From: AAAI-84 Proceedings. Copyright ©1984, AAAI (www.aaai.org). All rights reserved.
NON-MONOTONIC
REASONING
Matthew
Department
USING
St,anford
lahelletl
that
so as to reflect
represent
California
are thought
esi$ting
to combine
to he a natural
deduction,
the arcs
of semantic
detail.
of acceptable
con-
can be used
eflectively
developed
also seems
one in which
to describe
about
higher
and d is the extent
.98 = .02 above).
d, etc.,
so that
Rich
nets
[Rich,
[Quillian,
the deplh
sent.
SEMANTIC
birds
19681 with
of convict,ion
The non-monotonic
represented
factors”
rule
“birds
flyers
and
respectively.
semantic
c + d 5
indicating
fly” would
Since
1, with
complete
thus be
sponds
fib
flyers
knowledge
birds
the certainty
%
factor
birds 11~. Monotonic
flyers,
of .95 indicates
rules have certainty
ostriches
is also written
7
that
factors
95%
[Shafer,
but
of
(2)
ezncfly
that
between
conditional
95%
90%
3
thought
It seems
of all birds
and
98%
to
so c + d = 0 corre-
that
to no knowledge
unreasonable
do.
better
Instead
p(flyer(z)lbird(z))
to believe
sition to be confirmed
to which
of having
the
y (2)
to evaluate
x 5
of an x being
the minimum
z.
probability
reasons.
From
Thus
(3)
z,
the fact
that
of an x being
a z is at
a z is at least
the value of the arc x s
ud) and we have, for example,
z
birds
(1 0)
z
that
-3
(.9
the
a y is a, it follows
of an x not being
Tweety
flyers
.02)
conclusion
5
(.Q
a given propo-
evidence
jsa
and
y
gives rise to the non-monotonic
range as a pair (c n),
by the available
this rcpresenta-
to believe
= .95, we take
we believe
using
probability
Tweety
probabilities
of not as specific
E [.9, .98].
c is the extent
reasoning
an d want
is (ac
fly-much
We will write such a probability
where
simple
least UC. The probability
flyers.
are better
probability
p(flyer(z)jbird(z))
As c + d = 1 corresponds
minimum
that
as
19761 has argued
above
as ranges.
that
have
interval
we have
isa
ad for similar
ostriches
values,
if the probability
[0, l] and therefore
x (iZ)
of 1, as
non-flyers,
by Rich
suppose
(1)
in
Shafer
c _< d; we will always
only
of a probability,
to the interval
To perform
tion,
such as those
we need
equality
at all.
but as
which
3
flyers
(0 1)
[c, C1]is in fact a single point.
they repre-
not as
birds
where
is [c, Z] in general.
levels of
labelling
“certainty
(J for
interval
.02)
NETS
held in the properties
the probability
2
(.9
reasoning”.
19831 h as suggested
it is dis-
(1) and (2) now become
ostriches
I
to which
We will also write
to in the last paragraph
The beliefs
probabil-
methods
(1 -
c, d for 1 -
referred
they
If these
framework
such as “reasoning
1 -
nets be
in the propertics
of as ranges
The
example),
confirmed
in greater
statistical
them.
94305
above
confidence
is investigated
Gclcnccs
ities,
suggestion
Science
University
ABSTRACT
Rich’s
RULE
L. Ginsberg
of Computer
Stanford,
DEMPSTER’S
(-9 in the
126
flyers.
.02)
(4
II
The
DEMPSTER’S
difficulties
ing applications
conclusions.
with
RULE
this scheme
arise
when
differ-
If we have
(?;I
ostriches
with
result
we need is some
way
e.
admit
reasoning.
this
[Dcmpster
situation
exceptions;
conflicting
1968
or Shafer
in depth,
and
what
(0
a.
problem
(or other)
plication
attractive
inferences
and associativity
be able to overcome
(a
in which
are drawn
b) + (0
0) = (a
b).
any attempt
to apply
is critical,
example,
generate
since
another.
0)
that
an arc
The point
E
elephants
(0 1)
rule.
(a
ranges
(a
0) and
the corrcspondin
probabilities
0) =
(a + c 0)
(c
g arcs;
combine
each
in this
UC
is undefined.
has
ditficult
of determining
Such
been
proven
a conflict
If we denote
1) or (1
us to easily retract
lier inference
rule
of
a combination
both
valid
and
in the database.
without
tion
by -,
range
(0
from
cd - bee - ud2
(7)
of an ear-
conclusions
drawn using
AND
METARULES
approach
to non-monotonic
by McCarthy’s
formulation
A labnormall(x)
--+ flies(x)
deduc-
[McCarthy,
19841:
bird(x)
0)
The
from
We might,
*TX).
the conclusion
influencing
RULES
is implied
we will
ostrich(x)
--+ abnormall(x)
ostrich(x)
A labnormal2(x)
effect
of these rules
is an ostrich
invalidate
-+ Tflies(x).
is to have the fact
not
the conclusion
for
can fly, but
Ihe rule which
the
formulation
we want to deactivate
%
(.9
but the rule corresponding
we draw
that
Tweety
that
Tweety
led to that conclusion.
