From: AAAI-87 Proceedings. Copyright ©1987, AAAI (www.aaai.org). All rights reserved.
Filming
a Terrain
under Uncertainty
and Probabilistic
Raymond
Department
Using
Temporal
easoning
D. Gumb
of Computer
Science
University of Lowell
Lowell, Massachusetts
Abstract
We
address
under
problem
uncertainty,
text
seek
an
the
to
facilitate
to
identify
the
the
ambiguous
make.
and
the
a certain
been
type
object
sible
movement
a frame
idea
and
is to
an
ent
and
highly
one
type
does
and Lesser,
centralized
at a single node.
presented
Think
film
can
object
is of
properties
type
an
of a map
and
in
as
plausible
which
change
to TEMPRO
an
an
can be applied
only reasoning
whenever
19861 for probabilistic
logic (called
a language
having
ple, first-order
of
intensional
object
conditional
using, say, the methods
[Leblanc,
probabilistic
languages
semantics
1981]). A necessary conthe rules be expressible
a probabilistic
languages,
proba-
of [Nilsson,
of systems similar to TEMPRO
is that universal laws codifying
The
object
an
In the terminology
bilities can be ordered
dition for the application
being
(coher-
into
nature.
19861, we consider
in some of the literature
impos-
a film.
as well as of a spatial
The logical and probabilistic
features of TEMPRO
can be formalized in a natural manner. Systems similar
have
and
when
as being
mysteriously
objects
resolution,
of frames
in
conditional
an
of its
We
semantics.
in
For exam-
and most of the usual first-order
such as the one implicit
in this pa-
per, have a probabilistic
semantics, but, for second-order
languages, no probabilistic semantics is known.
type.
TEMPRO
l[ntroduction
.
In this paper,
some
probable)
of [Durfee
about
(that
a consistent
not
poral
con-
objects.
object
in conflict
is detected.
data
movements
alternative
a sequence
construct
of another
that
is used
is assigned
an
probability
given
recognized)
of
of
reasoning
possible
A conditional
and
identification
by
sensor
spatial
interpreting
temporal
type
image
probabilities
of
using
we describe
TEMPRO,
can be formalized
ities of alternative
a system
that em-
Hintikka
in terms of the probabil-
model systems in a quantified
ploys temporal reasoning and probabilities in conflict resolution. ’ TEMPRO
f ocuses on the information manage-
temporal logic with identity and a past tense operator.
In formal terms, we construct the most probable Hintikka
model system [Leblanc, 19811 (the most probable corrected
ment aspects of interpreting
TEMPRO
uses conditional
film) extending a given consistent
19781 (a given noisy film).
sensor data under uncertainty.
probabilities to order conflict-
ing rules, and diachronic inconsistencies (impossible
ments) to trigger the selection of alternative rules.
would be less reliable
if checks for
II.
dimensional
were ignored in conflict resolution.
The development of TEMPRO
objects
cerns similar to those motivating
19851 and
correction
[Ferrante,
[Durfee
and Lesser,
by con-
such works as [Ferrante, .
TEMPRO’s
error
19861.
facilities bear some similarity to the devices of
19851, which integrates techniques for reasoning
about uncertainty
the constraints
and constraint
embedded
within
propagation.
TEMPRO
However,
are of a tem-
other
are permanent
lished (i.e.
tional
research
was supported
Laboratories.
by grant
95-2931
from
Sandia
Na-
Automated Reasoning
The
i&own
objects
Icnown objects.
objects
An area containing
are called un-
one or more unknown
is said to be \occupied.
During
diagonal
whereas
are perma-
whose existence has been previously estabprior to the simulation).
Permanent objects
which are not known and all mobile objects
one unit of time, a mobile object
adjacent area (in a horizontal,
direction)
upon its type.
the terrain.
