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regular expression notation proposed by Asarin, Caspi, and Maler in [2], such
an automata-theoretic representation of the counterexamples can be achieved
through the mapping
)
de
=
FiniteModels
(
!
:
Counterexamples
(
)
of
fdPe;`
=
kg
-formulae to timed regular expressions, where
is the set of
min-terms over the free state variables of
and
W
a2;aj
=
P
a
)
de
=
dPe
)
;
FiniteModels
(
(0
;1
)
de
=(
)
[
k;k
]
;
FiniteModels
(
`
=
k
)
de
=
FiniteModels
(
FiniteModels
(
^
)
^
FiniteModels
(
)
;
)
de
=
FiniteModels
(
FiniteModels
(
;
)
FiniteModels
(
)
:
As the timed regular languages of variability
are furthermore eectively
closed under complementation (relative to the set of (
n
+
m
)-bounded trajecto-
ries) [45], this procedure can be extended to deal with inner negation also if only
the (
n
+
m
n
+
m
)-bounded counterexamples are of interest. Therefore, any formula
of the
fdPe;`
=
kg
fragment of DC can be eectively assigned a timed automa-
ton
)-bounded counterexamples. While the
deterministically recognizable timed regular languages are in general a proper
subclass of the non-deterministically recognizable ones, these two classes coincide
for the timed regular languages of variability
A
Traj
n
+
m
n
recognizing its (
n
+
m
n
+
m
. In particular,
A
Traj
n
+
m
n
can be made deterministic.
Now,
A
Traj
n
+
m
n
can be easily extended to an (again deterministic) timed
automaton
A
Traj
n
+
m
n
(
[Traj
O;m
)
\Traj
I;n
that, besides recognizing any (
n
+
m
)-
bounded counterexample of
, also recognizes all trajectories violating the
m
-
boundedness constraints on outputs, yet excludes trajectories violating the
-
boundedness constraint on inputs. This eectively reduces the controller synthe-
sis problem to a strategy construction problem in a timed regular game, where
A
Traj
n
+
m
n
(
[Traj
O;m
)
\Traj
I;n
is the game graph. As eective synthesis procedures
for timed regular games are known from the literature (cf. [3,26]), this solves the
controller synthesis problem.
n
t
f
R
P
kg
control prob-
lems, as their requirements formulae feature accumulated durations and thus
are considerably more expressive than the
Unfortunately, Theorem 14 does not generalize to
=
control problems: As
was shown in [14], page 35., by means of a real-time pumping lemma, it is in
general undecidable whether
Traj
n
M
fdPe;`
=
kg
R
. Thus, by
an argument akin to that used in the proof of Theorem 12, it follows that there
is no eective, sound, and
[[
]] f o r
f
P
=
k g
formulae
f
R
P
=
kg
-complete synthesis procedure even when
interface dynamics is restricted to
n
-bounded inputs and
m
-bounded outputs.
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