REASONING IN MULTIPLE BELIEF SPACES
Joao
State
P.
Martins
MBR
is
a
reasoning
system
which
allows
multiple
bellets
(beliets
trom
multiple
agents,
contradictory
beliefs,
hypothetical
beliefs)
to
be represented
s i m u l t a n e o u s l y i n t h e same k n o w l e d g e
base
and
performs
reasoning
within
sets
of
these b e l i e f s . MBR also contains provisos
to
detect
contradictions
and t o r e c o v e r
from them.
T h i s p a p e r d e s c r i b e s MBR's m e t h o d
of
detecting
and
recording
contradictions
within
beliefs
of
different
agents,
showing an example of such p r o c e s s .
Introduction
This paper r e p o r t s a small f e a t u r e of
a large system.
t h e MBR ( M u l t i p l e
Belief
Reasoner)
system
[3].
MBR
is
fully
implemented in
Franz L i s p ,
running on
a
VAX--11/7S0.
MBR
is
a
reasoning system
which
allows
multiple
beliefs
(beliefs
from
multiple
agents,
contradictory
beliefs,
hypothetical
beliefs)
to
be
represented
Simultaneously
in
the
same
knowledge
base
and
performs
reasoning
w i t h i n sets of these sets of b e l i e f s .
MBR
also
contains
provisos
for
detecting
contradictions
and
for
recovering
from
them.
The
problem
of
detecting
and
recording
contradictions
has
been
considered by
several researchers
(e.g.,
[2,
4,
5]).
The
p a r t of
MBR t h a t d e a l s
with
this
problem
differs
from
the
previous approaches because,
1)
It is
Current
address:
Engenharla
Mecanica,
T e c n i c o , Av.
Rovisco
Portugal
Stuart
C.
Shapiro
D e p a r t m e n t of Computer S c i e n c e
U n i v e r s i t y o f New Y o r k a t B u f f a l o
A m h e r s t , N.Y.
14226, U.S.A.
ABSTRACT
l.
and
Departamento
de
Instituto
Superior
Pais,
1000 L i s b o a ,
This work
was s u p p o r t e d
in part
by
the
National
Science
Foundation
under
Grant
MCS80-06314
and
by
the I n s t i t u t o
Nacional
de
Investigagaco
Cientifica
(Portugal)
under Grant No.20536.
based
on
a
logic
developed
for
such
purpose;
2) It
is implemented such
that
the detection of the hypotheses underlying
the
contradiction
is
done
by f o l l o w i n g
only two types of a r c s ;
there is no
need
to
explicitly
mark
propositions
as
believed or disbelieved;
t h e r e is no need
to worry about c i r c u l a r p r o o f s ;
there
is
no need to keep a s e p a r a t e d a t a
structure
to record previous contradictions.
T h e SUM s y s t e m i s t h e l o g i c a l
system
u n d e r l y i n g MBR.
It is
l o o s e l y based
on
the
logical
systems
of
[1]
and
[7].
Distinguishing
teatures
of
SWM
include
recording
dependencies
of
wffs,
not
allowing irrelevancies
to be
introduced,
and
providing
tor
dealing
with
contradictions.
The
SUM
system
deals
with objects
called
supported
wffs
which
are of the
f o r m F:' : t , α , p , in w h i c h £ is a w t t , T
(the
o r i g i n tag)
is an element ot the set thyp,
der, e x t > , a (the o r i g i n set) is a set
of
h y p o t h e s e s , and p
(the r e s t r i c t i o n set)
is
a set of
sets of hypotheses.
The
origin
t a g (OT) t e l l s w h e t h e r F i s a n
hypotheses
(x=hyp),
a normally derived wft
(x=der) or
a wff
w i t h an
extended OS
(T=ext)
(this
l a t t e r case w i l l not b e d i s c u s s e d i n
this
paper).
The o r i g i n s e t (OS) c o n t a i n s
all
the hypotheses which were a c t u a l l y used in
the d e r i v a t i o n ot F.