116
within an area. Some of the
(i.e. have a fixed location),
are mobile.
objects
nent objects
the types of obworld. The two-
world consists of areas laid out in a grid, with
zero or more objects contained
an (immediately)
’ This
[Gumb,
Objectives
TEMPRO
is designed to determine
jects situated within a two-dimensional
diachronic consistency were not in place, and both less reliable and more inefficient if the conditional probabilities
was motivated
theory
move-
TEMPRO
has been tested in a Monte Carlo simulation. Sensitivity analysis of the experimental results indicates that the system
evolving
or it can remain stationary,
An inspector
The
inspector
can move to
vertical, or
depending
is assigned the task of filming
is restricted
to moving
in a
straight line from one area on the grid to another at the
rate of one unit of distance per unit of time. In particular,
the inspector travels down row 0 and, at time t, is located
in area (0, t). The inspector films the terrain in his field
of vision, taking snapshots at the rate of one frame per
unit of time.
The
each snapshot
currently
inspector’s
field of vision is limited,
as
covers only the area where the inspector
located
and the adjacent
is
areas. The inspector
is
given (1) an initial map showing the location
of the known
objects
to traverse
on the terrain
and (2)
instructions
Figure
1: The Four Phases in the Simulation
a
path of length n. On his its path, the inspector’s objectives
are (1) to film as faithfully as possible both the permanent
ate the history and a correct film of the terrain.
and movable objects
introduced
and (2) to construct
map of the permanent
with TEMPRO
objects.
a more complete
The inspector
for use as an error correction
In the simulation,
the identification
is provided
system.
of unknown
jects takes place in the presence of uncertainty.
of an object
is determined
by the properties
error occurs when information
regarding
The type
it has, and an
an object’s
prop-
erties is lost in the sensor input, making the object’s
indeterminate.
moment
If the inspector
by reasoning
type
was located at the previous
in the same area where the object
now, TEMPRO
ob-
in question
is
might be able to correct the present frame
about
what objects
in the immediately
pre-
ceding frame could now be in that area. (Note that, in
the previous moment, the area where the object is now
and every
area adjacent
tor’s field of vision.
to that area were in the inspec-
Hence,
in the preceding
frame,
the
inspector could see every object which now could be in the
area in question, and the area in question is included in
the present frame.) Similarly, TEMPRO
might be able to
correct past frames based on the present and future frames,
Sometimes,
TEMPRO
is unable to identify
the type
of an unknown object with certainty.
However, knowing
some of the properties of an object allows TEMPRO
to
make informed
guesses about its probable
type.
TEMPRO
is given conditional probabilities
to facilitate its guesses.
Even when temporal reasoning can identify the type of an
object with certainty,
the conditional
probabilities
ful because they are used in conflict resolution,
the need for chronological
The
current
goal
minimizing
backtracking.
of TEMPRO
where, for any occupied
is to correct
errors
area, at most one of its objects
can be in error, and, for any object
of its properties
are use-
in error, at most one
cannot be identified.
and, in phase 3, TEMPRO
attempts
rors, producing
film and a more complete
map
In phase 4, TEMPRO’s
cor-
a corrected
of the permanent
objects.
film (phase
1) and evaluated
mance index.
corrected
PRO’s
with a grade and a perfor-
film, and the performance
index measures TEM-
efficiency.
In the first phase, generating
the history of the terrain,
the user is asked to supply the following
1. the number of time periods
2. the location
information:
(n),
of the known objects
on the terrain,
3. the average number of unknown objects
per area,
4. for each type, the absolute a priori probability
one of the unknown objects is of that type, and
of losing information
5. the absolute probability
ing an occupied area.
that
regard-
The information entered in step (2) provides the initial
map of the permanent objects.
