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A l l T h e y K n o w : A S t u d y i n M u l t i - A g e n t A u t o e p i s t e m i c Reasoning
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REPORT
Gerhard Lakemeyer
Institute of Computer Science I I I
University of Bonn
Romerstr. 164
5300 Bonn 1, Germany
[email protected]
Abstract
W i t h few exceptions the s t u d y o f n o n m o n o t o n i c
reasoning has been confined to the single-agent
case.
However, it has been recognized t h a t
intelligent agents often need to reason a b o u t
other agents and their a b i l i t y to reason n o n m o n o t o n i c a l l y . In this paper we present a form a l i z a t i o n of m u l t i - a g e n t autoepistemic reasoning, w h i c h n a t u r a l l y extends earlier work by
Levesque. In p a r t i c u l a r , we propose an n-agent
m o d a l belief logic, which allows us to express
t h a t a f o r m u l a (or f i n i t e set of t h e m ) is all an
agent knows, w h i c h m a y include beliefs a b o u t
w h a t other agents believe. T h e paper presents
a f o r m a l semantics of the logic in the possibleworld framework. We provide an axiomatizat i o n , w h i c h is complete for a large f r a g m e n t of
the logic and sufficient to characterize interesti n g f o r m s of m u l t i - a g e n t autoepistemic reasoni n g . We also extend the stable set and stable
expansion ideas of single-agent autoepistemic
logic to the m u l t i - a g e n t case.
1
Introduction
W h i l e the s t u d y o f n o n m o n o t o n i c reasoning f o r m a l i s m s
has been at the forefront of f o u n d a t i o n a l research in
knowledge representation for q u i t e some t i m e , work in
this area has concentrated on the single-agent case w i t h
only few exceptions.
T h i s focus on single agents is somewhat surprisi n g , since there is l i t t l e d o u b t t h a t agents, w h o have
been invested w i t h n o n m o n o t o n i c reasoning mechanisms,
should be able to reason a b o u t other agents and their
a b i l i t y to reason n o n m o n o t o n i c a l l y as well. For e x a m ple, if we assume the c o m m o n default t h a t birds norm a l l y fly a n d if J i l l tells Jack t h a t she has j u s t b o u g h t
a b i r d , then J i l l s h o u l d be able to infer t h a t Jack t h i n k s
t h a t her b i r d flies. M u l t i - a g e n t n o n m o n o t o n i c reasoning
is also crucial when agents need to coordinate their act i v i t y . P'or e x a m p l e , assume I promised a f r i e n d ( w h o is
always on t i m e ) to meet h i m at a restaurant at 7 P M . If I
leave my house k n o w i n g t h a t I w i l l not m a k e it there by
7 P M , 1 w i l l p r o b a b l y n o t change my plans and s t i l l go
to the restaurant. A f t e r a l l , I know t h a t my friend has
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no reason to believe t h a t I am n o t on my way to meet
h i m and t h a t he w i l l therefore w a i t for m e . N o t e t h a t I
reason a b o u t m y friends default a s s u m p t i o n t o w a i t i n
this case. O t h e r examples f r o m areas like p l a n n i n g and
t e m p o r a l p r o j e c t i o n can be f o u n d in [ M o r 9 0 ] .
One of the m a i n f o r m a l i s m s of n o n m o n o t o n i c reasoni n g is autoepistemic logic (e.g. [ M o o 8 5 ] ) . T h e basic idea
is t h a t the beliefs of agents are closed under perfect introspection, t h a t is, they k n o w 1 w h a t they know and
do n o t k n o w . N o n m o n o t o n i c reasoning comes a b o u t in
this f r a m e w o r k in t h a t agents can draw inferences on
the basis of their o w n ignorance. T h e f o l l o w i n g e x a m ple by M o o r e illustrates this feature: I can reasonably
conclude t h a t I have no older b r o t h e r s i m p l y because I
do n o t know of any older b r o t h e r of m i n e . A particular f o r m a l i z a t i o n of a u t o e p i s t e m i c reasoning is due to
Levesque [Lev90], w h o proposes a logic of only-knowing
{OL), w h i c h is a classical m o d a l logic extended by a new
m o d a l i t y to express t h a t a f o r m u l a is all an agent believes. An advantage of t h i s approach is t h a t , rather
t h a n h a v i n g to appeal to n o n - s t a n d a r d inference rules as
in Moore's o r i g i n a l f o r m u l a t i o n , OL captures autoepistemic reasoning using o n l y the classical n o t i o n s of logical
consequence and t h e o r e m h o o d .
