From: AAAI-86 Proceedings. Copyright ©1986, AAAI (www.aaai.org). All rights reserved.
IMPLEMENTATION
OF AND EXPERIMENTS
A VARIABLE
PRECISION
LOGIC INFERENCE
Peter
WITH
SYSTEM*
Haddawy
Intelligent
Systems
Group
Department
of Computer
Science
University
of Illinois
Urbana,
Illinois
ABSTRACT
A
approximate
system
capable
inferences
presented.
under
Censored
of
time
production
performing
is
are
used
to represent
tion.
These
both domain
and control
informaare given a probabilistic
semantics
a scheme
and
reasoning
is performed
using
Examples
based
on Dempster-Shafer
theory.
show the naturalness
of the representation
and
the
flexibility
further
of
research
the
system.
Suggestions
Certainty
refers
to
statement,
while
specificity
of detail
constraints
rules
61801
the
degree
of a description.
the
certainty
of an
varied
to
conform
to
sored
and
degree
inference
cost
Michalski
Precision
production
control
in
to the
a
The idea of a logic
which
presented
by
This Variable
of belief
refers
constraints
and
Logic
rules
be
was
Winston
[1985].
(VPL) used cen-
to encode
information.
in
could
both
The
rules
domain
take
the
form
for
are offered.
P->D[C
read
If P then D unless C.
part
of the rule
I INTRODUCTION
The unless
It is a Sunday
afternoon
and your
car is taking
you for a drive.
autonomous
fully
Suddenly
Your
to do.
If your
soning
technology,
reach
mine
best
any
better
than
because
course
decision,
there
are
extra-logical
time
and
In
cost,
resource
account.
certainty,
be used
any
and
to flexibly
what
it would
trying
reanever
you.
even
a rough
is needed
reasoning
constraints
limitations
which
tradeoff
specificity
/ SCIENCE
Whereas
to checking
in
In this
unless
guess,
is
is a sys-
exclusive-or
time
is
their
truth
that
of rule
resources
Thus,
only
is devoted
critical
to checking
The
interpreted
between
given
are
a lim-
situations.
logically
operator
the consequent.
of
than
unlimited
of resources
symbol
the censor.
the antecedents,
censors
the
as
censor
an
and
the rule
possi->
process
we
weather
is good,
Sunday
and the weather
as
must
be
the
of inferences
can
can
Go to the park
such
between
to these
priority
amount
conclude
the park.
constraints.
* Th’IS work was supported
in part by the Defense Advanced
Research
Project
Agency under grant N00014-K-85-0878,
in part
by the National
Science Foundation
under grant
NSF DCR 8406801, and in part by the Office of Naval Research
under grant
N00014-82-K-0186.
238
devoted
Sunday
cost
adjust
antecedents.
determination
a lower
the
limit.
practical
the
is given
ited
the best decision
The
Therefore,
values
is called
exception
conditions
and as
to be false most of the time.
hit the
to deter-
it would
and
What
time
ahead
to decide
itself
of producing
a given
into
while
the road
by current
are
of action
both
ble within
taken
powered
chances
indecision.
capable
into
5 seconds
car were
destroying
situation
out
car has
a decision
the
truck,
tem
pulls
a truck
of you.
Censors
represent
such are considered
if it is Sunday
I will go to the park;
is bad,
* - * +D [C,vC,v
interpreted
is bad,
and
the
and if it is
I will not go to
Formally,
P,AP,A
is logically
that
1 Weather
--.
as
P,AP,A
- - - A-( C,vC,v...)--+D
P,AP,A
’ ’ - A(C,VC,V...)--D
and
In order
to
make
the
exceptions
quantitative,
These
numerical
parameters
are associated
with each
rule, representing
the strength
of inference when
the truth
the
value
value
of the censor
is unknown.
precision
of inferences
This
of the notion
is known
These
values
by
the
following
allow
the
to be varied.
paper
presents
of uncertainty
to conform
constrained
and when
values
a formalization
in censored
When
the
Shafer
interval
value
according
produc-
to given
II FORMALISM
time
censor
is known,
the
D is computed
s(D) = @‘)[l - p(C)Io
p(D) = 1 - s(P)s(C)@
limits.
AND
of the
for the conclusion
to
tion rules and an implementation
of an inference
system capable of varying
the certainty
of inferences
are
restrictions:
THEORY
When
the censor
value
is unknown,
the formulas
are
Uncertain
tem
is
using
Dempster-Shafer
theory
information
and
inference
performed
A
fact
scheme
[Shafer
is represented
facts.
in the VPL
a
is
19761.
assigned
a
while
the p value
intervals
indicates
for A & 1A
The s and p values
as the
minimum
and
the proposition.
proposition
values.
the interval
disjunction
formulas
for
respectively
values
of
The
of
vals is similar
probabilities
of uncertainty
the
or
the
product
or
and
plausibility
support
across
rules
the
employed
by Dubois
rule
A
using
an
approach
[1984]
& Prade
->
[1985].