The number of unknown
objects is determined by the information given in steps (1)
and (3), and the total number of objects is the sum of
the known
and unknown
objects.
given in step (4), the system
of each unknown object,
and then each object
random on the grid, which, together
permanent
objects,
of the objects
Using the information
chooses at random
on the terrain.
the type
is placed at
with the initial map of
gives the initial (time
Proceeding
1) configuration
inductively,
the
next configuration (time 2,. . . , n) of objects on the terrain
is obtained by choosing, for each object, one of its possible
to the inspector’s
noise in a frame,
field of vision.
in time t (1 < t < n), the restriction
the objects
testing
TEMPRO’s
error correction abilities. In phase 1, the user
supplies a (possibly incomplete) map of the permanent objects and other information
with the correct
moves at random.
Simulation
the four phases in the simulation
these er-
The grade gives the accuracy of TEMPRO’s
In phase 2, generating
1 depicts
to eliminate
rected film is checked for correspondence
restricted
Figure
Errors are
at random in phase 2, resulting in a noisy film,
which the system uses to gener-
attention
is
For each point
of the configuration
of
on the terrain to these areas gives the correct
frame at time t. A noisy frame is generated
frame by selecting
occupied
using the error rate specified
ing an unknown
object
and losing one property
from a correct
areas (at random)
for an error
in step (5) of phase 1, select-
for an error in each selected
area,
of each selected object.
Gumb
117
In phase 3, correcting errors using temporal and probabilistic reasoning, TEMPRO’s
corrective action depends
upon the time.
First, without using any temporal reasoning, TEMPRO
determines all possible object configurations that are compatible with the information provided
in the noisy frame.
Second, TEMPRO
orders the possible configurations on a list using conditional probabilities
computed
from the information
(4) of phase 1. There
figurations
1 to n.
for each time from
the moment
at least)
Third,
con-
an error in its program
frame.
If
P4
t3
P2
P3
t4
I P2
P4
logic.
termiIf this is
pro-
t, t > 1, and the list for time
backtracks
corrected
and the list for time t is not empty, the first configuration
on the list is removed and checked for compatibility
with
the corrected
frame for time
it is rejected,
and the next configuration
<t,(X)
Ifn(X),
then h(X) iff
not t3(X).
<t,(X)
then t2(X) iff
not t4(X).
<t,(X)
and
proceeds to time
t = n, TEMPRO
the permanent
objects
on the terrain
the simulation
proceeds
to phase 4.
V.
on the list is re-
frame, and checked.
and t < n, TEMPRO
If it is compatible
Figure 3: TEMPRO’s
Laws
If it
t4(X)
if 232(X)>
if ps(X);
t3(X)
if p3(X)>
if p4(X);
t4(X)
if p4(X)>
1
from Universal
nent objects
are printed.
The
To illustrate TEMPRO’s
error correction techniques, we
consider the following simple universe: There are only four
types of objects
more complete
the location
of the known objects
of
t + 1.
reports a map of
along his path, and
The final corrected film (i.e. the final sequence of corrected frames) and the more complete map of the permamap gives
as well as the location
that are judged to be of permafilm represents a consistent and
plausible (highly probable if not completely correct) evolving theory [Gumb, 19781. The user is given the option of
tracing the corrected
PI(X)>
if pz(X);
Rules are Extracted
Types
(tl,. . . ,t4) and four properties (PI,. . . ,p4),
which characterize
of those unknown objects
nent. The final corrected
if
t2(X)
then h(X) iff
not t4(X).
Pair
if pi(X);
t - 1. If it is not compatible,
moved, taken to be the corrected
is compatible
Rule
<t,(X)
iff
If m(X),
If p4(X),
to time t - 1, the config-
t - 1 is removed and taken
frame. If this is time t, t > 1,
Law
PI(X), then tl(X)
uration first on the list for time
to be the (new)
1
and their Properties
not t2(X).
