2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/30885We show that it is decidable whether a transitive mixed linear relation has an $ω$-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.26 pagesLogic in Computer ScienceD.2.4; F.1.1text