The r e s t r i c t i o n
set
(RS) c o n t a i n s s e t s o t h y p o t h e s e s , e a c h
of
w h i c h when u n f o n e d w i t h t h e h y p o t h e s e s
in
the OS forms
a set which
is known to
be
inconsistant.
An
inconsistent set,
is a
s e t o t w f f s f r o m w h i c h a c o n t r a d i c t i o n may
be d e r i v e d .
RSs a r e v e r y d i f f e r e n t e n t i t l e s
trom
OTs a n d O S s .
Whereas
t h e OT a n d OS
of a
proposition
reflect
the
way
the
proposition
was
derived,
the
RS
of
a
p r o p o s i t i o n r e f l e c t s the c u r r e n t knowledge
a b o u t how
the hypotheses
underlying that
p r o p o s i t i o n r e l a t e to the other hypotheses
in
the
system.
Once
a
p r o p o s i t i o n is
d e r i v e d i t s OT and OS remain c o n s t a n t .
J. Martins and S. Shapiro 371
whereas
its
inconsistencies
system.
3.
Contexts
RS
are
changes
uncovered
as
in
new
the
and Bellet Spaces
MBR i s
to be
used as
the deduction
system
in
a
knowledge
base
which
may
c o n t a i n i n f o r m a t i o n e n t e r e d b y many u s e r s ,
with
different
and
even
conflicting
interests.
We
assume t h a t
each user
of
t h e Knowledge base
h a s some b a s i c
set of
b e l i e f s w h i c h h e / s h e t o l d MBR a b o u t .
Such
b e l i e f s are
the user's
basic assumptions
and were e n t e r e d
i n t o the knowledge
base
as hypotheses.
Every p r o p o s i t i o n
derived
f r o m t h i s s e t o f a s s u m p t i o n s i s assumed t o
be be b e l i e v e d by the u s e r .
We d e f i n e
a context
to be
a s e t of
hypotheses.
A context represents the
set
of a s s u m p t i o n s
o f some
user.
A
context
determines a
B e l i e f Spaces (BS) which
is
the set of a l l the hypotheses d e f i n i n g the
context
and
all
the
propositions which
were d e r i v e d
from them.
Within the
SWM
f o r m a l i s m ( t h e l o g i c u n d e r l y i n g MBR),
the
propositions
in
a
given
BS
are
characterized
by
having
an
OS which is
contained in the context.
At
any
point,
hypotheses believed is
context
(CO,
which
belief space
(CBS)
the
set
of
all
termed the
current
defines the current
.
Contexts
delimit
smaller
knowledge
bases
(called
Belief
Spaces) w i t h i n the
knowledge
base.
The
knowledge
base
retrieval
operations
only
retrieve
the
p r o p o s i t i o n s w i t h i n t h e CBS,
ignoring
all
other
propositions.
Based
in
these
two
rules
of
inference,
whenever
MBR
finds
a
contradiction
it
takes
one
of
the
following
actions:
1.
I f o n l y one o f t h e c o n t r a d i c t o r y
wtfs
belongs to
t h e CBS
the c o n t r a d i c t i o n
is recorded
(through the
application
o f URS) b u t n o t h i n g m o r e h a p p e n s .
The
e f f e c t of doing
so is to
record that
some
set
of
hypotheses,
s t r i c t l y
c o n t a i n i n g t h e CC, i s now Known t o
be.
inconsistent.
2.
If both
contradictory wffs
belong to
the
CBS.
Then
the
rule
o f URS i s
applied but,
in addition, the rule
of
1
is
also
applied.
This
has t h e
effect
of
adding
new
wffs
to
the
knowledge base and a l s o w i l l cause t h e
CC to be r e v i s e d .
5.
An A n n o t a t e d Example
we p r e s e n t in
this section a
sample
r u n u s i n g MBR.
S u p p o s e t h a t MBR i s
being
used
by
some
university
as
a
meeting
scheduling
system.
The
knowledge
base
contains,
in this case, general statements
reflecting
policies
for
scheduling
meetings
and
also
statements concerning
the p a r t i c u l a r schedules
of the users
of
the system.