In t h i s paper, we propose a p r o p o s i t i o n a l m u l t i - a g e n t
extension of OL. We p r o v i d e a f o r m a l semantics w i t h i n
the possible-world f r a m e w o r k and a p r o o f theory, which
is complete for a large f r a g m e n t of the logic. T h e new
logic also leads us to n a t u r a l extensions of notions like
stable sets and stable expansions, w h i c h were o r i g i n a l l y
developed for the single-agent case.
General m u l t i - a g e n t n o n m o n o t o n i c reasoning form a l i s m s have received very l i t t l e a t t e n t i o n u n t i l
recently. 2
A n o t a b l e exception is work by M o r g e n stern and G u e r r e i r o [ M o r 9 0 , M G 9 2 ] , w h o consider b o t h
m u l t i - a g e n t a u t o e p i s t e m i c reasoning a n d m u l t i - a g e n t circ u m s c r i p t i o n theories. On the a u t o e p i s t e m i c side, they
propose m u l t i - a g e n t versions of stable sets, w h i c h a l 1
While we are concerned w i t h belief and, in particular,
allow agents to have false beliefs, we nevertheless use the
terms knowledge and belief interchangeably.
2
There has also been work applying nonmonotonic theories to special multi-agent settings such as speech acts
(e.g. [Per87, AK88]). Unlike our work, these approaches are
not concerned with general purpose multi-agent nonmonotonic reasoning.
low agents to reason about other agents' nonmonotonic
inferences.
In contrast to our work, however, these
stable sets are not justified by an independent semantic account. Recently, and independently of our work,
Halpern [Hal93] also extended Levesque's logic OL to
the multi-agent case. While Halpern's logic and ours
share many (but not all) properties, the respective model
theories are quite different. In particular, while our approach remains within classical possible-world semantics,
Halpern uses concepts very much related to the so-called
knowledge structures of [FHV91]. Using the same technique, Halpern also extends the notion of only-knowing
proposed in [HM84] to the multi-agent case. There, however, agents are not capable of reasoning about other
agent's nonmonotonic inferences because only-knowing
is used only as a meta-logical concept.
The rest of the paper is organized as follows. Section 2
defines the logic OLn, extending Levesque's logic of onlyknowing to many agents. Besides a formal semantics we
also provide a proof theory, which is complete for a large
fragment of 0Ln. Furthermore, we look at the properties of formulas in 0Ln which uniquely determine the
beliefs of an agent and are thus of particular interest to
knowledge representation. Section 3 considers examples
of multi-agent autoepistemic reasoning as modeled by
0Ln Section 4 shows how 0Ln yields natural multiagent versions of stable sets and stable expansions. Finally, we summarize the results of the paper and point
to some future work in Section 5.
2
T h e Logic 0Ln
After introducing the syntax of the logic, we define the
semantics in two stages. First we describe that part of
the semantics that does not deal with only-knowing. In
fact, this is just an ordinary possible-world semantics for
n agents with perfect introspection. Then we introduce
the necessary extensions that give us the semantics of
only-knowing. Finally, we present a proof theoretic account, which is complete for a large fragment of the logic,
and discuss properties of the logic which are important
in the context of knowledge representation.
Lakemeyer
377
of a world w, which corresponds intuitively to the real
world, and a set of worlds W', which determine the beliefs
of the agent. There is no need for an explicit accessibility relation, since the worlds in M are globally accessible
from every world and a sentence is believed just in case
it is true at all worlds in W. Unfortunately, such a simple model does not extend to the multi-agent case and
we are forced to a more complicated semantics with explicit accessibility relations as defined above.7 For this
reason, the extension of Levesque's logic OL to many
agents turns out to be a non-trivial exercise.
2.3
The Canonical M o d e l
It is well known that, as far as basic formulas are concerned, it suffices to look at just one, the so-called canonical model [HC84, HM92]. This canonical model will be
used later on to define the semantics of only-knowing.
