N o n - o m n i s c i e n t b e l i e f as c o n t e x t - b a s e d reasoning*
Fausto Giunchiglia1,2 Luciano Serafini1 Enrico Giunchiglia2 Marcello Frixione2
Mechanized Reasoning Group
1
RST, 38050 Povo, Trento, Italy
2
D I S T - University of Genoa, V i a Opera Pia 11 A, Genova, Italy
phone:+39 461 814440
{fausto,serafini}@irst.it
{enrico,frix}@dist.unige.it
Abstract
T h i s paper describes a general framework for
the f o r m a l i z a t i o n o f m o n o t o n i c reasoning a b o u t
belief in a m u l t i a g e n t e n v i r o n m e n t . T h e agents*
beliefs are modeled as logical theories. T h e
reasoning a b o u t their beliefs is formalized in
s t i l l another theory, w h i c h we call the theory of
the c o m p u t e r . T h e f r a m e w o r k is used to model
non-omniscient belief and shown to have m a n y
advantages. For instance, it allows for an exhaustive classification of the "basic" forms of
non logical omniscience and for t h e i r "composit i o n " i n t o the s t r u c t u r e o f the system m o d e l i n g
m u l t i a g e n t omniscient belief.
1
T h e approach
T h i s paper describes a general framework for the f o r m a l ization of m o n o t o n i c reasoning about belief in a m u l t i a gent e n v i r o n m e n t . T h e most c o m m o n solution is to take
a first order ( p r o p o s i t i o n a l ) theory, to extend it using
a set of m o d a l operators,
and to take
as
meaning t h a t an agent
believes A (see for instance
[Halpern a n d Moses, 1985]). T h e r e is only one theory
of t h e w o r l d , however this theory proves facts a b o u t the
agents' beliefs. A c c o r d i n g to a first i n t e r p r e t a t i o n , this
theory is t a k e n to model t h i n g s how they really are. It is
therefore a finite (and possibly incomplete) presentation
of w h a t is t r u e in the w o r l d , and the fact t h a t B i A is a
theorem means t h a t it is, in fact, the case t h a t a i believes
A. A c c o r d i n g to another i n t e r p r e t a t i o n , this theory is
taken to be the perspective t h a t a generic reasoner has
of the w o r l d . It is therefore a finite presentation of the
reasoner's beliefs, and the fact t h a t
A is a theorem
means t h a t the reasoner believes t h a t
believes A.
Once one accepts the second i n t e r p r e t a t i o n (as we
d o ) , a mechanized theory is n a t u r a l l y taken as representing the beliefs of the c o m p u t e r where it is implem e n t e d . Moreover, in the case of m u l t i a g e n t belief, a
f u r t h e r step is to have, together w i t h the theory of the
c o m p u t e r , one theory ( a t least, see later) for each agent.
*Alessandro C i m a t t i and K u r t Konolige have provided
very useful feedback and suggestions. The work at IRST
has been done as part of the M A I A project. This paper is a
short version of the I R S T technical report #9206-03.
548
Knowledge Representation
T h e t h e o r y of the c o m p u t e r plays the same role as the
unique theory in the m o d a l logics a p p r o a c h . T h e agents'
theories are the ( m e n t a l ) representations t h a t the computer has of the agents themselves. T h e c o m p u t e r has
beliefs a b o u t the beliefs of the agents n o t because, as
it happens in the single theory approaches, it " s i m u lates" t h e m in its o w n theory b u t , r a t h e r , because it
can infer, say,
Bi("A")
f r o m the fact t h a t A is a theorem in the a i ' s theory. ( I n this f r a m e w o r k , theories
are called " c o n t e x t s " and the f o r m a l systems, which
are basically defined as sets of c o n t e x t s , are called multicontext systems ( M C systems) a n d , sometimes, multtlanguage systems ( M L systems) [ G i u n c h i g l i a , 1991;
G i u n c h i g l i a , 1993].)
