FAULT
DIAGNOSIS
UNDER
TEMPORAL
CONSTRAINTS
John W. Sheppard and William R. Simpson
ARINC Research Corporation, 2551 Eva Road, Annapolis,
MD
21401
ABSTRACT
Currently, intelligent diagnostic systems
are
applied
to problems in which the state of the system
is
fmd. Unfortunately, with systems increasing in
complexity at
high
rates coupled with aichitectures
promoting concurrence and fault tolerance,
diagnosis in a temporal context
is
becoming
imperative.
In
this
paper we describe an approach
to associating temporal knowledge with a diagnostic
system using the propositional calculus to represent
temporal relationships that allows efficient
propagation of time constraints through the
knowledge network. We conclude the paper with a
brief discussion of
bo*.-:-.
this work
can
be
applied to
the diagnostic domain.
INTRODUCTION
Research in artificial intelligence
has
advanced to
the point where we can develop software capable of
diagnosing complex systems at whatever level of
detail required. The most common forms of
artificial intelligence systems include rule based
approaches, in which
if
then
structures
are
used to
describe the diagnostic problem, and model based
systems, in which either the structure or the
behavior of the system
is
represented
mathematically to facilitate efficient and effective
fault diagnosis. Several diMerent problems relating
to fault diagnosis concern representation
of
information and knowledge about the system to be
diagnosed.
A
signXcant representation problem
relates to how one represents information about
time constraints in relation to the diagnostic
situation.
Typically reasoning systems operate
on
an
instantiation or a “snapshot” of some problem
domain and deal specifically with logical relations
between facts in thc !.::-ledge
base.
These logical
relations often take the form of production rules
that define linkages
in
a
howledge network. Yet
typically these knowledge networks omit information
151
about time. Specifically, some of the concerns
relating to temporal knowledge include efficiently
ordering tests
so
that various time constraints
can
be met. For example, suppose a piece of test
equipment has a video display that requires a
warming up period. If some tests
can
be
performed
immediately without the display,
the
desirability of
performing these tests while we
are
waiting
increases.
This
construct would be typically given
as
Test Group
A
cannot
be
performed prior to
x
minutes. Other constructs would deal with Test
Group
A
before/after/during Test Group
B
and
Test Group
A
within/before/after time interval
y.
When considering the task
of
reasoning about time,
two problems arise. The first
is
the problem of
representing the temporal information. What
is
the
primitive unit of time? How do these primitives
combine to form an event? How do temporal
events relate to one another? How
are
events and
their interrelationships represented in a computer?
These questions are fundamental to the problem of
developing a temporal knowledge base. Once we
determine initial interrelationships between events,
we can pursue the propagation of constraints
imposed by these events and their basic
interrelationships throughout the network.
“his
is
the second problem and
is
referred to
as
constraint
propagation. Once we determine a method for
representing time and temporal relationships,
together with an algorithm for performing the
constraint propagation, we
can
construct a
knowledge network
of
temporal events for an
inference system such as a diagnostic engine.
REPRESENTING TEMPORAL
INFORMATION
Primitives used in representing time events are
either time points or time intervals. If we assume
that points
in
time are primitive, then we can
combine the points into a time interval with the end
points delimiting the interval.
On
the other hand,
many have felt that time interval
is
more reasonable
primitive unit.
In
this
case,
the primitive
is
the
CH3148-4/
92/0000-015
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992
IEEE
event. There
is
some disagreement among
researchers over which primitive
is
best. We
will
therefore
begin
with
a
brief
discussion
of
each type
of
representation.
Point
Based Representations
If
we assume the time point is primitive, then we
can
define an interval of time to consist of the set
of
all
pointsp between to endpoints. Let
Z
=
<I;>
=
(p
I
I
<
p
4
r}
where
Z
represents a time interval
and
<It>
is
an ordered pair such that
I
is
the
left
endpoint and r
is
the right endpoint. (Note that, by
transitivity,
I
5
r.)
Also,
we
can
define a time
interval
as
the set of points
p
that occur within
some time limit following a start point. Let
Z
=
<s,6>
=
@
I
s
s
p
4
s
+
6)
where
Z
represents
a time interval
as
above and
cs,6>
is
an ordered
pair such that
s
is
the starting point and
6
is
the
interval’s duration.
Interval Based Representations
James Allen raised several questions regarding the
time-point primitive and decided to assume the
interval
was
primitive. (See reference
1
for
his
discussion
on
the point/interval issue.)
He
then
developed a set of
13
binary
relations on these
intervals and a constraint propagation algorithm to
compute the transitive closure of these relations
through a network of intervals.
A nice feature of the interval primitive
is
that the
interval
is
completely self-contained.
Thus,
the
concern
is
simply with events, and specific points in
time become irrelevant unless they, again, are
specific events. Another nice feature
is
the ease in
representing ambiguity. Consider the two examples
in
the previous section. These two situations may
be
represented
as
Za
(Equals Starts Is-Started-By)
Zb
and
Za
(Before After)
1,.
