One of the key aspects in the design of an architecture for autonomous agents is the way in which beleifs are revised in the face of new information. An intelligent agent must be prepared to handle new information (be it sensory input, communication with other agents, etc). The areas of Knowledge Representation and Reasoning and Belief Revision deal with the issues of how beliefs are represented, how they are used in reasoning processes, and how a current body of beliefs reacts to the appearance of new information. There are many problems that must be solved in order to e®ectively manage a body of knowledge which, more often than not, may be incomplete or even inconsistent. The adequate solution to these problems is an essential feature of autonomy, because an agent must always be up to date in order to behave accordingly.Eje: Inteligencia artificial distribuida, aspectos teóricos de la inteligencia artificial y teoría de computaciónRed de Universidades con Carreras en Informática (RedUNCI

Falappa, Marcelo Alejandro

Simari, Gerardo I.

English

El Servicio de Difusión de la Creación Intelectual

Non Prioritized Belief Revision with AnsProlog*
Gerardo I. Simari 1 Marcelo A. Falappa
Laboratorio de Investigacio´n y Desarrollo en Inteligencia Artificial (LIDIA) 2
Departamento de Ciencias e Ingenier´ıa de la Computacio´n
Universidad Nacional del Sur. Av. Alem 1253, (8000) Bah´ıa Blanca, Argentina
Tel: ++54 291 4595135 - Fax: ++54 291 4595136
{gis,mfalappa}@cs.uns.edu.ar
1 Introduction and Background
One of the key aspects in the design of an architecture for autonomous agents is the way in
which beleifs are revised in the face of new information. An intelligent agent must be prepared
to handle new information (be it sensory input, communication with other agents, etc). The
areas of Knowledge Representation and Reasoning and Belief Revision deal with the issues of
how beliefs are represented, how they are used in reasoning processes, and how a current body
of beliefs reacts to the appearance of new information. There are many problems that must be
solved in order to effectively manage a body of knowledge which, more often than not, may be
incomplete or even inconsistent. The adequate solution to these problems is an essential feature
of autonomy, because an agent must always be up to date in order to behave accordingly [12].
Belief Revision was mainly introduced by Ga¨rdenfors and later extended by Alchourro´n,
Ga¨rdenfors, and Makinson [1, 4]. Its main goal is to model the dynamics of knowledge, i.e.,
the way in which an agent’s knowledge reacts before new information. Daily life offers many
examples in which humans perform belief revision: watching the stock quotes on TV may inform
us that a certain stock is doing well, or a phone call from an old friend may tell us that he is in
town for the holidays, and that he is now married. In this example, the basic operations of belief
revision, known as expansion, contraction, and revision, are mapped respectively into finding
out that the stock is doing well, that our friend is no longer out of town, and that he is now
married. These operations are also related to environments closer to the computational realm,
such as the use of communication protocols where new protocols can be added (expansion),
withdrawn (contraction), or a subset of them be replaced by one or more (revision).
Revisions are the most commonly used change operators because they allow a sentence α to
be included into a set K, generating a new set K ′, preserving consistency in the new set. The
traditional revision models are prioritized, that is, they give priority to new information over
the information that is already part of the knowledge. This property does not seem plausible
This work is partially supported by the Agencia Nacional de Promocio´n Cient´ıfica y Tecnolo´gica (PICT
13096)
1Partially supported by Comisio´n de Investigaciones Cient´ıficas, Gobierno de la Provincia de Buenos Aires.
2Member of the IICyTI (Instituto de Investigacio´n en Ciencia y Tecnolog´ıa Informa´tica).
in the real world, because in many cases it is not reasonable to give priority to information just
because it is new.
In non prioritized models, it is possible for new information not to be totally accepted.
Such new information can be rejected or accepted only after a debate process. In this sense,
there exists a variety of different non prioritized belief revision models, among which are David
Makinson’s Screened Revision [11, 10], Sven Ove Hansson’s semirevision operators [9], Andre´
Fuhrmann’s merge operations [3] and Falappa et al.’s [2] recently formulated revisions by sets of
sentences. In this last work, a new kind of non prioritized revision operator based on the use of
