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Table 7.
Timedprotectionsystem
Π
begin create object
ω
;enter
r
0
into
A
(
σ, ω
)
;enter
r
1
into
A
(
σ, ω
)
;enter
r
2
into
A
(
σ, ω
)
end
if
r
0
is in
A
(
σ, ω
)
since at least duration
2
then enter
r
3
into
A
(
σ
,ω
)
if
r
0
is in
A
(
σ, ω
)
since at least duration
2
then enter
r
4
into
A
(
σ
,ω
)
if
r
3
is in
A
(
σ, ω
)
since at least duration
3
and
r
4
is
in
A
(
σ
,ω
)
since at least duration
3
then enter
r
5
into
A
(
σ
,ω
)
The concept of timed protection system and the adjectives conditional, monotonic,
monoconditional, and mono-operational are defined as in section 3. For all
i
∈
{
C
timed
(
i, j
)betheclassofalltimed
protection systems such that none of their timed commands contain more than
i
elementary condition or more than
j
primitive operations and
0
,
1
,
2
,
∞}
and for all
j
∈{
1
,
2
,
∞}
,let
+
C
timed
(
i, j
)be
C
timed
(
i, j
). For example,
the timed protection system
Π
shownintable7isintheclass
the class of all monotonic timed protection systems in
+
C
timed
(2
,
∞
). A
timed history is a mapping that assigns to every non-negative real number a pro-
tection state. Timed histories will be denoted by the letters
h
,
h
, etc, possibly
with subscripts. We are interested in non-Zeno timed histories, i.e. timed histo-
ries changing at most a finite number of times in any finite interval. Hence, we
assume that for all timed histories
h
, there exists a strictly increasing sequence
v
0
,
v
1
,
...
of real numbers such that:
-
v
0
=0,
-
lim
n→∞
v
n
=
,
-
for all non-negative integers
n
, there exists a protection state
∆
n
such that
h
(
v
)=
∆
n
for all
v
∞
∈
[
v
n
,v
n
+1
[.
We shall say that the sequence (
v
0
,∆
0
), (
v
1
,∆
1
),
...
is a timed sequence for
h
.
In such sequences, the three components of protection state
∆
n
will be denoted
S
n
,
O
n
,and
A
n
for each non-negative integer
n
. To gain some intuition, the
reader may easily see that the sequence (0
,∆
), (1
,∆
), (2
,∆
), (3
,∆
), (4
,∆
),
(5
,∆
), (6
,∆
), (7
,∆
(4)
), (8
,∆
(4)
),
...
is a timed sequence for the timed history
h
shown in table 8, where
∆
,
∆
,
∆
,
∆
,and
∆
(4)
are the protection states
defined by tables 1, 2, 3, 4, and 5. Let
θ
be a substitution and
C
be an elementary
condition. Suppose there is no variable in
θ
(
C
), i.e. every variable in
C
is replaced
by an individual through the use of
θ
.If
h
is a timed history with timed sequence
(
v
0
,∆
0
), (
v
1
,∆
1
),
...
and
v
is a non-negative real number then we shall say that
θ
makes
C
true in
h
at
v
,insymbols
h, v
|
=
θ
C
, iff the following condition is
satisfied:
-
C
is “
r
is in
A
(
σ, ω
)
since at least duration
d
”,
d
≤
v
, and for all non-negative
integers
n
,if
v
n
<v
and
v
−
d<v
n
+1
then
θ
(
σ
)isin
S
n
,
θ
(
ω
)isin
O
n
,and
r
is in
A
n
(
θ
(
σ
)
,θ
(
ω
)).
