Abstract: 
We develop a theory of equivalences for timed systems. Two systems are equivalent iff external observers cannot observe differences in their behavior. The notion of equivalence depends, therefore, on the distinguishing power of the observers. The power of an observer to measure time results in untimed, clock, and timed equivalences: an untimed observer cannot measure the time difference between events; a clock observer uses a clock to measure time differences with finite precision; a timed observer is able to measure time differences with arbitrary precision.
We show that the distinguishing power of clock observers grows with the number of observers, and approaches, in the limit, the distinguishing power of a timed observer. More precisely, given any equivalence for untimed systems, two timed systems are kclock congruent, for a nonnegative integer k, iff their compositions with every environment that uses k clocks are untimed equivalent. Both kclock bisimulation congruence and kclock trace congruence form strict decidable hierarchies that converge towards the corresponding timed equivalences. Moreover, kclock bisimulation congruence and kclock trace congruence provide an adequate and abstract semantics for branchingtime and lineartime logics with k clocks.
Our results impact on the verification of timed systems in two ways. First, our decision procedure for kclock bisimulation congruence leads to a new, symbolic, decision procedure for timed bisimilarity. Second, timed trace equivalence is known to be undecidable. If the number of environment clocks is bounded, however, then our decision procedure for kclock trace congruence allows the verification of timed systems in a trace model.
