of Peter. 
(quantification) 
(6) There are three men in a room. Every man 
loves a woman, 
~here are clearly two possible paraphrases of (6): 
(?) "Phere are three men in a room. q?here is 
a woman such that she is loved by each 
of them, 
(8) There are three men in a room. For each 
man there is a woman such that he loves 
her. 
Applying his "ordering principles", based on the 
observation that humans are used to handle infor- 
mation from left to right, IIintikka in (Ilintikka, 
1977) convincingly suggests that (8) is prag- 
matically preferred reading for (6). Neverthe- 
less, this conclusion may be overturned, lPor 
instance, it (6) is extended to 
(9) "l~here are three men in a room. Every man 
loves a woman. Her name is Melinda. 
Example 4: (quantification) 
(10) A rifle has been given to all soldiers, 
From among two possible readings of (10), the 
one asserting that different rifles have been 
given to different sotdiers should be semantic- 
ally preferred° But this interpretation is imme- 
diately inv~lidated when (10) is replaced by 
(11) A rifle has been given to all soldiers. 
It has turned out to be broken. 
General conclusions 
'l~he following examples of presupposition il- 
lustrate that semantic-based as well as prag- 
matic-based general conclusions are subject to 
cancellation. 
(presupposition) 
(3.2) John regrets that he didn't win the prize. 
(12) presupposes that John did not winthe prize. 
Nevertheless, this general conclusion is imme- 
diately invalidated when (12) is extended to 
ed to 
(3.3) John regrets that he didn't win the prize. 
In fact, he doesn't know that he did. 
Example 6_ (presupposition) 
(24) "If I had money I would buy a rifle to de- 
fend yeu from the Wolf", Red Riding Hood 
maid to her Grrandmother. 
(14) presupposes that RRH had no money. 
This follows from the pragmatic rules of interpreta- 
tion of counterfactuals in English. "\]2he cancel~ 
ability of this conclusion is easily seen, if (14) 
is extended to 
(15) "If I had money I would buy a rifle to de- 
fend you from the W'olf", RRH said to her 
CTrandmother. In fact, RRH WaS cheating her. 
5he had money, but she wanted to buy 
herself a new dress. 
According to the presented examples, nonmo- 
notonicity is not a local phenomenon. It exists on 
various levels of discourse analysis. It seems, 
therefore, that no high quality discourse under- 
standing system can i~nore this fact. 
q?OWARDS A DI~SCOURSE-ORIEN'J?ED 
NONMONOG)ONIC FORMALISM 
Although a great number of various nonmono- 
tonic (nm, for short) formalisms can be found in 
AI literature, none of them seems to be fully 
appropriate for the purpose of discourse under- 
standing. In our opinion, only default logic pare 
tially captures the expressive power of natural 
language. \]Deriving from Minsky's frame concept, 
this formalism has turned out to be useful for 
various natural language applications (see (Dunin- 
-K~plicz, 1984), (Mercer, Reiter, 1984). 
Default logic has been introduced by Relier 
in (P.eiter, 1980) to model default reasonin_q, i.e., 
the drawing of plausible conclusions from incom- 
plete information in the absence of evidence to 
the contrary, f typical example of default reason- 
ing is the inference rule "<Pypically birds fly". If 
"l~weety is e bird, then in the absence of evi- 
dence to the contrary, we normally assume that 
q~weefy flies. But we are prepared to reject this 
conclusion, if we learn theft (Dweety is a pinguiru 
In default logic the rule about birds is repre- 
sen\[ed by the following default: 
bird(x) : M flies(x) / flies(x) 
with the intended interpretation: "for each individ- 
ual x, if x is a bird and i{ is consistent to as- 
sume that x flies, then it may be assumed that 
x flies". 
In default logic lhe world under consideration 
is represented as a default the~, i.e. a pair 
consisting of a set of first-order formulae, the 
axioms of the theory, and a set of defaults. De- 
faults extend the information contained in axioms 
by sanctioning plausible, but not necessarily 
true, conclusions. 
A set of formulae derivable from a given de- 
fault theory is; called an extension of the theory 
and is interpreted as a set of beliefs about the 
world being *h~delled. (see (Relier, 3_980) fop details). 
