The Existence of ω-Chains for Transitive Mixed Linear Relations and Its Applications

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We 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.