If this is
time 1 and the list for time 1 is not empty, TEMPRO
t is empty, TEMPRO
P3
Pl
Uwiversal
first on the list, taking this (for
to be the corrected
ceeds to time 2. If this is time
Pl
t2
Figure 2: Types
TEMPRO
time 1 and the list for time 1 is empty, TEMPRO
nates, reporting
Properties
t1
entered by the user in step
is one such list of all possible
removes the configuration
Type
frames as they are selected.
the four types.
each type is characterized
If information
In Figure 2, note that
by two properties.
regarding
a property
of an object is lost
in the sensor input, TEMPRO
can narrow the object’s possible type down to two types. For example, if an unknown
object is really of type tl and property pl is lost, the inspector
can determine
that the object
is either of type tl
or t3 because the inspector knows that property p3 holds.
The information in Figure 2 determines four universal laws
(Figure 3) that state that, if one of the four properties hold
of an object o, then o is of exactly one of two types. From
each universal law, a pair of conflicting TEMPRO
rules is
In phase four, the correct and corrected frames are
compared, and TEMPRO
is assigned a grade and a performance index. The grade is the number of errors in the
noisy film that were properly corrected in the corrected
film divided by the total number of errors in the noisy film.
The performance
index is an ordered pair (b, T), where b
is the number of backtracks and T is the number of rejected configurations.
performance
ranging
under various conditions
performance
number of backtracks
constituting
debilitates
118
A rough ranking of the TEMPRO’s
indices
order.
The
b is the first item in the ordered pair
the performance
efficiency
can be had by ar-
in lexicographic
index because backtracking
as well as real-time
Automated Reasoning
veracity.
extracted as shown in Figure 3. The
said to codify TEMPRO’s
rules.
Within
universals
each pair of rules, conditional
solve conflicts.
For example,
regarding
laws are
probabilities
re-
the first pair of
rules, if object Q is observed to have property pl and the
conditional probability of an object’s being of type tl given
that it has property pl is greater than the conditional probability of its being type t2 given ~1, then object Q is taken
to be of type
tl.
Permanent
objects are of type tl. Objects of types
t2 - t4 are mobile and, during one time period, can remain
stationary
or move to an adjacent
area.
time
1
2
3
Suppose k errors occur in a noisy frame, and, if Ic 2 1,
that the i-th object in error (1 5 i 2 Ic) is observed to
have exactly
one property.
Then,
the unknown object
can
Figure 4: A Noisy
be of one of two possible types. In general, there are 2k
possible configurations
of the objects (i.e.
possible corrected
frames).
Each
possible
ble with the information
a possible
configuration
ity checks),
(prior
objects
to making
in error,
sumed types, and P(tij
given
tij
is compati-
in the noisy frame.
J&j, then
observed
ti,, . . . , ti,
of
are their as-
, pij ) is the conditional
(assuming
In
the compatibil-
if pi,, . . . , pik are the properties
the k unknown
Of
configuration
provided
probability
independence)
Figure 5: A Corrected
we have
p(til7
Pi,)
X ’ ’’ x P(ti,, pi,) as the probability of this configuration. The ordering of the possible configurations induced by these probabilities
is used in conflict resolution
as described
Each
earlier.
might
viewed
in temporal
tally)
when
viewed
be (synchronically)
consistent.
t-1, t - 11, t-1, t>, C-4t (1, t - l), and (1, t)).
Film
21, V-4t - l>, to, t>, (1, t - 21,
t > 1 and the corrected frame for time t is not compatible
frame for time t - 1, TEMPRO
rejects
the corrected frame for time t.
If
of two frames,
lation,
Film
succession,
in temporal
consistent,
might
but,
isowhen
with the corrected
not be (diachroni-
The possible movements
of objects serve
to determine compatibility
checks for adjacent frames in a
film. For example, an unknown mobile object located in
area (i, j) has 9 possible movements available to it if it is
not on an edge of the grid. The nine areas to which it can
move are (; - 1,j - 1), (i - l,j),
(i,j>,
(i,j+l),
TEMPRO
cerning
(i+Lj-I),
employs
the possible
(; - 1, j + l),
movements
of objects
tor’s field of vision.