On the other hand, a major disadvantage appears
when one performs the transitive closure. Vilain
and
Kautz
showed that the closure of intervals
is
reducible to the satisfiability problem which
is
known
to
be
NP-complete? Although Allen’s
algorithm for propagating time constraints executes
in polynomial time, Allen took certain shortcuts
to
achieve this result.
He
avoided the combinatorial
explosion by verifying consistency
through
only three
adjacent intervals.
Viain
and
Kautz,
on
the other
hand, adopt algorithms employing a point-based
representation and find the algorithms to
be
more
efficient than those operating
on
an interval-based
representation.
REPRESENTING END POINT
RELATIONS
WITH
INEQUALITIES
Because of the intractability of closure
on
interval-based networks, we
will
proceed from a
point-based representation.
In
particular, we
will
represent a time interval by its endpoints and, for
the time being, we
will
not concern ourselves with
the problem of two intervals overlapping
by
a single
point
or
the problem of representing
all
ambiguities.
Matrix Representation
OF
Point Relations
We are
now
ready to address the question, “How
do temporal events relate to one another?” First,
consider two arbitrary points,
po
and
p,.
There
are
only three waysp, is able to relate top,
@,
before
p,p,
afterp, andp, at the same time asp,). Thus,
given
pa
re1
pb
where
re1
E
{
<,
>,
=},
we
can
define the following.
Pa
<
Pb
Pa
’
Pb
jE
Pa
beforepb
SE
pa
afterpb
pa
=
Pb
pa
at the same time
asp,.
Next, consider two intervals,
Za
=
<Ida>
=
I
la
We
can
specify all possible relations between the
endpoints
as
la
rel,
l,
1,
rel, r,,
ra
rel,
1,
and
ro
re14
r,
where re&
E
{
c,
>,
=}
and
i
=
1
...
4.
Without
considering the limitation imposed that
I,
5
r, (from
our definition of an interval), with four relations and
three choices for each relational operator, the total
possible combinations of relations (which we
will
call
relation signatures)
is
3‘
or
81.
If
we impose
the constraint that
I,
S
r,,
we find that the number
of possible relation signatures reduces to
18.
4
p
<
r,}
and
=
=
@
I
1,
<
p
5
rb}.
BINARY RELATIONS
ON
TIME INTERVALS
Proceeding from the
18
relation signatures,
which
we will call the set of relevant relation signatures,
we
will
divide the set into
two
major subsets: the set
of interval relations and the set of point-interval
relatiom. Had we proceeded from the assumption
that the interval
was
primitive we would have found
only
U
relations. These
U
relations
are
shown
with graphical representations and relation
signatures in Figure
1.
Because we combined point
152
Figure
l.
Temporal Relations
and interval constructs we were able to identify
5
additional relations that are special cases of
5
of the
interval relations. These special cases arise when
one or both
of
the intervals are time points. These
signatures are
also
shown in figure
1.
We chose
the relation labels shown in Figure
1
because they
uniquely characterize the corresponding relations.
CONSTRAINT PROPAGATION ON TEMPORAL
INTERVALS
Previous sections described a relational algebra
on
the temporal intervals. Given
two
intervals we have
described
18
ways
to
specify
how these intervals
may
relate to
one other
in
terms
of
their end
points.
The problem discussed in this section
k,
given
a
set
of intervals and relationships between a subset
of
these intervals, how are any
two
intervals related?
This
problem
can
further be divided into
two
subproblems. The first may be stated
as
follows:
Given two intervals,
Ii
and
4
in set
I
such that their
endpoints are
known,
how are
Ii
and
4
related?
The second problem
is
similar.
Given three
intervals,
I,,
&
and
I,
in set
I
such that the endpoints
are unknown,
I,
re1
4
is
known, and
4
re1
I,
is
known, how are
I,
and
I,
related? Due to space
constraints, an algorithm for solving only the latter
problem
will
be discussed.
Let
us
represent the relation signatures
as
2x2
matrices. Fist, we define the addition and
multiplication operators
on
our relational algebra in
Figure
2.2
With these operators available, we
can
propagate the intervals' relational constraints by
multiplying the
two
intervals' relation matrices
together (using standard matrix addition and
subtraction). The matrix multiplication algorithm
may be used to define a transitivity table for
all
pairwise combinations
of
interval and point-interval
relations. We provide this table (Table
1)
using our
relation symbols and including the point-interval
relations. (Note: Allen gives this table for the
12
interval relations and excludes Equals.) We
can
then use
a
constraint propagation algorithm, such
as
the one given in reference
1,
to determine higher
order relations between the intervals.
Unfortunately,
as
mentioned earlier, this algorithm
has exponential complexity which
is
unacceptable
for large problems.
In
the next section, we will
describe a simpler algorithm that, though
incomplete, provides a polynomial time solution that
provides excellent coverage.
O=
0
?=
(<>=)
Figure
2.
Relational Algebra Operators
153
REPRESENTING
TEMPORAL
RELATIONS
WITH PROPOSITIONAL
CALCULUS
Now that we have
an
algebra for representing and
operating
on
temporal relations, the next step
is
to
map this algebra into a knowledge based inference
system. Given knowledge about how interval end
points relate to one another, we would like to
propagate this information throughout the resulting
knowledge network.
