It is interesting and important to compare, analyze and assess the alternative tools that could be used in the area of Knowledge Representation. In this paper we present a reasearch line associated to this goal: to formally establish the relation among Knowledge Representation formalisms in order to make a sensible use of them. As a part of this main task, we present a comparison between Normal Default Theory and Defeasible Logic Programming. This comparison is achieved introducing a DELP variant, called DELP;, which allows to associate the answers of a DELP interpreter to the consequences, credulous and skeptical, of a Normal Default Theory.Eje: Agentes y Sistemas InteligentesRed de Universidades con Carreras en Informática (RedUNCI

Delladio, Telma

Rotstein, Nicolás D.

Simari, Guillermo Ricardo

English

El Servicio de Difusión de la Creación Intelectual

A Comparison Between Non-Monotonic Formalisms
Telma Delladio Nicola´s D. Rotstein Guillermo R. Simari
Laboratorio de Investigacio´n y Desarrollo en Inteligencia Artificial
Departamento de Ciencias e Ingenierı´a de la Computacio´n
Universidad Nacional del Sur
{td,ndr,grs}@cs.uns.edu.ar
ABSTRACT
It is interesting and important to compare, analyze and as-
sess the alternative tools that could be used in the area
of Knowledge Representation. In this paper we present a
reasearch line associated to this goal: to formally establish
the relation among Knowledge Representation formalisms
in order to make a sensible use of them. As a part of
this main task, we present a comparison between Normal
Default Theory and Defeasible Logic Programming. This
comparison is achieved introducing a DELP variant, called
DELP∅, which allows to associate the answers of a DELP
interpreter to the consequences, credulous and skeptical, of
a Normal Default Theory.
Keywords: Knowledge Representation, Non-
monotonic Reasoning, Default Logic, Defeasible
logic programming.
1 INTERESTS AND MOTIVATION
In general, it is interesting and important to compare,
analyze and assess the alternative tools that could be
used to confront a specific problem. In particular,
in the area of Artificial Intelligence there are several
research lines dedicated to the development of for-
malisms and tools regarding Knowledge Representa-
tion. These formalisms are so diverse that many times
it is difficult to recognize their advantages, disadvan-
tages and differences in order to make a plausible use
of them. For this reason, it is interesting to analyze the
relation among knowledge representation formalisms
to evaluate their differences and similarities. Sev-
eral works relating diverse approaches of defeasible
and non-monotonic reasoning have been developed
[6, 5, 4, 3, 1, 2].
In this paper, we present, as a part of a major
work, a preliminary analysis about the relation be-
tween Default Logic and Defeasible Logic Program-
ming (DELP). The current goal is to establish the pos-
sibility of defining a set of conditions and transfor-
mation rules that allows us to study this relation in
Partially supported by Agencia Nacional de Promocio´n
Cientı´fica y Tecnolo´gica, SGCyT - UNS, and Consejo Nacional
de Investigaciones Cientı´ficas y Te´cnicas (CONICET)
a formal way. In this first stage, we analyze several
scenarios and specific examples that evidence the dif-
ferences between these two formalisms, in order to
establish their causes. This is the first step to clarify
the relation under study.
2 DEFAULT LOGIC AND DELP
In this section, we briefly describe the theoretical
background of both Default Logic and DELP. We will
also introduce a variant of DELP that will be used for
comparison.
2.1 Default Logic
A Default Theory T = 〈W,D〉 consists of a set of
facts W of ground sentences. Each default rule in
D has the form a : b1,...,bnc (sometimes written a :
b1, . . . , bn/c), where a is called the prerequisite, bi
are the justifications and c is the consequent of the de-
fault. The intuitive meaning of a default is: if a can be
derived and it is possible to consistently assume each
bi, then conclude c. In this first work, we will consider
default theories with the following characteristics:
(C1) The theories are normal. The defaults have the
form a : c/c; for short, we write a ↪→ c.
(C2) Only propositional sentences are allowed.
(C3) For every default a ↪→ c, the sentence c is a sin-
gle literal.
2.2 DELP and DELP∅
Defeasible Logic Programming (DELP) is a formal-
ism that combines Logic Programming and Defeasi-
ble Argumentation. DELP allows the representation
of defeasible information in the form of two kinds
of rules: defeasible rules and strict rules. The first
