  The paper is focused on temporal logics for the description of the behaviour
of real-time pushdown reactive systems. The paper is motivated to bridge
tractable logics specialized for expressing separately dense-time real-time
properties and context-free properties by ensuring decidability and
tractability in the combined setting. To this end we introduce two real-time
linear temporal logics for specifying quantitative timing context-free
requirements in a pointwise semantics setting: Event-Clock Nested Temporal
Logic (EC_NTL) and Nested Metric Temporal Logic (NMTL). The logic EC_NTL is an
extension of both the logic CaRet (a context-free extension of standard LTL)
and Event-Clock Temporal Logic (a tractable real-time logical framework related
to the class of Event-Clock automata). We prove that satisfiability of EC_NTL
and visibly model-checking of Visibly Pushdown Timed Automata (VPTA) against
EC_NTL are decidable and EXPTIME-complete. The other proposed logic NMTL is a
context-free extension of standard Metric Temporal Logic (MTL). It is well
known that satisfiability of future MTL is undecidable when interpreted over
infinite timed words but decidable over finite timed words. On the other hand,
we show that by augmenting future MTL with future context-free temporal
operators, the satisfiability problem turns out to be undecidable also for
finite timed words. On the positive side, we devise a meaningful and decidable
fragment of the logic NMTL which is expressively equivalent to EC_NTL and for
which satisfiability and visibly model-checking of VPTA are EXPTIME-complete.

Logic in Computer Science