Propositional Calculus Representation
of
Inequality
Our
relational algebra is a three value logic that
is
cumbersome to work with. Since digital computers
are binary machines it is desirable for the sake of
Table
1.
Transitivity Table for All
18
Relations'
efficiency
to
map
this
three valued logic into
a
two
value
logic.
As
we
will
soon
see this
will
allow
for
a rapid closure algorithm with respect to the end
points of the temporal network. Unfortunately,
mapping from a three space to a two space
will
result in some loss of information. Fortunately,
there
is
a way to recover much
of
this lost
information.
As
we begin the process of mapping
our
three-valued logic into the two-valued logic, we
consider
two
points in time,
a
and
b,
such that
a
occurs before
b.
We represent this,
using
our
temporal algebra,
as
(a
<
b).
We also know that
the inverse of
(a
<
b)
is
(b
>
a);
however,
this
still
5Mmp
hghl~gbts
ambigutlm
mdiated
by
(he
mmrponding
Inter. A
questton
mark
(?)
mdiatn
m
mfomutmo
may
bc
inferred
from
the
corresponding
relation
Innsitmy.
leaves
us
with the three-valued
logic. Assume now that we use
the binary relation
"e"
(i.e., less
than) to represent the temporal
relation between
a
and
b.
Then,
if
we wish to consider the inverse
of
(a
e
b),
we say
b
is not before
a.
Therefore,
c
3ay be after
b,
or
a
may occur at the same time
as
b.
Under this specification, the
inverse of
(a
e
b)
is
(b
2
a).
Using propositional calculus, the
above discussion translates
into
the following. We are interested
in propagating a truth value
through a logic network. This
propagation occurs by chaining
rules of inferencL together. These
rules of inference correspond to
an implication
in
propositional
calculus. Thus,
if
we want to say
"If
A
is true, then
B
is
true," we
write
(A
+
B).
At the same the,
we know from propositional
calculus that (A
4
B)
is equivalent
to
(1B
-.
TA).
In other words,
if
B
is
false, then
A
is
also false.
Returning to our temporal system,
we know
if
(a
c
b),
then for
b
to
have occurred,
U
must
also
have
occurred. Thus, the truth
of b
implies the truth
of
u.
Conversely,
if
U
has
not
occurred,
b
cannot
have occurred. Therefore, the
falsity of
a
implies the falsity of
b.
154
This
is
equivalent to saying
(b
+
U).
Note
also
that
if
(b
=
U),
the above implication still holds.
In
fact,
equality would translate into equivalence, i.e.,
(U
i
b).
Therefore, we may use this mapping
to
determine propositional rules for
our
18
temporal
relations.
Mapping Temporal Relations
As
we begin to determine the propositional rules
corresponding to each temporal relation, we recall
the relation signatures defining the relations
between each of the end points. First we will state
a couple
of
obvious rules based
on
our definition
of
a temporal interval. Then we
will
delineate the
propositions that arise from the relation signatures.
Red from our definition that we declared the left
endpoint of an interval either to be equal to the
right endpoint (a point interval) or to be before the
right endpoint (a non-point interval). For point
intervals, we have
(1,
--
r,)
which translates in
propositional calculus to
(I,
ss
ri)
or, using
implication,
(Z,
-
r,)
A
(r,
-
1,).
For non-point
intervals,
(II
c
rl);
therefore,
(r,
-.
I[).
Another
obrious observation
is
that any given point equals
itself; therefore,
(I,
+
Ii)
and
(ri
-.
ri).
The reason
for explicitly stating this tautology will become
apparent below. The rules corresponding to
mapping each
of
the relation signatures are given in
Table
2.
Table
2.
Propositional Rules for Temporal Relations
Table
3.
Bit Signatures for Temporal Relations
Bit Matrix Signatures
Having specified the propositional rules
corresponding to each of
18
relation matrices, we
will now represent these logical relations
in
the
machine. These logical relations may be
represented as a two dimensional binary matrix.
For any given cell in the binary matrix, a zero
indicates
it
is not known
if
the point specified by the
roh implies the point specified by the column. A
one,
on
the other hand, indicates
an
implication
is
known.
For example, a one in row
2,
column
3,
indicates
(r,,
-
Zb).
In other words,
(I,
e
r,,).
If
a
one
is
in
row
2,
column
3
and row
3,
column
2,
then
we
can
say
(Z,
=
ra).
This
is
because
(Zb
+r,,)
A
(r,,
-
I,),
i.e.,
(Z,
=
rJ.
All the bit matrix signatures for
the 18 temporal relations are shown in Table
3.
CONSTRAINT
PROPAGATION OF
BINARY
SIGNATURES
In
order to represent a temporal network we
will
now specify a data structure for representing
t-mnn-al
intervals
in
our system.
This
data
structure consists of a
2n
x
2n
binary
matrix where
n
is
the number
of
intervals in the system. The
relations between the intervals are then entered into
the matrix according to the relation signatures given
155