Timed Pushdown Automata Revisited

Abstract

This paper contains two results on timed extensions of pushdown automata (PDA). As our first result we prove that the model of dense-timed PDA of Abdulla et al. collapses: it is expressively equivalent to dense-timed PDA with timeless stack. Motivated by this result, we advocate the framework of firstorder definable PDA, a specialization of PDA in sets with atoms, as the right setting to define and investigate timed extensions of PDA. The general model obtained in this way is Turing complete. As our second result we prove NEXPTIME upper complexity bound for the non-emptiness problem for an expressive subclass. As a byproduct, we obtain a tight EXPTIME complexity bound for a more restrictive subclass of PDA with timeless stack, thus subsuming the complexity bound known for dense-timed PDA.

Timed pushdown automata and branching vector addition systems

“…Contributions: Our headline result answers positively the question raised by Clemente and Lasota [9]: we prove that non-emptiness remains decidable when the assumption of orbit-finiteness of trPDA is dropped. The resulting class of automata strictly subsumes all pushdown extensions of timed automata mentioned above (with the exception of RTA [4], [19] 1 ), and is the first one to allow timed stacks without bounding the differences of state and top-of-stack clocks.…”

“…Motivated by verification of programs with both procedural and timed features, several extensions of timed automata by a pushdown stack have been proposed, including pushdown timed automata (PDTA) [6], recursive timed automata (RTA) [4], [19], densetimed pushdown automata (dtPDA) [1], and timed register pushdown automata (trPDA) [9].…”

“…Surprisingly, Clemente and Lasota showed that, as a consequence of the interplay of the stack discipline and the monotone elapsing of time, dtPDA is in fact not more expressive than PDTA, and the two models are strictly subsumed by orbit-finite trPDA [9]. Moreover, subsumption still holds if trPDA are restricted to timeless stack, and in this case there is nothing to pay in terms of the complexity of non-emptiness, which is the central decision problem for model checking: it is EXPTIMEcomplete for both PDTA, dtPDA, and trPDA with timeless stack; for orbit-finite trPDA, the best known upper bound rises to NEXPTIME (ibid.).…”

“…The main question posed in the latter work is whether the heavy restriction of orbit finiteness, which bounds the differences between state and top-of-stack clocks, can be lifted while keeping non-emptiness decidable. The proofs of the NEXPTIME and EXPTIME upper bounds for orbit-finite and timeless-stack trPDA (respectively) [9] involved translations to systems of equations in which variables range over sets of integers, and available operations include addition, union, and intersection with the singleton set t0u. Similar systems have been studied in a variety of contexts, and extensions quickly lead to undecidability: e.g., already over the naturals, when arbitrary intersections are permitted, decidability is lost since this model subsumes unary conjunctive grammars [14].…”

“…The first one is a sequence of transformations that translates, in exponential time, trPDA to systems of equations over sets of integers with addition, union, and intersections with N and´N, and is considerably more involved than the corresponding translation for orbit-finite trPDA [9].…”