ones, also called weak rules, are used in the represen-
tation of tentative information, and the second ones
for representing strict knowledge. The language used
allows strong negation, and DELP uses an argumen-
tation formalism to deal with contradictory knowl-
edge. In DELP, a literal h is warranted if there ex-
ists a non-defeated argument A supporting h. An
argument 〈A, h〉 for a literal h is a minimal consis-
tent set of defeasible rules that infers h. In order to
establish whether 〈A, h〉 is a non-defeated argument,
argument rebuttals or counter-arguments that could
be defeaters for 〈A, h〉 are considered, i.e., counter-
arguments that, by some criterion, are preferred to
〈A, h〉. Since counter-arguments are arguments, there
may exist defeaters for them, and defeaters for these
defeaters, and so on. Thus, a sequence of arguments
called argumentation line may appear, where each ar-
gument defeats its predecessor in the line. Usually,
each argument has more than one defeater and more
than one argumentation line exists. Therefore, a tree
of arguments called dialectical tree is constructed,
where the root is 〈A, h〉 and each path from the root
to a leaf is an argumentation line. A dialectical anal-
ysis of this tree is used for deciding whether h is
warranted. This dialectical analysis is carried out la-
belling the arguments conforming the dialectical tree.
The arguments in the leaves of the tree are considered
undefeated. Every inner node with at least a child
marked as undefeated, is considered and marked as a
defeated argument. In the other case, it is undefeated.
Following this analysis, a literal h is said warranted if
there is a dialectical tree where the root is an argument
for h that has been marked as undefeated.
In this formalism, several elements can be adjusted
to define variants of DELP; for instance, the notions
of attack and defeat, as well as the conditions required
for acceptable argumentation lines. We will consider
a DELP variant, that we call DELP∅, observing the
following condition:
• The relation defining the comparison criterion is
the empty set. This criterion turns attack into
defeat.
In general, given two conflicting arguments A and
B, they can be compared using some criterion. In that
case, if A is better than B, A is a proper defeater for
B. But, if neither of the two is better than the other, A
is a blocking defeater for B, and vice versa. Note that,
in DELP∅ every attack is a blocking defeat and there
are no proper defeaters.
3 EXTENSIONS AND ANSWERS
Default Logic allows the existence of multiple exten-
sions; therefore, in these cases, it is necessary to de-
fine what pieces of information (or beliefs) will be
accepted. For instance, a reasoner may choose to be-
lieve only in those pieces of information present in
every extension. Several semantics had been defined,
but we will consider the classical notions of conse-
quence: skeptical and credulous. If a literal l belongs
to every extension of a default theory, l is said to be
skeptical, denoted l ∈ SKEP. Otherwise, if there ex-
ists an extension E such that literal l does not belong
to E, we say that literal l is a credulous consequence,
denoted l ∈ CRED. In the general case, it is possible
for a default theory to have no extensions, but, in the
case of normal theories, the existence of at least one
extension is ensured.
In a defeasible logic program (de.l.p.) P it is possi-
ble to associate each literal in P to one of the follow-
ing three sets:
• YES = {L : L is warranted from P}
• NO = {L : L is warranted from P}
• UND = {L : L,L are not warranted from P}
We want to determine the relation between Nor-
mal Default Theories and DELP∅. The de.l.p. asso-
ciated to a default theory will be determined through
an transformation from default rules (in the default
theory) into defeasible rules (in the de.l.p.).
Other research works had established the relation
between Logic Programs and Default Logic. For this,
the standard translation considers: rules of the form
c← a1 . . . , an, not b1, . . . , not bm
can be viewed as the default rule
a1, . . . , an : ¬ b1, . . . ,¬ bm/c
where not represents negation as failure.
We will show that, for our purpose, a default rule
can be expressed as a defeasible rule. In both for-
malisms the existence of this kind of rules (defaults
and defeasible) follow a similar goal: to represent a
relation among pieces of knowledge that could be de-
feated if new or global information is considered. For
instance, we can accept the general rules:
- if it is Sunday and there is no reason against going
shopping, we go.
