On the initialization of clocks in timed formalisms

Abstract

Constraint LTL over clocks (CLTLoc) is an extension of LTL allowing for atomic formulae of the form x<c or x=c, which constrain the time delay measured by clock x with respect to constant value c. In a previous work, we showed that CLTLoc is equivalent to Timed Automata. The result was proven by considering a clock semantics that conforms to the original definition by Alur and Dill, i.e., when clocks are reset (i.e., equal to 0) in the origin, both CLTLoc and Timed Automata define the class of Timed ω-Regular languages. In this paper, we show that if we allow the clocks to have any value in the origin, the power of the formalism to express timed languages does not change, as long as non-Zeno languages are considered. If Zeno languages are allowed, then CLTLoc is strictly more powerful than TA. As a consequence of these results, we also show that non-Zeno Timed ω-Regular languages are closed with respect to the left quotient operation with timed regular languages over finite words.