B,
prob(A)EISAPA].
it
can
be
prob(B) E [S,S,, I--s,+S,P,].
prob(d3 / A)E[S,P,]
th en P, = l-S,
it follows
this
that
scheme,
represented
prob(B)EISAS,,
the
we have
certainty
by four values:
of
the
that
Now
if
from
a
to
and
shown
l-S,,%].
conditionally
to combine
conclu-
to that
independent.
two
in [Ginsberg
which
To
use
rule
is
Q p 7 S, where
The
Shafer
inter-
19841:
above
logic,
but
logic
representation,
instances
and
discussion
for
proposi-
the
system
uses
a typed
VPL
The
containing
to
logic
How-
a semantics
for expressions
is needed.
Rules
entire
with
of the
a short-hand
domain
certainty
with
form
certainty
as Vx,y p(B(x)lA(x,y))
preted
=
Similarly,
[s p] is interpreted
free variA(x,y)
->
[s p] are inter-
for listing
of x and y.
ground
problems.
no additional
an associated
only
infor-
a finite
In this
propositional
ables
essentially
type
the elements
of
predicate
argument.
equivalent
present
presented
scheme
ever,
B(x), with
has
inference
terms
are
thus
l-(ad+bc)
representation.
enumerates
for each
1
bd
“_”
approximate
mation
domain
=
a=S,forPAlC->D
l-
I-(ad+bc)
predicate
derived
prob(l3~A)~[S,P,]
where
Th en
Suppose
similar
and
argued
--
an
as expressing
are propogated
by Ginsberg
are
used
[u b]@[c d] =
tional
conditional
events
formula
sum
separately.
Rules are interpreted
probabilities.
Beliefs
that
as
by applying
probabilistic
to
evidence
of the
of a conjunction
is calculated
for multiply
in a
difference
is represented
Th e value
of facts
The
Evidence
1-
as the
1 - s(P)/3.
sions is combined
using Dempster’s
orthogonal
sum rule. This requires the assumption
that the
can also be thought
of ‘unknown’
[0 l].
s
by p(A) =
The amount
A value
p(D) =
certainty
its plausibility.
maximum
is defined
s(D) = s(P)a
of rules
[s p]. The
a proposition,
are related
$(-IA).
on
Domain
in the form
represented
by a Shafer
interval,
value indicates
the support
for
sys-
based
[s p]. This
rules
is
over the
a fact
A(x)
as Vx p(A(x))
Is PI.
p=S;forPAC->lD
7=
S, for P ->
s=S;forP->
D
YD
Uncertainty
and Expert
Systems:
AUTOMATED
REASONING
/ 239
III
SYSTEM
OVERVIEW
approximate
that
The VPL
components:
the
knowledge
and
base,
the
consists
interface,
the unifier,
rule-base
implemented
the
bird
such
as a penguin
parser,
tion
such
as dead.
Lisp
the
engine,
The
system
and
runs
is
on
The
system
for
is designed
incremental
The user
may
assert
base
type
declarations,
followed
some
Following
example
runs.
rules
the
system
from
develop-
or retract
and
Next,
changing
our
When
may
perform
may
The
rule-base
sible
query
analyzer
the
chain
is a rule
from
a censor.
depths
determines
inference
depths
time
of censor
chain
required
chaining.
of times
query
times.
for each
for
tree
with
is stored
posall
A censor
in the search
A list
for each
inference
time
limit
system
the
the
possibility
our belief
is neither
bird,
leading
in the time
associated
searches
censor
time
limit
The
inference,
with
possibly
and
breadth
first
is achieved
ground
instances
If none
with
the goal.
query
is given
search
process,
certainty
satisfy
a fact
is found,
any
it tries
If both
When
on the
calculation
on a value
of the stack
attempts
stack
are
fail, the
the
the
on a calculation
This
ties.
/ SCIENCE
The
section
first
is
rover
jane
tweety
road-runner))
(animal))
(bird))
(animal))
(bird))
(bird))
(bird))
(is-kiwi
(bird))
(is-domestic-turkey
(bird))
(animal))
(animal))
(has-broken-wing
(bird))
assert
((is-bird
evaluated
and
put
on the bottom
8x) =>
(flies 8x)
1 (is-special-bird
8x)
(is-in-unusual-condition
((is-dead
8x) 1.0 1.0 .9 .05)
$x) = >
(is-in-unusual-condition
$x) 1.0 0.0)
to the user query.
presents
intended
$x) = >
(is-in-unusual-condition
two
the system’s
to
simple
capabili-
highlight
the
with
decrease
road-runner))
(is-emu
((is-sick
to demonstrate
to further
the entries
I-V EXAMPLES
examples
(spot
(is-penguin
(is-sick
performing
The computation
tweety
(flies (animal))
system
During
terminates,
(is-bird
(is-dead
unify
put
that
combines
in his ability to fly. Finally,
if tweety
in an unusual
condition
nor a special
(tweety
after
to find rules which
are
corresponds
the
of [0 11.
the search
list.
exhaustive
with
for
of his death
to fly propor-
is told
information
fly.
death
he is able to fly.