A l l the previous work on the f o r m a l i z a t i o n of proposit i o n a l a t t i t u d e s , namely m o d a l logics and the syntactic approach (where belief is a first order predicate,
see for instance [ K o n o l i g e , 1982]) makes use of a single theory. However, ideas s i m i l a r to ours, w h i c h suggest the use of m u l t i p l e d i s t i n c t theories, have been exploited in m u c h applied w o r k in c o m p u t a t i o n a l linguistics and AI (see for instance [ W i l k s and B i e m , 1979;
G i u n c h i g l i a and W e y h r a u c h , 1988]). T h e r e are in fact
m a n y i r n p l e m e n t a t i o n a l advantages, all d e r i v i n g f r o m
the fact t h a t reasoning becomes i n t r i n s i c a l l y localized
[ G i u n c h i g l i a , 1993] (for instance: the m o d u l a r i z a t i o n of
the knowledge base, c o m p u t a t i o n a l efficiency ~ due to
the division of the search space in smaller search spaces
- and the p o s s i b i l i t y of p a r a l l e l i z a t i o n of the reasoning
process).
O u r f o r m a l i s m seems q u i t e close, arguably
closer t h a n the previous approaches, to the current practice in the more applied i r n p l e m e n t a t i o n a l w o r k . I n deed this is one of the m a i n m o t i v a t i o n s u n d e r l y i n g our
work and MC systems have been i m p l e m e n t e d inside the
GETFOL system, [ G i u n c h i g l i a , 1992] an extension of a rei m p l e m e n t a t i o n of the FOL system [ W e y h r a u c h , 1980].
However in this paper we focus on the representational
issues and argue t h a t our approach allows for more n a t u ral and more i n t u i t i v e f o r m a l i z a t i o n s (see also [Dinsmore,
1991]).
In [ G i u n c h i g l i a and Serafini, 1991] we have described
and m o t i v a t e d the basic s y s t e m , w h i c h models logically
omniscient agents. In t h i s paper we take a step further a n d treat the p r o b l e m of non s a t u r a t e d belief, t h a t
is the fact t h a t real agents never believe all the logical consequences of their basic beliefs. O u r t r e a t m e n t
is based on a d i s t i n c t i o n between ideal and real reasoners. We show how ideal and (different) real reasoners can
be modeled by contexts w i t h different s t r u c t u r a l properties (section 2) a n d / o r different connections among t h e m
(section 3). O u r approach presents various advantages.
F i r s t , it allows for an exhaustive classification of all the
basic forms of n o n logical omniscience (see sections 2, 3).
Second, a l l t h e various f o r m s of non logical omniscience
can be identified as a d i s t i n c t s t r u c t u r a l p r o p e r t y of the
s y s t e m . T h i r d , complex forms o f non logical omniscience
can be " c o n s t r u c t e d " by composing simpler forms i n t o
the s t r u c t u r e p r o v i d e d by the system for s a t u r a t e d belief
(see sections 2, 3). A l l the cases studied in the past plus
new i n t e r e s t i n g ones are c a p t u r e d by this classification
(see section 4 ) . F o u r t h , deductions can be performed
very n a t u r a l l y by e x p l o i t i n g the fact t h a t each reasoner
is modeled as a d i s t i n c t theory (section 5 ) 1 .
1
For lack of space, these ideas are described only partially.
For instance, we state theorems but we do not prove them.
We do not consider nested belief. We do not give semantics
([Giunchiglia et al., 1992] presents a somewhat old semantics
for the basic saturated case). We do not give any complexity
argument; although it is intuitive that reasoning with contexts saves time, we have not done any in depth analysis on
this issue.
Giunchiglia et al.
549
Figure 2: M B I E - Ideal and real reasoner
3.2
I m p l i c i t and Explicit belief
One approach to the formalization of non logical
omniscience is based on the distinction between explicit beliefs and implicit beliefs.
