We consider pushdown timed automata (PTAs) that are timed automata (with dense clocks) augmented with a pushdown stack. A configuration of a PTA includes a state, dense clock values and a stack word. By using the pattern technique, we give a decidable characterization of the binary reachability (i.e., the set of all pairs of configurations such that one can reach the other) of a PTA. Since a timed automaton can be treated as a PTA without the pushdown stack, we can show that the binary reachability of a timed automaton is definable in the additive theory of reals and integers. The results can be used to verify a class of properties containing linear relations over both dense variables and unbounded discrete variables. The properties previously could not be verified using the classic region technique nor expressed by timed temporal logics for timed automata and CTL for pushdown systems. The results are also extended to other generalizations of timed automata.

Abstract. We investigate the Presburger liveness problems for nondeterministicreversal-bounded multicounter machines with a free counter (NCMFs). We show the following:-The 9-Presburger-i.o. problem and the 9-Presburger-eventual problem areboth decidable. So are their duals, the 8-Presburger-almost-always problemand the 8-Presburger-always problem.- The 8-Presburger-i.o. problem and the 8-Presburger-eventual problem areboth undecidable. So are their duals, the 9-Presburger-almost-always prob-lem and the 9-Presburger-always problem. These results can be used to formulate a weak form of Presburger linear tem-poral logic and develop its model-checking theories for NCMFs. They can also be combined with [12] to study the same set of liveness problems on an extendedform of discrete timed automata containing, besides clocks, a number of reversalbounded counters and a free counter. 1 Introduction An infinite-state system can be obtained by augmenting a finite automaton with oneor more unbounded storage devices. The devices can be, for instance, counters (unary stacks), pushdown stacks, queues, and/or Turing tapes. However, an infinite-state sys-tem can easily achieve Turing-completeness, e.g., when two counters are attached to a finite automaton (resulting in a "Minsky machine"). For these systems, even simpleproblems such as membership are undecidable.