  We show that it is decidable whether a transitive mixed linear relation has
an $\omega$-chain. Using this result, we study a number of liveness
verification problems for generalized timed automata within a unified
framework. More precisely, we prove that (1) the mixed linear liveness problem
for a timed automaton with dense clocks, reversal-bounded counters, and a free
counter is decidable, and (2) the Presburger liveness problem for a timed
automaton with discrete clocks, reversal-bounded counters, and a pushdown stack
is decidable.