- if we have homework and nothing is contrary to keep
working at home, we stay at home.
and the strict rule:
- if we stay at home, we do not go shopping.
This information could be represented by a normal
default theory with the rules:
(sunday ↪→ go) (homework ↪→ stay)
(stay → ¬ go) (sunday) (homework)
and through a de.l.p. with the rules:
(go —≺ sunday) (stay —≺ homework)
(¬ go← stay) (sunday ←) (homework ←)
Considering the fact “we have homework”, both
formalisms conclude reasonably “do not go shop-
ping”, but additional knowledge about the fact “it
is Sunday” makes no conclusion drawn about going
shopping or not.
Therefore, we use a defeasible rule:
b —≺ a
to represent each default rule
a : b/b.
Then, given a normal default theory T = 〈W,D〉,
such that D = {δ1, . . . , δn}, where δi = ai ↪→ bi, the
associated defeasible logic program is
PT = (Π,∆)
where ∆ = {bi —≺ ai : δi ∈ D} and Π consists of
strict rules originated from clauses in W .
In the rest of the article we will show that every lit-
eral belonging to the set SKEP of a normal default the-
ory T will be in the set YES of the associated de.l.p.
PT , and vice versa.
4 EXAMPLES
In this section we present some examples in order
to clarify the relation between Normal Default Logic
and DELP∅.
Example 1
Consider the default theory T1 = 〈W1, D1〉, where
W1 = {a, b}
D1 = {(a ↪→ x), (b ↪→¬ x), (b ↪→ w)}
The corresponding de.l.p. will be PT1 = (Π1,∆1),
where
Π1 = {(a←), (b←)},
∆1 = {(x —≺ a), (¬ x —≺ b), (w —≺ b)}
Theory T1 has two extensions. Literal x belongs to
one of these extensions, whereas ¬ x belongs to the
other, and w is in both extensions. For this reason,
we say that {x,¬ x} ⊆ CRED and w ∈ SKEP. In
PT1 , literal w is warranted, but literals x and ¬ x are
undecided. That is, w ∈ YES and {x,¬ x} ⊆ UND.
Example 2
Consider the following theory T2 = 〈W2, D2〉 where
W2 = {a, b, c}
D2 = {a ↪→ x, b ↪→ x, c ↪→ ¬ x}
The corresponding de.l.p. is PT2 = (Π2,∆2), where
Π2 = {(a←), (b←), (c←)}
∆2 = {(x —≺ a), (x —≺ b), (¬ x —≺ c)}
In example this example, we can also see that for
theory T2, {x,¬ x} ⊆ CRED and for PT2 , {x,¬ x} ⊆
UND.
Note that, in example 2 there are two default deriva-
tions for literal x, one using the default rule a ↪→ x
and the other using b ↪→ x, whereas there is only a
single derivation for ¬ x. Each default derivation in
T2 can be associated to a defeasible derivation in the
associated de.l.p. P2. Then,there are two arguments
for x and only one argument for ¬ x. However, since
in DELP∅ every attack is a defeat, and all defeaters
are blocking defeaters, there is no chance of defense.
That is, if an argument A is defeated by argument B
there is no chance of defeating B using a third argu-
ment C. This is because of the condition that avoids
blocking-blocking situations in the formation of argu-
mentation lines.
c
:x!
a
x!
b
x!
A B C
Figure 1: Blocking situation en DELP∅
It is important to note that, in normal default the-
ories, the existence of two applicable conflicting de-
fault rules makes their conclusions to be in different
extensions. Since DELP∅ avoids any opportunity of
defense among arguments (see figure 1), the literals
supported by conflicting arguments are undecided.
The next example shows that the concept of “de-
fense among arguments” arises just when a prefer-
ence criterion among arguments can be established.
If an argument A is preferred over an argument B, A
could defend an argument C from being attacked by
B.
Example 3
Given the theory T3 = 〈{x,w, z}, {δ1, δ2, δ3}〉where
δ1 = x ↪→ b,
δ2 = x,w ↪→ ¬ b,
δ3 = x,w, z ↪→ b
The associated de.l.p. is PT3 = (Π3,∆3), where
Π3 = {(x←), (w←), (z ←)}
∆3 = {(b —≺ x), (¬ b —≺ x,w), (b —≺ x,w, z)}
Considering the normal default theory T3,
{b,¬ b} ⊆ CRED, and considering DELP∅, we have
{b,¬ b} ⊆ UND. Nevertheless, suppose that the
information supplied by the rule δ3 is preferred to the
information given by δ2, and this is preferred to δ1.
Normal Default Logic and DELP∅ cannot express
this preference.
Observe that the de.l.p. of example 3 allows the
construction of arguments A = {b —≺ x,w, z}, B =
{(¬ b —≺ x,w)}, and C = {(b —≺ x)}. Suppose that
we are considering a DELP variant with a preference
criterion based on a relation establishing that A is bet-
ter that B, and B is better than C. In others words, ar-
gument A is a proper defeater forB, and B is a proper
defeater for C. This situation allows the construction
of an acceptable argumentation line C − B − A (see
figure 2), and the literal b, which is supported by C, is
finally warranted.
x,w
:b!
B
(defeated)
A
b!
x,w,z
(undefeated)(undefeated)
C
b!
x