If the simulation
so the time is t =
1, then TEMPRO
patibility
provide
context.
three compatibility
- l),
in the inspec-
is just beginning
and
can make no com-
checks because there is not (yet)
temporal
(;,j
(i+l,j),
and(i+l,j+l).
three compatibility
checks con-
If the time is
a past frame to
t 2 2, there are
checks:
In each of the 6 areas covered in both the frame for
time t - 1 and and the frame for time t, there must
be the same number of permanent
(The6commonareas
(type
are(-l,t-1),
(-l,t),
tl) objects.
(&t-l),
Consider
view.
2 in area (1,3),
truck (T)
the number of objects of that type located in the area
must be less than or equal to the sum in the
frame for time
and adjacent
t of the objects of that type in that
the areas (-1, t - 1), (-1, t),
(?‘s)
For each type from
(1, t - 11, (1, 0,
indicates
those areas in which
a car (C)
at time 2 in area (0, l),
at time 3 in area (1,4).
TEMPRO’s
and a
compatibilobjects
to be determined:
1. In frame 1, a car (C) is in area (-1,O)
because the
car at time 2 in area (0,l) must have come from there
(compatibility
check (3)).
2. In frame 2, a truck (T)
have moved
is in area (0,3)
at time 3 to area (1,4)
truck is the only object
because it must
and, at time 3, a
in area (1,4)
(compatibility
check (2)).
3. In frame 3, a boulder (B) is in area (1,3) because it
was there at time 2 (compatibility
check (1)).
TEMPRO
areas (i.e.
t-v
+ I>, to, t - l>, to, t>, (0, t + l),
and (1, t + 1)).
mark (7)
ity checks enable the types of all three unidentified
t2 to t4, in the frame for time t - 1,
(0,t)
A question
information about an object has been lost. The following
objects are observed with certainty: A boulder (B) at time
(0, t>, (1, t - 11, and (1, t)).
For each type from
the noisy film in Figure 4 consisting of the frames
for times 1, 2, and 3. In each of the three frames, the inspector (1) is in the middle of the areas in his field of
Figure
constructs
the corrected
film as shown in
5.
t2 to t4, in the frame for time t,
the number of objects of that type located in the area
(0,t - 1) must be less than or equal to the sum in
the frame for time t - 1 of the objects of that type
in that and adjacent areas (i.e. the areas (-1, t - 2),
To
facilitate
ennabling
sensitivity
analysis,
an option
is provided
the user to run three variants of TEMPRO
Gumb
with
119
the same terrain history and the same noisy film. First,
TEMPRO
can be run with the standard conflict resolu-
of the current 9 large areas.
tion and compatibility
checks as described above (STANDARD).
Second, TEMPRO
can be run with the stan-
more effective,
dard compatibility
checks but with conflict resolution
done
patibility
checks (suitably
expected
plausibly
The second and third com-
modified)
and, with
should become
the resolution
much
fine enough,
the
number of unknown objects per area might
restricted to a maximum of one.
be
by inversely ordering each list of possible configurations
(REVERSED-PROBABILITIES).
Third, TEMPRO
can
The algorithm could be made more efficient
jecting into the future the number of permanent
be run with standard conflict resolution but with no compatibility checks (NO-COMPATIBILITY-CHECKS).
in each previously observed areas. Regarding extensions
of TEMPRO
(incorporating,
for example,
more types
Under a variety
mance indices
pared
with
PRO,
yielding
poral
reasoning
those
resolution.
a grade
of conditions,
achieved
for
the grades and perfor-
by STANDARD
the
other
some insight
two
In eleven sample
probabilities
chance.)
(A
had
STANDARD
one time and rejected
REVERSED-PROBABILITIES
much and rejected
DARD’s
grade
CHECKS
while
10 times as
STAN-
move”),
as expected,
other two compatibility
2. Compatibility
caught more
checks.
(<.2
3. Conditions
that:
objects
errors
never
than
of unknown
the
when
objects
that overwhelm
the com-
patibility checks (resulting in poor grades and performance indices) are large error rates (>.8 per area),
dense unknown
object
and a large number
distributions
of time periods
(>2
per area),
(>lO).