505 
Specifying a nm discourse-oriented formalism 
is a difficult problem. "\])he first step is to deter- 
mine all its relevant properties. In this section 
we shall try to accomplish this point and to dis- 
cuss some weaknesses of default logic in this 
respect. 
The first problem is to choose an appropriate 
standard logic as a basis of the constructed 
system. Although the majority of existing nm for- 
malisms are based on classical first-order logic, 
it is well recognized that to capture the struc- 
ture of natural languages, at least intensional 
logic, preferable with tense operettors, is required. 
It should be, however, stressed that for computa- 
tional reasons some compromise between ade- 
quacy and simplicity is necessary. 
The next step is to add a nonmonotonic de- 
ductive structure to the chosen monotonic logic. 
This amounts to the following task. Given a the- 
err A, i.e., a set of formulae describing relevant 
information about a world, determine a set E(A) 
of conclusions nonmonotonically derivable from 
A. The general idea is to identify this set with 
the set of conclusions monotonically derivable 
from some extension of A. For this reasonlD(A) 
is called an extension of A, and is interpreted 
as the set of plausible conclusions about the 
world under consideration. 
Defining a language and nm deductive struc- 
ture, determines a nm system. But to make the 
system applicable for discourse processing, 
a number of additional factors should be taken 
into consideration. 
~irst, a semantics mus\[ be specified. In par- 
ticular, a model regarded as a nm description of 
a real world under consideration should be 
defined. 
Second, a method of determining whether a 
conclusion can be inferred from a given set of 
premises should be specified. Because nm deriv- 
ability depends not only on what can be proved, 
but also on what cannot be proved, there are 
technical problems involved here. In particular, 
even if the system is based on semi-decidable 
first-order logic, its proof theory is generally 
undecidable. "\])his means that some kind of 
heuristics is necessary, and the system will 
sometimes arrive at mistaken conclusions. 
Nevertheless, it is often sufficient to consider 
only models with finite domains limited to the 
individuals explicitely occurring in the discourse. 
In such a case, the logic we are dealing with 
is, in fact, the propositional one. "I~he nm formal- 
ism based on propositional logic is, of course, 
decidable. 
"l~hird, belief___.~s \[evision~ that is a method of 
reorganizing world model when new information 
leads to inconsistency should be specified. (\['his 
very difficult problem has been marginally treat- 
ed in the most of existing nm systems, On the 
other hand, since each discourse utterance mod- 
ifies the actual world model, beliefs revision 
seems one of the central problems of discourse 
analysis. 
Fourth, the existence of extensions should 
be guaranteed. Although this demand seems ob- 
vious, it iS not satisfied in many existing nm 
formalisms, in particular~ in Reite~s default logic, 
An example of a discourse, whose representation 
in lqeiter's system lacks an extension, can be 
found in (Lukaszewicz, 1984). "l~his paper pre~- 
sents also an alternative formalization of default 
logic which satisfies the above postulated proper- 
ty. 
Fifth, the system should ~ model com- 
mon sense reasoning. In terms of extension, this 
means that such an extension should contain 
those discourse-connected conclusions which 
would be drawn by humans. In our opinion, none 
of existing nm formalisms fully captures this 
requirement. The detailed discussion of this hypo- 
thesis can be found in (Lukaszewicz, 1986). 
In the majority of nm systems the notion of an 
extension is defined in a way admitting the exist- 
fence of many different extensions for a given 
theory, q~his poses the question of how to define 
the set of conclusions derivable from such a the- 
ory. 'l~here are two possibilities. First, to view 
each extension as an alternative set of beliefs 
one can hold about a world under consideration. 
In fact, this solution has been adopted in default 
logic. Second, to identify the set of conclusions 
with the intersection of all extensions. Both solu- 
tions are reasonable but have different interpreta- 
tions, q~he first represents what an agent beliefs 
about a world, the second, wh~t an outside ob- 
server would know about the agent~ beliefs, 
given the set of the agent's premises about the 
world. 
In our opinion, the second solution is better 
motivated in discourse processing, q~his follows 
from the fact that both the speaker' and the 
hearer try to achieve a common representation 
of a discourse. "l~his would be impossible, if 
each of them accepted a different extension. 
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