Figure 2: Preference in DELP
In Default Logic, conflict between default rules ap-
pears either directly or indirectly. A direct conflict
occurs when two applicable defaults rules conclude
complementary literals, which will end up in sepa-
rate extensions. Regarding the associated de.l.p., one
argument for each literal is built and since these argu-
ments are in conflict, both literals will be undecided.
The other kind of conflict we are considering is the
indirect conflict. This one occurs when the strict in-
formation of the theory concludes a literal that blocks
the application of a default rule.
Next, we describe an example that evidences the
occurrence of an indirect conflict in a normal default
theory, and we show how DELP∅ models this conflict,
and gives the expected answers.
Example 4 (Buy)
Consider the theory T4 = 〈W4, D4〉 where W4 con-
sists of the facts and rules:
(pricy)
(studyBook)
(mandatory→ buy)
and D4 consists of the default rules:
δ1 = interesting ↪→ buy
δ2 = pricy ↪→ ¬ buy
δ3 = studyBook ↪→ mandatory
Figure 3 shows that the associated DELP∅ deter-
mines a mutual attack between arguments for ¬ buy
and buy. However, the attack is indirectly determined
by literals buy and mandatory.
expensive
:buy!
A B
studyBook
mandatory!
buy
Figure 3: Indirect conflict
Example 5
Given the theory T5 = 〈W5, D5〉 where
W5 = {p, x, z}
D5 = {p ↪→ a, x ↪→¬ c, z ↪→ c, c ↪→ ¬ a, },
The corresponding de.l.p. has the rules:
Π5 = {(p←), (x←), (z ←)}
∆5 = {(a —≺ p), (¬ c —≺ x), (c —≺ z), (¬ a —≺ c)}
The theory T5 has three extensions and the sets of
conclusions {c, a}, {c,¬ a} and {¬ c, a} are subsets
of CRED. In DELP∅ literals a,¬ a, c and ¬ c are un-
decided. Figure 4 depicts extensions and arguments
defined by this theory.
:a
:a
:c
:c
!
p
p
a
a
c
c
z
z
x
x
!
!
!
Figure 4: Extensions and Arguments
Example 6
Given the theory T6 = 〈W6, D6〉 where W6 consists
of the following rules and facts:
anniversary
rain
noon
(overcast→ ¬ sunny)
(anniversary→ winter.day)
and D6 consists of the default rules:
δ1 = noon ↪→ sunny
δ2 = winter.day ↪→ overcast
δ3 = ¬ sunny ↪→ cold
δ4 = rain ↪→ ¬ sunny
δ5 = overcast, cold ↪→ take.a.taxi
The associated de.l.p. determines, for instance, that
take.a.taxi ∈ UND, because of the presence of an ar-
gument for sunny that attacks (and defeats) the argu-
ment for take.a.taxi. The default theory T6 in a simi-
lar way, sanctions take.a.taxi ∈ CRED, because there
is an extension E such that sunny ∈ E and both lit-
eral are incompatible, since take.a.taxi is derived from
¬ sunny. Hence, take.a.taxi ∈ E and it is a credulous
consecuence.
Example 7 (A classical example)
Given the theory 〈W7, D7〉 where:
W7 = {evidenceA, evidenceB}
D7 = {↪→¬ guilty,
evidenceA ↪→ responsible,
responsible ↪→ guilty,
evidenceB ↪→¬ responsible}
In this final example, we can see again that de-
fault theory sanctions literals guilty and responsible
as credulous conclusions, and the de.l.p. associated to
this theory classifies both literals as undecided.
Note that the default rule (↪→¬ guilty) states the
presumtion of innocence. This rule is associated to
the defeasible rule (¬ guilty —≺ .)
5 DELP∅ AND DEFAULT THEORY:
FUTURE WORK
In this article, we have presented a variant of DELP,
which we call DELP∅, which allows to model the be-
havior of a Normal Default Logic via defeasible ar-
gumentation. The relation under study maps the dif-
ferent kinds of consequences from the default logic to
the different answers given by DELP∅. In this way,
every skeptical consequence of a particular normal
default theory will be a warranted literal in DELP∅,
whereas every credulous consequence of a default
logic will be an undecided literal in DELP∅. The vari-
ant we are considering defines the argument prefer-
ence criterion as the empty relation. This decision im-
plies that no argument is better than any other. Thus,
there are no proper defeats since every defeat is a
blocking defeat. The definition of DELP∅ follows the
behavior of the normal default theories. In this for-
malism, the application of a default rule blocks the
derivation of a literal that conflicts with the conclu-
sion of the rule being applied. In default logic, when-
ever this blocking situation arises, a new extension is
generated. In DELP∅, this situation turns the conflict-
ing literals into undecided.
Future work in this research line includes the study
of the relation between DELP and general default
theories, prioritized default theories, and other exten-
sions.
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A comparison between non-monotonic formalisms

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