Coarse Abstractions Make Zeno Behaviours Difficult to Detect

Abstract

Abstract. An infinite run of a timed automaton is Zeno if it spans only a finite amount of time. Such runs are considered unfeasible and hence it is important to detect them, or dually, find runs that are non-Zeno. Over the years important improvements have been obtained in checking reachability properties for timed automata. We show that some of these very efficient optimizations make testing for Zeno runs costly. In particular we show NP-completeness for the LU-extrapolation of Behrmann et al. We analyze the source of this complexity in detail and give general conditions on extrapolation operators that guarantee a (low) polynomial complexity of Zenoness checking. We propose a slight weakening of the LU-extrapolation that satisfies these conditions.

Why Liveness for Timed Automata Is Hard, and What We Can Do About It

“…It is possible to convert every TBA into a strongly non-Zeno TBA. This strongly non-Zeno construction could lead to an exponential blowup [6,9] to the abstract zone graph (which is defined below), but we prefer to employ this commonly used assumption in order not to divert from the main subject.…”

“…Therefore, at depth N + 2, the BFS generates exactly one node for each state in A B,w containing the associated maximal zone. This takes polynomial time since up to depth N + 1 the automaton consists just of a linear number of sequences as in ( 5), (6), and (7). Then, due to subsumption, BFS stops at depth N + 3 after one transition from (1), (2) or (3).…”

Efficient emptiness check for timed Büchi automata

“…This emphasizes that time must elapse from node (a, x = 0, {x}) in order to take a transition with guard x ≥ 1. An optimized guessing zone graph construction is given in [19]. Figure 8: A TBA A 1 and the guessing zone graph GZG a (A 1 ) (with τ self-loops omitted for clarity).…”

“…One way to treat Zeno runs would be to say that a timed automaton admitting such a run is faulty and should be disregarded. This gives rise to the problem of detecting the existence of Zeno runs in an automaton [9,16,19]. The other approach to handling Zeno behaviours, that we adopt here, is to say that due to imprecisions introduced by the modeling process one may need to work with automata having Zeno runs.…”

Coarse abstractions make Zeno behaviours difficult to detect

“…A shorter version of this paper appeared at the 22 nd International Conference on Concurrency Theory in the year 2011 [12]. The current version includes the missing proofs, a new discussion (Section 5) about two observations arising out of the complexity analysis, and the new result about the Pspace-completeness of the Zeno-related problems when the only input is the automaton (Section 6).…”