For ex-
ample, in one run with an error rate of .9 and a average density
of 2 objects
per area, TEMPRO
a grade of -76 (respectable)
(poor).
performance
(and details of the Franz LISP
volves making the resolution
able, so that the inspector’s
more finely
index
analysis of TEMPRO’s
can be found in [Gumb,
One of the most promising
120
and a performance
of (16,4096)
tation)
Further
received
implemen-
system
enhancements
in-
of the inspector’s sensor varifield of view could be carved
into as many as, say, 25 small areas instead
Automated Reasoning
for his suggestions
for their
LISP,
Arun
on the present paper, and Gary David-
Katsaounis,
and Peter F. Patel-Schneider
a hand in the preparation
for
of this paper.
References
[Durfee and Lesser, 19861 E. H. Durfee
Incremental
planning
to control
and V. R. Lesser.
a black-board
based
problem solver. In Proceedings of the Fifth National
Conference
on Artificial
Intelligence,
pages 58-64,
1986.
and Lesser,
Using Partial
19871 E. H. Durfee
Global Plans
and V. R. Lesser.
to Coordinate
Distributed
Problem Solvers. Technical Report 87-06, Computer
and Information
Science Department,
University of
Massachusetts, Amherst, MA, January 1987.
[Ferrante, 19851 R. D. Ferrante.
The characteristic
approach to conflict resolution. In Proceedings
Ninth International
Joint Conference
telligence, pages 331-334, 1985.
error
of the
on Artificial
In-
[Gumb, 19781 R. D. Gumb.
Summary of research on
computational
aspects of evolving
theories.
ACM
SIGART
[Gumb,
Newsletter,
(67):13,
1978.
19861 R. D. G umb. Filming
certainty
Using
Temporal
a Terrain
Under Un-
and Probabilistic
Reason-
ing. Technical Report 172, Computer Science Department, NMIMT,
Socorro, NM, August 1986.
[Leblanc,
order
19861.
in Franz
19861, Ric D avis and Rick Craft for their advise on
as well as technical matters, Pierre Bieber
son, Christos
[Durfee
per area).
(in combination)
[Gumb,
likely.
checks 2 and 3 were more effective
expected
and Alex Trujillo
TEMPRO
administrative
is more (less)
(“Permanent
there was a sparse distribution
go to Sarah Bottomley
on implementing
lending
NO-COMPATIBILITY-
and
) were .81, .62, and 56.
check (1)
Acknowledgements
Arya and Alex and Sarah for their assistance in preparing
chronologically
of these and other runs revealed
film.
by
when the types are, roughly, equally
1. Compatibility
corrected
be expected
The average grade, number of temporal backtracks,
and number of rejected configurations in 93 runs of (STANDARD)
TEMPRO
(without also running (REVERSED-
Analysis
probable
Thanks
58 configurations,
PROBABILITIES)
(NO-COMPATIBILITY-CHECKS)
substantial
work
(REVERSED-PROBABILITIES)
pronounced
checks),
.58
backtracked
over
compatibility
of
5 times as many configurations.
advantage
most
achieved
a grade
grade of .5 might
and more sophisticated
changes in the underlying algorithm are required to guarantee that, in the general case, TEMPRO
will produce the
in conflict
REVERSED-PROBABILITIES
On the average,
backtracked
of TEM-
runs, STANDARD
(NO-COMPATIBILITY-CHECKS)
(.68, respectively).
variants
into the value of using tem-
and conditional
of .73, whereas
have been com-
by proobjects
19811 H. Leblanc.
semantics.
editors,
Handbook
274, Reidel,
[Nilsson,
Alternatives
In D. Gabbay
of Philosophical
Dordrecht,
28~71-88,
Logic,
pages 189-
1981.
19861 N. J. Nilsson.
Intelligence,
to standard firstand F. Guenthner,
Probabilistic
1986.
logic. Artificial