Improving search order for reachability testing in timed automata

Standard algorithms for reachability analysis of timed automata are sensitive to the order in which the transitions of the automata are taken. To tackle this problem, we propose a ranking system and a waiting strategy. This paper discusses the reason why the search order matters and shows how a ranking system and a waiting strategy can be integrated into the standard reachability algorithm to alleviate and prevent the problem respectively. Experiments show that the combination of the two approaches gives optimal search order on standard benchmarks except for one example. This suggests that it should be used instead of the standard BFS algorithm for reachability analysis of timed automata.

💡 Research Summary

The paper investigates a subtle but significant source of inefficiency in reachability analysis of timed automata: the order in which transitions are explored. Standard symbolic algorithms work on the abstract zone graph, where each node is a pair (state, zone) and zone inclusion is used to prune the search. If a small zone is discovered before a larger one that subsumes it, the algorithm must still explore the entire subtree of the small node before it can discard it, leading to redundant work. The authors call such situations “mistakes” and demonstrate, using a simple example (Figure 1) and a constructed family of automata (Figure 2), that the number of visited nodes can grow exponentially when the worst exploration order is taken, even on well‑known benchmarks such as the FDDI model.

To mitigate this problem, two heuristics are proposed:

1. Ranking System – Each state and its outgoing transitions are assigned a numeric rank that estimates the size of the zone that will be produced. The rank may combine clock bounds, the number of resets, and the length of the path to the node. The waiting list of the reachability algorithm is turned into a priority queue ordered by rank, so that transitions likely to generate larger zones are explored first. By bringing large zones to the front, the algorithm can prune many smaller zones early, reducing both the number of visited nodes and the size of the visited set.

2. Waiting Strategy – Independently of ranking, the algorithm maintains a “waiting” pool of nodes that have been generated but not yet expanded. When a new node (q, Z′) is discovered that subsumes an existing waiting node (q, Z), the smaller node and all of its descendants still in the waiting pool are removed immediately. This requires a fast inclusion test and a data structure (e.g., a tree‑based index) that can locate all subsumed nodes efficiently. The strategy is simple to implement and works as a complement to the ranking system.

The authors integrate both heuristics into the classic BFS‑style reachability algorithm (Algorithm 1.1) used in tools such as UPPAAL. They evaluate four configurations on a suite of standard benchmarks: (i) plain BFS, (ii) BFS with ranking, (iii) BFS with the waiting strategy, and (iv) BFS with both heuristics combined. The experimental results show that the combined approach yields the smallest number of explored nodes and the lowest memory consumption on almost all benchmarks, notably eliminating the exponential blow‑up observed on the FDDI model. In a single benchmark the combined method incurs a modest overhead, which the authors attribute to the cost of rank computation and extra bookkeeping.

The paper situates its contributions within related work on state caching, state‑space fragmentation, and the empirical observation that BFS often outperforms DFS for timed‑automata reachability. Unlike prior work, which mainly focuses on the choice between BFS and DFS, this study directly addresses the interaction between zone inclusion and exploration order, providing concrete mechanisms to favor the discovery of larger zones early.

In conclusion, the authors argue that the “rank + waiting” combination should replace the default BFS in timed‑automata reachability tools. They acknowledge that the design of the ranking function may need to be tuned for specific domains and that the waiting strategy could, in pathological cases, delay the exploration of useful large zones. Future research directions include automatic learning of ranking parameters, dynamic adaptation of the waiting policy, and extending the approach to distributed model‑checking settings.

Overall, the paper makes a clear case that careful ordering of transition exploration, guided by lightweight heuristics, can dramatically improve the practical performance of symbolic reachability analysis for timed automata.