explanations is presented. It is argued that an agent, before incorporating information which is
inconsistent with its knowledge, should request an explanation that supports this information.
An explanation is characterized as being composed of an explanans (set of sentences supporting
a certain belief), and an explanandum (the final conclusion).
An example of this situation is the following: Suppose a person believes that (α) all felines
can climb and (β) Moebius is a feline. Thus, he will believe (δ) Moebius can climb. Later on,
another person states that Moebius cannot climb. If δ is dropped, then α or β will have to be
dropped as well. However, it does not seem rational to incorporate external beliefs without
pondering them. An explanation should be required in order to incorporate such information,
especially in the case in which it contradicts the previously maintained set of beliefs.
In our case, the person should demand an explanation for ¬δ. One possibility is that he is
given an explanation such as: Moebius is a cat, but he cannot climb because he is hurt, and cats
that are hurt cannot climb. Now, the sentences in the explanans can be used to evaluate the
new piece of information before it is incorporated. In [12], we proposed the use of AnsProlog*
as a language for the representation of beliefs, as well as for explanations.
2 Belief Revision with respect to Explanations
An agent’s beliefs can be represented by means of belief sets (sets of sentences closed under
logical consequence) or belief bases (arbitrary sets of sentences). In computational applications,
one must opt for finite belief bases. New information, which is called epistemic input, is
sometimes represented by a sentence of the language or an arbitrary set of sentences. The
revision operator proposed in [2] allows a non prioritized revision in the following way:
• The epistemic input is a single sentence with an explanation for it.
• The explanation (a set of sentences with some constraints) is added to the original set,
maybe resulting in a temporarily inconsistent set.
• The consistency is then restored by a contraction by falsum.
It should be noted that this operator incorporates the possibility of partial acceptance, i.e.,
even though the proposed explanation for α may be rejected, parts of it may still be added to
the knowledge base in the process.
An explanation can be defined as follows [2]. The set A is an explanation for the sentence
α if, and only if, the following conditions are satisfied,
1. Deduction: A ` α.
2. Consistency: A 0 ⊥
3. Minimality: if B ⊂ A then B 0 α.
4. Informational Content: Cn(A) 6⊆ Cn(α).
Deduction guarantees that the explanans (support for a given belief) implies the explanan-
dum (the belief being explained). Consistency prevents having an inconsistent explanation
(which would explain any belief). Minimality establishes that every belief in the explanation
is needed to obtain the explanandum, and Informational Content avoids cases in which the
explanandum implies every sentence in the explanans. In particular, it avoids a sentence being
an explanation for itself. As we can see from this description, and explanation can be seen as
an external argument supporting a given belief.
Assume we want to revise a given belief base Π with respect to a given explanation A for a
belief α. The revision involves:
1. Construction of counter-explanations for A from Π. These counter-explanations are min-
imal subsets of Π which are inconsistent with A.
2. A is compared with respect to its counter-explanations.
3. If A is (in some sense) “better” than its counter-explanations, then A is incorporated into
Π and its counter-explanations (or part of them) are eliminated. In any other case, Π
will remain unaltered.
The last step of the revision involves a decision between the proposed explanation, and the
counter-explanations that can be built from Π. This decision will depend upon the particular
(typically partial) ordering that is imposed over the set of beliefs.
3 AnsProlog* and the Answer Set Semantics
We will now briefly introduce the language of logic programming with respect to the answer set
semantics, which is usually referred to as AnsProlog*. This name is short for “Programming
in Logic with Answer sets” [6]. An AnsProlog* program is a finite collection of rules of the
form:
L0 or ... or Lk ← Lk+1, ..., Lm, not Lm+1, ..., not Ln.
where the Li’s are literals (in the sense of classical logic). Intuitively, a rule of this form means
that if Lk+1, ..., Lm are true and if Lm+1, ..., Ln can be assumed to be false, then the sentence
L0 or ... or Lk is true (i.e., at least one of its literals is true). The symbol ‘*’ in AnsProlog*
means that no restriction is applied to the structure of the program’s rules; a variety of syntactic