MBR
is
asked
to
schedule meetings
among a c e r t a i n number o f
i t s users and it
does
so
either
by
finding
a time slot
which is compatible with t h e i r
particular
schedules
or
by
reporting
that
the
schedules of
the users
do not
allow the
scheduling
of
the
desired
meeting.
In
t h i s example w e w i l l assume t h a t :
1.
Meetings
are
being
scheduled within
one
day
only,
therefore information
about
dates
is
absent
from
our
representation;
2.
Meetings
can
not
both
be
in
the
morning
and
in
the afternoon (hypl,
Fig.i).
3.
Two
different
meetings
can not f i l l
t h e same t i m e
slot,
i . e . , morning
or
afternoon
(hyp2 ,
F i g . l ) .
We w i l l
f o l l o w MBR's
behavior using
the
information contained in the schedules
of
two of i t s u s e r s , Stu and Tony.
Both
Stu
and
Tony
already
have
some
scheduled
meetings:
1.
Stu's
schedule;
Stu teaches a seminar
in the morning
(hyp6,
Fig.l).
2.
Tony's
schedule;
Tony
has a t e n n i s
372 J. Martins and S. Shapiro
t h i s session Stu
concludes that, the
best
t i m e , f o r him, t o r scheduling the
faculty
m e e t i n g i s i n t h e a f t e r n o o n (wff 2 ) .
S u p p o s e now t h a t
Tony a l s o t r i e s
to
f i n d t h e most c o n v e n i e n t t i m e , f o r h i m , t o
have a f a c u l t y m e e t i n g .
In t h i s case,
he
does r e a s o n i n g
in the
BS d e f i n e d
by the
context
Tony-schedule=<nypl,
hyp2, hyp4,
hyp 5, hyp
7).
Some
results
of
such
wffs
derived
figure 3
from "Tony-schedule"
Suppose
that
someone
now
wants to
schedule a
f a c u l t y meeting
with all
the
members o f t h e f a c u l t y , w h i c h i n c l u d e b o t h
Stu and Tony.
When t h a t r e q u e s t
i s made
considering
a
context
containing
"Stu-schedule"
and
"Tony-schedule"
the
system
immediately
reports
that
such
context is inconsistent.
Notice that this
context
contains,
possibly
among
other
hypotheses,
the
hypotheses
hypl,
hyp2,
hyp3, hyp4, hyp5,
hyp 6 and h y p 7 .
The R S
of
hypl,
tor
example,
is
(hyp2,
hyp3,
hyp4, hyp5, hyp6, hyp7)) (Figure 4 ) , which
records that
the set
of hypotheses
hypl
t h r o u g h hyp7 i s i n c o n s i s t e n t .
The
system
responds t h a t such c o n t e x t i s I n c o n s i s t e n t
and
a
revision
of
the
CC
should
be
performed.
J. Martins and S. Shapiro 373
Suppose
now
that
s t a r t i n g from the
knowledge base r e p r e s e n t e d i n F i g u r e 1 t h e
request is
made t o
schedule the
taculty
meeting
in
a
BS
defined
by
a context
containing
"Stu-schedule"
and
"Tony-schedule"
In t h i s case,
there are no
recorded
i n c o n s i s t e n c i e s and t h e system w i l l t r y t o
schedule the f a c u l t y
meeting in that
BS.
Among
the
results
derived
are the w f f s
represented in
Figure 5.
In this
case,
time faculity-meet
wff 2 ' :
time
faculity-meet,
wffs
,morning
>
der. (hyp1 ,hyp2 ,hyp3 .hyp5,hyp6> , ( )
morning)
der.
<hyp2 , hyp4 , hyp5 ,hyp7> , ()
From
then
on,
all
a r e d i s r e g a r d e d b y MBR.
such
F i n a l l y t h e d e f i n i t i o n o f RSs
waives
t h e need t o keep a s e p a r a t e d a t a
strucure
t o record a l l the previous
contradictions
( e . g . , t h e NOGOOD l i s t ( [ 2 ] ) .