(is-ostrich
is
goal.
instructions
stack.
stages:
the
of these
this
(is-special-bird
on
strategy
variables
goal,
in tweety’s
in his ability
(is-in-unusual-condition
all consistent
unifies
a value
calculations
depth
The
a
which
If no
chaining
in two
free
certainty
bird,
cannot
pred
is unlim-
search
by generating
is a dead
tweety
the
will
time.
depth
exhaustive.
To
unification.
searches
for
which
search
The
of
case
backward
is performed
and
search
depth
limited
calculation.
(bird
base to determine
chaining
performs
Inference
censors.
it, in which
data
declara-
data
an optional
in the requried
is specified,
system
have
chaining
a response
ited.
search
with
the time
maximum
guarantee
may
tweety
that
the system
be a kiwi,
type
(animal
query
by predicate
domain
associated
base.
A user
condi-
file shows
the input file is a log of
After the rules are loaded,
that
the certainty
of
or is in an unusual
it concludes
changes
uniform
240
is told
which
tionately.
analysis
of
followed
define new types and predicates,
and make
Once a rule base is complete,
the user
an
is
kind
by rules.
queries.
facts,
idea
a
to be fully
rule
The
it is a special
The input
tions,
3640.
interactive
methods.
can fly unless
of six main
the inference
analyzer.
in Common
Symbolics
ment.
system
user
inference
a bird
((has-broken-wing
(is-in-unusual-condition
8x) 0.9 0.06)
$x) = >
8x) 1.0 0.0)
((is-penguin
8x) =>
((is-ostrich
(is-special-bird
$x) =>
(is-special-bird
8x) 1.0 0)
((is-emu
$x) = > (is-special-bird
8x) 1.0 0)
((is-kiwi
$x) =>
tf;x) 1.0 0)
(is-special-bird
((is-domestic-turkey
0.027635619
<RESULT>
Example
Tweety
Runs
is a dead bird ---
file shows
in time
tion
> assert
Command
> ((is-bird
ENTER
tweety)
Command
> ((is-dead
tweety)
of its brakes
limit.
or assertion
the system
the
1 1)
ENTER
Command
or make
query
and
run shows
With
a censor
cannot
depth
seconds
elapsed
chaining
version
A log of the
of varying
the time
of 1 or less
depth
the truth
censor
approximate
on the condi-
road.
determine
and
values
thus
uses
of
the
of the rule.
We
of system
(level (low medium
chaining
the
of censor
The input
if a car can stop
based
the effect
road-condition
more
> (flies tweety)
<RESULT>
an obstacle
1 1)
or assertion
0.103377685
for determining
sample
Command
censor
rules
to avoid
or assertion
ENTER
using
[l.OO 1.001
ity of the system
to vary the depth
chaining
in response
to time limits.
Command
ENTER
time
The
next
example
is
the
one
in the introduction.