sub-classes can be defined when such rules are restricted. The importance of defining a set of
sub-classes lies in the varying degree of complexity of the rules in each class. This complexity
has a profound impact on the computational cost of operations that may be performed on
programs, such as Answer Set checking and verifying if a given belief is entailed by a given
program [7]. Part of our work lies in the study of how the computational complexity of these
and other problems is related to belief dynamics under this representation.
An AnsProlog* program can be used to represent an agent’s beliefs by means of its answer
set semantics. Such semantics can be defined as follows [5]:
1. A program Π cautiously entails a literal L (Π |= L) if L belongs to all answer sets of Π.
2. A program Π bravely entails a literal L (Π |=b L) if L belongs to some answer sets of Π.
Where answer sets are minimal and consistent sets of literals that obey the rules of Π. It should
be noted that, for programs having only one answer set, there is no difference between these
two relations.
In the same way, any explanation for a given belief can be represented by an AnsProlog
program; a subset of the program’s rules can be interpreted as the explanans, and the desired
explanandum will be entailed by the program.
An example of a belief base is the following program Π1:
climbs(X)← feline(X).
feline(X)← cat(X).
cat(moebius)← .
Π1 has as its only answer set:
S1 = {feline(moebius), cat(moebius), climbs(moebius)}
An explanation for climbs(moebius) represented by an AnsProlog* program, related to this
example, is the following program Ψ1:
climbs(X)← feline(X).
feline(fluffy)← .
This program explains that Fluffy climbs because he is a feline, and has the sole answer set:
E1 = {cat(fluffy), climbs(fluffy)}. This program can then be used as an explanation for
climbs(fluffy), where Ψ1 is the explanans and climbs(fluffy) is the explanandum. An
example of an explanation that conflicts with Π1 is the following program Ψ2:
¬climbs(X)← hurt(X).
hurt(moebius)← .
which has as its only answer set: E2 = {hurt(moebius),¬climbs(moebius)}. This answer set
clearly disagrees with Π1, which entails the belief climbs(moebius).
4 Research Goals
Our work involves research with the goal of defining revision operators based on this framework.
This includes defining preference criteria that allows an agent to decide between a proposed
explanation and a counter-explanation that can be obtained from its own knowledge. Further-
more, we are also studying how kernel contractions [8] can be constructed using this framework,
in order to eliminate inconsistencies that arise due to the acceptance of explanations that con-
tradict prior knowledge.
References
[1] Alchourro´n, C., Ga¨rdenfors, P., and Makinson, D. On the Logic of Theory Change: Partial Meet
Contraction and Revision Functions. The Journal of Symbolic Logic 50 (1985), 510–530.
[2] Falappa, M. A., Kern-Isberner, G., and Simari, G. R. Belief Revision, Explanations and Defeasible
Reasoning. Artificial Intelligence Journal 141 (2002), 1–28.
[3] Fuhrmann, A. An Essay on Contraction. Studies in Logic, Language and Information, CSLI Publications,
Stanford, California, 1997.
[4] Ga¨rdenfors, P. Knowledge in Flux: Modelling the Dynamics of Epistemic States. The MIT Press,
Bradford Books, Cambridge, Massachusetts, 1988.
[5] Gelfond, M., and Leone, N. Logic programming and knowledge representation—the A-prolog perspec-
tive. Artificial Intelligence 138, 1–2 (2002), 3–38.
[6] Gelfond, M., and Lifschitz, V. Logic Program with Classical Negation. In Proceedings of the 7th Int.
Conf. on Logic Programming (June 1990), D. H. D. Warren and P. Szeredi, Eds., MIT, pp. 579–597.
[7] Gottlob, G. Complexity and expressive power of disjunctive logic programming. In Logic Program-
ming - Proceedings of the 1994 International Symposium (Massachusetts Institute of Technology, 1994),
M. Bruynooghe, Ed., The MIT Press, pp. 23–42.
[8] Hansson, S. O. Kernel Contraction. The Journal of Symbolic Logic 59 (1994), 845–859.
[9] Hansson, S. O. Semi-Revision. Journal of Applied Non-Classical Logic 7 (1997), 151–175.
[10] Hansson, S. O. Theoria: Special Issue on Non-Prioritized Belief Revision. Department of Philosophy,
Uppsala University, 1997.
[11] Makinson, D. Screened Revision. In Theoria [10].
[12] Simari, G. I., and Falppa, M. A. Belief dynamics and explanations in ansprolog*. In Proceedings of
the IX Argentine Congress on Computer Science (CACIC) (La Plata, Buenos Aires, Argentina, 2003),
M. Naiouf, Ed., Universidad Nacional de La Plata, pp. 589–600.


Non prioritized belief revision with ansprolog

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Non prioritized belief revision with ansprolog

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