ACKNOWLEDGEMENTS
Many t h a n k s t o G e r a r d D o n l o n , D o n a l d
McKay, E r n e s t o M o r g a d o , T e r r y N u t t e r , B i l l
Rapaport
and
the
other
members
of the
SNePS
Research
Group
t o r t h e i r comments
and
criticisms
concerning
the
current
work .
figure 5
d e r i v e d w i t h i n the CC
both
wffl'
and
wft2'
b e l o n g t o t h e CBS
(the
CC
contains
the
hypotheses
hypl,
hyp2,
hyp3,
hyp4,
hyp5,
hyp6,
hyp7).
Therefore,
not
only
the
r u l e o f URS i s
applied, recording
the i n c o n s i s t e n t
set,
but also
~I is
applied in
order to rule
out
some
hypothesis
(or
hypotheses)
d e f i n i n g t h e CC.
6.
the
CC.
propositions
C o n c l u d i n g Remarks
MBR
has
been
implemented
in F r a n z
Lisp (runing
on a
VAX-11/750) u s i n g
the
SNePS s y s t e m
[6].
The
example p r e s e n t e d
h e r e was o b t a i n e d from a n a c t u a l r u n
just
just
by
slightly
changing
the
output
syntax.
One
of
the
main
distinguishing
c h a r a c t e r i s t i c s o f MBR i s t h a t i t i s b a s e d
on a l o g i c
(SUM) e s p e c i a l l y d e s i g n e d
for
B e l i e f Revision systems.
I n MBR
p r o p o s i t i o n s are
represented
b y n e t w o r k nodes
and a r e l i n k e d
w i t h the
hypotheses
in
their
OS
and t h e s e t s i n
their
RS.
This
way
of
representing
p r o p o s i t i o n s makes i t
p o s s i b l e t o know
a
p r i o r i t h e number o t
a r c s t h a t has t o
be
traversed to find
out a l l the
hypotheses
underlying a contradiction.
Another
characteristic
of
MBR
concerns
the
way
contexts
and
BS
are
defined.
By d e f i n i n g
a c o n t e x t as
a set
o f h y p o t h e s e s w e c a n h a v e a s many c o n t e x t s
in
tha
system
as
the
power s e t o f t h e
hypotheses introduced.
Also,
the
network
retrieval
functions
only
consider
the
propositions
in
the
CBS.
Whan
a
contradiction is detected, after selecting
one h y p o t h e s i s ( o r s e v e r a l h y p o t h e s e s )
as
tha
culprit
for
the
c o n t r a d i c t i o n , the
disbelief
in
all
tha
propositions
depending on such h y p o t h e s i s
(hypotheses)
i s done
just by
dropping it
(them) f r o m
REFERENCES
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and B e l n a p N . ,
Entailment:
The L o g i c QF Relevance ana
Anderson A.
Necessity.
Vol.1,
1975.
Princeton
University
Press,
[2]
Doyle
J.,
"A
Truth
Maintenance
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12:3
(1979) 2 3 1 - 2 / 2 .
[3]
Martins
J.,
"Reasoning
in
Multiple
Belief
Spaces",
Ph.D.
Dissertation,
D e p a r t m e n t of
Computer S c i e n c e ,
SUNY
a t B u f f a l o , May 1 9 8 3 .
[4]
McAliester
D.,
"An
Outlook on T r u t h
Maintenance",
A . I . Lab.,
M.I.T.,
AI
Memo 5 5 1 , 1 9 8 0 .
[5]
McDermott
D.,
"Contexts
and
Data
Dependencies:
A
Synthesis",
Department ot
Computer S c i e n c e ,
Yale
U n i v e r s i t y , 1982.
[6)
Shapiro
network
S.,
"The
processing
Associative
(ed.).
1979.
[7)
Networks,
Academic
SNePS
semantic
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in
Press,
N.v.Findier
pp.179
203,
S h a p i r o S . a n d Wand M . , " T h e R e l e v a n c e
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Science
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U n i v e r s i t y , I n d i a n a , 1976.