Tweety
(should
the inverse
0),
means.
%
flyers.
(0 1)
here is that such an inference
0) + (c
(5) again.
a non-monotonic
This enables
the ap-
that
z
flyers
(0 0)
it) will have no effect on our eventual
c.
is simply
reasoning--that
d) # (0
pair
Tweety
the
invertible.
III
non-
at all and will result
the arc Tweety
that
we have, for (c
The commu-
The probability
an inapplicable
rule can
(4)
(a b)- (c d) = ( F$--e!;!;dd
by
non-monotonic
of (6) guarantees
to no knowledge
for
result
is no danger
rule to obtain
and as such represents
this difficulty.
corresponds
There
0);
such as (5) will bc invalidated;
to apply
1) + (1
prop-
In many
associative.
of one rule may invalidate
tativity
b.
and
the order
systems,
(1
This
is in
If we denote
has the following
It is commutative
=
1).
of a non-monotonic
the two results
A more efficient
monotonic
d)
(0
certainty.
f. + is (nearly)
other
formulation
d) =
conclusions
by corn bining the
(a b) + (c d) the inference obtained
two inferences
(a 6) and (c d), Dempster’s
rule gives us
This
0) + (c
a non-monotonic
conclusion
it is legitimate
invalid
19763 has dis-
our
case of his investigations.
a special
a logical
of non-monotonic
indicates
such as (4) and (5).
Dempster
(1
no application
of combining
when
of non-monotonic
to combine
l),
1) -t (c
inference.
(4).
rules by their very nature
(0
This allows us to avoid the most computationally
2
flyers,
(0 1)
is typical
This situation
(0
0),
applying
an eventual
%+ flyers,
(0 1)
d) #
(1
that
aspect
in contradiction
fact
(c
outweigh
when
Tweety
cussed
d) f
ever
we obtain
Default
For
(c
implies
of the rule used in (3) lead to different
Tweety
d.
In our
not the arc
flyers
.02)
to
conclusions.
0).
The
indicate
case,
the
no
birds
probability
disbelief
*%
(.9
flyers
(8)
.02)
in
itself.
(independent)
describe
in the usila.1 fashion.
127
In order
to see how
to do this,
the rule (8) in greater
detail.
we need
first t.o
We will think
of a rule as a triple
a is a list of antecedents,
(cz
probability
interval.
antecedents
are satisfied,
The
then the consequent
probability
range
example
p.
An
the best clarification:
intention
list consists
is (isa x flyers),
Thus
ensuing
Returning
to a subset.”
provide
(((isa
then z can
multiple
with confidence
at the same
antecedents,
if the value
(.9
a product
case,
(rule
(rule
should
((isa
Now suppose
we read an article
(11~) and (llb)
Rules
the metarule
article
1))
(12)
is now believed
repeats
cannot
fly. The second,
itself.
If the rule has been
accurate;
rule ensures
that
of the
rule
we will reach
we apply
(9)
PW
ostriches
the reversability
the conclusions
The
of
tially
we have been
ostriches
will remain
the same
t o an ostrich.
considering,
conclusion
(12),
But
the cer-
consider
or not
are true.
in the National
(.9
Enquirer
If I read something
probablity
such
applied
in the Nat,ionnl
and I will believe
.05)
for example,
range
interval
(.92
.07).
as (lob)
that
ensures
when
(llb)
Enquirer,
(.5
.4)
to be true,
appearance.
(a
maximum
extent
RULES
we have described
is probably
range
examples
in applying
weight
given
such as
as a way to assign
b), we should
poten-
we have
of a metarule
Thus
a proba-
a rule with
its conclusion
any other probabilities,
to which
since b is the
the rule may be applicable.
both
(11~)
not
Firstly,
it enables
without
obtaining
being
rules
do need
will
of this idea will require
a list of rules
rule assumes
which
have been
There
reapplying
information.
independence
combined,
multiple
us to mainused
are thrne advantages
us to avoid
new
either
or actito this.
a single
Since
rule
Dcmpster’s
of the probability
estimates
used of a single rule need to be
avoided.
to be true with
Here we really
that
FOR
the
to a rule itself.
v&ted by other rules.
(114
are true.
the story
Now rule (11~)
is believed
the following
W)
can be applied
if we later read
interpretation
Implementation
Articles
rule
best
updating
tain
articles
.4).
Times.
by the Enquirer
of the methods
by 6 before
example:
Newspaper
article
well beyond
The
probability
fly guarantees
whether
power
thus far.
WC will be saved
do not
the
The
(.5
happens
PROBABILITIES
extends
bility
that
is what
to some extent
IV
the rule (9)
so.
In the example
that
applied,
if the rule has not been applied,
the work of doing
tainty
deactivates
and
(11~).