According to
Levesque [Levesque, 1984], while explicit beliefs are the
set of effective beliefs of the agent, implicit beliefs model
what the world would be like if what he believed were
true. Implicit belief is used as a limit notion which gives
an upper bound to what a real agent can ever believe. No
matter what it explicitly believes, it will never explicitly
believe more than its implicit beliefs. In this context, a
natural step is to take the agent's implicit and explicit
beliefs as the beliefs of an ideal reasoner 1 and of its realized version E, respectively. The intuition is in fact the
same, that is that I/ implicit belief is the limit idealized
version of E/ explicit belief. However the emphasis is
slightly changed as we do not care of how the world is
but only of the subjective view of /. Implicitly we assume that the world is how / assumes it is, but this is
irrelevant from a technical point of view.
Thus, if we want to model a situation where c can reason about the idealized and real capabilities of an agent,
550
Knowledge Representation
3.3
L o c a l reasoning
Underlying all the above definitions is the intuition that
a reasoner can use all its construction rules on all its basic
facts. However, it is well known that human beings have
difficulty in putting together all the information they
possess. Human memory seems structured in frames of
mind hardly communicating between them [Stalnaker,
1984). Reasoning happens locally to each frame and
hardly interacts with the reasoning performed in other
frames. Thus, for instance, we never think of Africa and
a flat tire at the same time. The obvious way to model
this situation is to associate to each agent a set of reasoners E i , . . . , E n , one for each frame of mind, variously
interconnected among themselves and with c. This can
be done in many different ways, each modeling a different
intuition. In the following we discuss one very general
possibility.
As usual c is connected to I via reflection up and reflection down. We suppose that the various Ei have different
languages and different sets of inference rules. We have
a reflection up rule between c and each Ei. This guarantees the completeness of c. However we have only one
predicate EB for all frames and no reflection down rules
into the frames. This models the intuition that c has a
correct view of what is explicitly believed but without
knowing in which frame theorems are proved. E B ( " A " )
means that there is at least one frame of mind in which
Definition 3.3 generalizes definition 3.2 of MBIE in the
sense that E has been substituted with n frames.
Notice that, starting from the intuition that a reasoner
can be modeled as a logical theory, we have introduced
various forms of non logical omniscience and associated
them to a particular structural part of E. The resulting
system MBIE(n) models all of them. This process is
exhaustive in the sense that all the structural subparts of
E have been considered (that is its language, its theorems
and E itself). If one accepts the abstract characterization
Giunchiglia et al.
551
552
Knowledge Representation
A first observation is t h a t the agents' contexts are used
to prove theorems which are then reflected up (e.g. line
30). T h i s is a practical consequence of the fact t h a t
agents are not simulated in c but are rather "used" to
e x t r a c t w h a t it is proved in their contexts. In this perspective reflection down is used as a means to impose a
set of facts i n t o a context (e.g. lines 8, 13, 17). Notice
t h a t all the applications of reflection d o w n are above all
the applications of reflection up. T h i s is in fact the general way of p e r f o r m i n g reasoning w i t h MC systems for
belief: (i) first some conjectures are made in c; (ii) then
these conjectures are " i m p o s e d " on the reasoners (via
reflection d o w n ) ; (iii) then the reasoning capabilities of
the reasoners are used io compute the consequences of
the conjectures; ( i v ) finally the result is exported back to
c via reflection up. Some amount of propositional reasoiling can be performed in c before step (i) and after
step ( i v ) .
6
Conclusion
In this paper we have proposed a new approach to the
f o r m a l i z a t i o n of reasoning about belief where agents are
modeled as sets of i n t e r a c t i n g contexts. In this framework the properties of each agent can be directly f o r m a l ized by imposing local suitable conditions on the theories
modeling it ( t h a t is, on the language, on the axioms, on
the deductive machinery, on the bridge rules).
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