The central idea behind canonical models are maximally consistent sets.
D e f i n i t i o n 6 Maximally consistent sets
Given any proof theory of K45n and the usual notion of
theoremhood and consistency, a set of basic formulas T
is called m a x i m a l l y c o n s i s t e n t iff T is consistent and
f o r every basic
either
is contained in T.
The canonical K45n-model Mc has as worlds precisely
all the maximally consistent sets and a world w' is i n accessible from w just in case all of i's beliefs at w are
included in w'.
D e f i n i t i o n 7 The Canonical K45n-Model Mc
The canonical model Mc = (Wc, Tl, R 1 , . . . , Rn) is a
Kripke structure such that
The following (well known) theorem tells us that nothing is lost from a logical point of view if we confine our
attention to the canonical model.
T h e o r e m 1 Mc is a
model and for every set of
basic formulas T, T is satisfiable iff it is satisfiable in
Mc.
2.4
T h e Semantics of A l l T h e y K n o w
Given this classical possible-world framework, what does
it mean for an agent i to only-know, say, an atom p at
some world w in a model Ml Certainly, i should believe
p, that is, all worlds that are i-accessible from w should
make p true. Furthermore, i should believe as little else
as possible apart from p. For example, i should neither
believe q nor believe that j believes p etc. Minimizing
knowledge using possible worlds simply means maximizing the number of accessible worlds. Thus, in our example, there should be an accessible world where q is
false and another one where j does not believe p and
so on. It should be clear that in order for w to satisfy
7In essence, if we have more than 1 agent and a global
set of worlds for each agent, the agents would be mutually
introspective, which is not what we want.
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Lakemeyer
379
What are examples of determinate formulas? In the
single-agent case, it has been shown that all objective
formulas (no modalities at all) are determinate. Not
surprisingly, objective formulas are also i-determinate.
In fact, this result can be generalized to include all basic
i-objective formulas.
Theorem 6
All basic i-objective formulas are i-determinate.
Other examples of i-determinate formulas, which are
not i-objective, include
which allow agents to reason nonmonotonically and are
discussed in more detail in Section 3.
So far we have only considered basic i-determinate
sentences. Does the above result extend to non-basic
i-objective formulas as well? The answer, surprisingly,
is: sometimes but not always! For example, it is not
hard to show that the formula
(where p is an atom)
is also i-determinate, that is, the beliefs of agent i are
uniquely determined if all i knows is that all j knows is p.
However, the formula
is not i-determinate because
In other words, it is impossible for i to
only-know that j does not only-know p. Intuitively, for
i to know that j does not only-know p, i needs to have
some evidence in terms of a basic belief or non-belief of
j. It is because of properties like
that our
axiomatization is not complete for all of
3
Multi-agent Nonmonotonic
Reasoning
While our axiomatization is complete only for a subset of
0 L n , it is nevertheless strong enough to model interesting cases of multi-agent nonmonotonic reasoning. Here
are two examples:
1. Let p be agent i's secret and suppose i makes the
following assumption: unless 1 know that j knows my
secret assume that j does not know it. We can prove
in 0Ln that if this assumption is all i believes then he
indeed believes that j does not know his secret. Formally
A formal derivation of this theorem of 0Ln can be obtained as follows. Let
. The justifications in the following derivation indicate which axioms
or previous derivations have been used to derive the current line. PL or
indicate that reasoning in either
standard propositional logic or
is used is without
further analysis.
To see that i's beliefs may evolve nonmonotonically given
12
In contrast,
is satisfiable in Halpern's logic.
Note that if we replace
we obtain regular singleagent autoepistemic reasoning.
13
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5
Conclusion
We proposed a multi-agent logic of only-knowing that
extends earlier work by Levesque regarding the singleagent case. Our logic gives a semantic and proof theoretic account of autoepistemic reasoning for many knowers. Notions like stable set and stable expansion fall out
as natural extensions of single-agent autoepistemic logic.
As for future work, it would be interesting to obtain a complete axiomatization for all of OL. Also, as
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Acknowledgements
I would like to thank Joe Halpern and Hector Levesque
for fruitful discussions on this subject.
Lakemeyer
381