1)).
the rule that
however,
with (116) activating
deactivating
in the New York
is applied
(12’)
Enquirer.
to be true with confidence
important
corroborated
.02))(0
are activated,
.05))
1)).
in the National
and therefore
.4)))
2) (.9
(0
(( (isa x ostriches))
simply
z) (.5
(accurate
.02a).
case, we have the rules
( isa 2 flyers) (.9
(accurate
x newspaper-article))
is (CL b), the
will be (.9a
alone
x birds))
newspaper-article)
((isa x Enquirer-article))
Equally
Dempster’s
(12)
1)).
we have
(( (isa Enquirer-article
.02).
level as the
of (isa x birds)
to (isa x flyers)
to the ostrich
first of these
this as
x y)
As a special
the same article
The
write
a
can
(9)
.OZ)).
(isa x flyers)(O
((isa
WC could
apply
rule which
(rule (a (isa z x) b) c d))
(( (isa x ostriches))
(rule
“Never
with
of the single arc (isa x birds).
is activated
increment
be the metarule,
x birds))
The antecedent
be used).
holds
would
(rule (u (isa z y) 6) c e)(O
The consequent
(for
bc applied
if all of the
will probably
(isa 5 flyers){.9
antecedent
is that
still
rule to a set when there is a corresponding
as
(((isa
rule itself
Better
p) where
and p is a
The rule “if z is a bird,
fly” will be represented
The
c
c is a consequent,
Secondly,
a
be
can be.
128
this approach
activate
a rule.
Returning
pcrlmps
all wc should
enables
to
us to
purtiully
our newspaper
say is that the rule (11~)
de-
example,
is not as
‘l‘he
(rule ((isa
5 IlcwsI)aI)er-n~t,iclc))
(accurate
x)( .9 .05))
(0
of (1 Ib)----note
If we use this rule instead
d i~~nppcars -- an article
believed
to be true with
Bote
also that
of Dempster’s
fore
in tllc
(1 la)
we store
probability
in order
will have no difficulty
invocation
(12’)
will
reversing
we
and
Izave used
the inference
that
(1 la),
advantage
For a rule which
the attention
has probability
range
attention
((isa
( isa x flyers)
a rule will itself need
(.9
If we are maintaining
they
a list of rules
are expected
rule such as (14) can be used to ensure,
system,
that
the inference
fly will be drawn
generally,
an element
to that
(((isa
Mike
stimulating
group,”
think
were the only
(1
(14)
to
any given
about
one in which
them
probabilistic
reasoning
can be
This framework
to describe
general
examining.
would
like to thank
Gencsereth
John McCarthy,
and Sally Greenhalgh
Ben
for many
discussions.
bird
PI
early.
about
McCarthy,
considering
properties
which
are
Quillian,
( isa x z) a) (-5
flyers,
so that
flyers
reproduce
As it stands,
0)).
%
birds
had
(14))
with
y =
the result
and flyers will be somewhat
of applying
weaker.
Sot.
of Bay&an
B, 30 (1968))
“Applications
Infer-
205--247
of Circumscription
Sense Knowledge,”
PI
“Semantic
Processing,
Press,
1968, pp.
Rich,
E. “Default
ing.”
In Proc.
PI Shafer,
x y))
Stat.
Common
M.R.
formation
William
(15)
J.
Formalizing
MIT
“n7hen
“A Generalization
A.P.
J. Roy.
to
to appear
(1984)
a
into the metarule
0)’ this would
and z = flyers.
PI
and
x y)
(rule ((isa
to birds
them
can be focussed.
and useful
ence.”
for any forward-
(isa 2 y))
(15)
the rules
fly,
to be useful,
that
we can translate,
of a group,
value
and
by including
assigning
of the type WC have been
The author
Grosof,
.02))( .5 0)).
a high level of activation
which
with
More
by allowing
both
Reasoning
to be a promising
PI Dempster,
x birds))
can probably
by
easily,
REFERENCES
the levels
birds
b),
the rule is likely
x birds))
chaining
and
own.
have
be of any use).
truth
of
(u
on the fact that birds
(rule
can be obtainctl
and attention
knowledge
of this
To focus
If birds
seems
rules
of their
discussed,
them.
n-ill1
described
counterparts.
thcmsclves to be treat t:d as arcs,
other
ire dealing
be
their monotonic
power
in sewhich
ACKNOWLEDGKMENTS
unexplored)
to be nscflrl.
(((isa
can
to arcs
of’ the pi-oblcnls
after a sub-
to which
unique
we
with
Further
rah:;es
sonic
11~: c~;countcrcrl
infcrcnccs
of (13).
of u as the extent
(Such
otjherwisc
weights
(13) be-
of conlidcnrc
sccn~s to solve
mesh neatly
within
invertability
we can think
we might
be
(.54 .03).
range
is that it may allow us to focus
the system.
(13)
now
this result--provided
that
A final (but currently
approach
that
that we need not apply
to obtain
would
assignnicnt
nets
Non-monotonic
4.
Enqllircr
the commutativity
rule mean
the information
sequent
Natioin~l
mantic.
Reasoning
Inc.,
G. A Mathematical
Princeton
in Semantic
M. Minsky.
In-
Cambridge:
216-270.
AAAI-83.
Kaufmann,
Princeton:
Memory,”
ed.
as Likelihood
Los
Altos,
1983, pp.
Theory
University
ReasonCalifornia:
348-351
of Evidence.
Press,
1976