It shows the abil-
$x) 1.0 0)
described
ENTER
elapsed
$x) = >
(is-special-bird
--
seconds
$x) 1.0 0)
(rating
of UNLIMITED
(good
high))
fair poor))
(substance
(gravel ice))
(place (road ground))
time
(temp-type
[O.OO 0.001
(looks
(below-freezing
(shiny
moderate
hot))
rough))
-- reduce certainty in Tweety’s death ENTER
Command
> (assert
ENTER
or make
((is-dead
tweety)
Command
query
pred
of system
(speed-distance-ratio
.7 .8))
(road-condition
> (? (flies tweety))
using
censor
0.1013306
chaining
seconds
depth
elapsed
(rating))
(brake-condition
of UNLIMITED
(rating))
(on (substance
time
place))
(temperature
<RESULT>
---
[O.OO 0.3Oj
suspect
ENTER
that Tweety
Command
> (assert
ENTER
((is-kiwi
Command
or make
tweety)
is a kiwi --query
censor
0.10849539
chaining
seconds
(temp-type))
(road-appearance
(looks))
(construction-site
0)
(sound-of-pebbles-hitting-underside-of-car
assert
.3 .5))
; rules
or assertion
( (- speed-distance-ratio
depth
elapsed
of UNLIMITED
(can-stop-in-time)
time
1 (road-condition
high)
Tweety
ENTER
> (assert
ENTER
is healthy
Command
or make
query
Command
Command
( (on gravel
of system
tweety)
censor
chaining
= > (road-condition
poor)
1.0 0)
( (temperature
or assertion
tweety)
road)
=>
(road-condition
poor)
.9 .l)
0 0))
below-freezing)
(road-appearance
1 1)
shiny)
= > (on ice road)
.9 .I)
or assertion
( (construction-site)
> (? (flies tweety))
using
1.0 1.0 .85 .l)
and normal --
(( is - in - unusual-condition
> ((I is-special-bird
ENTER
poor)
[O.OO 0.211
( (on ice road)
---
=>
poor)
(brake-condition
<RESULT>
0)
of system
> (? (flies tweety))
using
(level))
(can-stop-in-time())
or assertion
depth
of UNLIMITED
Uncertainty
(sound-of-nebbles-hitting-underside-of-car1
and Expert
Systems:
AUTOMATED
REASONING
/ 24 1
=>
(on gravel
road)
.9 .l)
systems
for operating
a rule’s
censor
rooms
In the current
; facts
((speed-distance-ratio
((temperature
.05 .15)
high)
.2 .3)
below-freezing)
((road-appearance
shiny)
((construction-site)
values
0 0)
system,
is known,
are used.
to domestic
if the value
the a! and
If the value
robots.
is unknown,
the
and 6 values are used. A better approach
be to look at the degree to which the
1.0 1.0)
((sound-of-pebbles-hitting-underside-of-car)
value
.8 .85)
is
known
between
the
and
use
this
to
7
would
censor
interpolate
7 & S rule
o & p and
of
/3 certainty
certainty
values.
Example
Runs
--- time limit of 1 second --
expressed
ENTER
nomies
Command
or input
> (? ((can-stop-in-time)
using censor chaining
0.08085977
seconds
<RESULT
>
file
1))
depth of 2
elapsed
carry
I am working
> (? ((can-stop-in-time)
using
censor
0.031729784
The
or make
query
of system
trade-offs
This
depth
of 1
seconds
elapsed
time
is
a
the
between
direction
simple
system
special-
investigated
time
cost
colmore
in taxonomies.
has only
inference
best
Taxo-
on incorporating
paper
learning
than
make
and specificity
machine
is
taxonomies.
for reasoning
between
certainty
.05))
chaining
To
rules
This
knowledge
of
information
effective,
ized inference
time
.05 second ---
Command
more
of rules.
[O.OO 0.451
time =
ENTER
world
form
the
lections
trade-off
---
Much
in
and
and
the
certainty.
specificity
and
need yet to be explored.
in
which
research
the
results
hold much
of
promise.
ACKNOWLEDGEMENTS
<RESULT
> [0.72 0.921
I would
-
time =
ENTER
.03 second ---
Command
or make
> (? ((can-stop-in-time)
using
censor
0.017470836
query
of system
.03))
chaining
depth
of 0
seconds
elapsed
time
<RESULT>
---
for his guidance,
[0.72 0.921
Command
Cannot
perform
Minimum
of this
Kelly
for help on an earlier
D., Prade,
inference
query
of system
August,
in requested
time
Ginsberg,
time.
is 0.018423745
set
CONCLUSIONS
It has
of censored
probabilistic
the
constraints.
cations
made
tion.
242
/ SCIENCE
been
semantics
time
that
allows
the
when
a system
to conform
a system
in situations
Examples
John
Wiegand
on a
and
Jim
implementation.
H. “Combination
and Propo-
1985, pp. 111-113.
M.L.
Dempster’s
Austin,
Texas,
R.S.,
has numerous
to time
appli-
decisions
must
with
uncertain
informa-
and
from
medical
(accepted
1984, pp. 126-129.
August,
Winston,P.H.,
“Variable
857, Artificial
Laboratory,
MIT, August,
for AI Journal,
Preci1985
1986).
to adjust
where
range
a
Reasoning
Rule” In Proc. AAAI-84.
MIT AI Memo
Intelligence
formalgiven
“Non-monotonic
Using
Michalski,
rules
of inferences
Such
in real
shown
production
certainty
Michalski
.Ol))
sion Logic”
ism
and
Prof.
for comments
gation of Uncertainty
with Belief Fuctions”
In Proc. IJCAI-85.
Los Angeles, California,
or make
guaranteed
V
draft
Dubois,
> (? ((can-stop-in-time)
paper,
Wolf
REFERENCES
time limit too low ---
ENTER
like to thank
Lisa
be
expert
Shafer,
G. A Mathematical
Princeton
Jersey,
University
1976.
Theory of Evidence.
Press,
Princeton,
New