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Abstract:
A specification formalism for reactive systems defines a class of
$\omega$-languages.
We call a specification formalism {\em fully decidable\/} if it is
constructively closed under boolean operations and has a decidable
satisfiability (nonemptiness) problem.
There are two important, robust classes of $\omega$-languages that are
definable by fully decidable formalisms.
The {\sc $\omega$-regular languages\/} are definable by finite automata,
or equivalently, by the Sequential Calculus.
The {\sc counter-free $\omega$-regular languages\/} are definable by
temporal logic, or equivalently, by the first-order fragment of the
Sequential Calculus.
The gap between both classes can be closed by finite counting
(using automata connectives), or equivalently, by projection
(existential second-order quantification over letters).
A specification formalism for real-time systems defines a class of
{\em timed\/} $\omega$-languages, whose letters have real-numbered time
stamps.
Two popular ways of specifying timing constraints rely on the use of
clocks,
and on the use of time bounds for temporal operators.
However, temporal logics with clocks or time bounds have undecidable
satisfiability problems, and finite automata with clocks
(so-called {\em timed automata\/}) are not closed under complement.
Therefore, two fully decidable restrictions of these formalisms have
been
proposed.
In the first case, clocks are restricted to {\em event clocks}, which
measure
distances to immediately preceding or succeeding events only.
In the second case, time bounds are restricted to {\em nonsingular
intervals},
which cannot specify the exact punctuality of events.
We show that the resulting classes of timed $\omega$-languages are
robust,
and we explain their relationship.
First, we show that temporal logic with event clocks defines the same
class
of timed $\omega$-languages as temporal logic with nonsingular time
bounds,
and we identify a first-order monadic theory that also defines this
class.
Second, we show that if the ability of finite counting is added to these
formalisms, we obtain the class of timed $\omega$-languages that are
definable by finite automata with event clocks, or equivalently, by a
restricted second-order extension of the monadic theory.
Third, we show that if projection is added, we obtain the class of timed
$\omega$-languages that are definable by timed automata, or
equivalently,
by the full second-order extension of the monadic theory.
These results identify three robust classes of timed $\omega$-languages,
of
which the third, while popular, is not definable by a fully decidable
formalism.
By contrast, the first two classes are definable by fully decidable
formalisms from temporal logic, from automata theory, and from monadic
logic.
Since the gap between these two classes can be closed by finite
counting, we
dub them the {\em timed $\omega$-regular languages\/} and the
{\em timed counter-free $\omega$-regular languages}, respectively.
