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Parallel reachability analysis of hybrid systems in XSpeed

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Abstract

Reachability analysis techniques are at the core of the current state-of-the-art technology for verifying safety properties of cyber-physical systems (CPS). The current limitation of such techniques is their inability to scale their analysis by exploiting the powerful parallel multi-core architectures now available in modern CPUs. Here, we address this limitation by presenting for the first time a suite of parallel state-space exploration algorithms that, leveraging multi-core CPUs, enable to scale the reachability analysis for linear continuous and hybrid automaton models of CPS. To demonstrate the achieved performance speedup on multi-core processors, we provide an empirical evaluation of the proposed parallel algorithms on several benchmarks comparing their key performance indicators. This enables also to identify which is the ideal algorithm and the parameters to choose that would maximize the performances for a given benchmark.

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Acknowledgements

The authors would like to thank National Institute of Technology Meghalaya, for providing the computational facilities and infrastructure for carrying out this work. This work was supported in part by the National Institute of Technology Meghalaya, India and DST-SERB, GoI under project grant No. YSS/2014/000623. This work was also partially supported by the Doctoral Program Logical Methods in Computer Science (W1255-N23) and the Austrian National Research Network RiSE/SHiNE (S11405-N23 and S11412-N23) project, both funded by the Austrian Science Fund (FWF) and by the Air Force Office of Scientific Research under award no. FA2386-17-1-4065, and by the ARC project DP140104219 (Robust AI Planning for Hybrid Systems).

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Authors and Affiliations

  1. National Institute of Technology Meghalaya, Shillong, India

    Amit Gurung & Rajarshi Ray

  2. Vienna University of Technology, Vienna, Austria

    Ezio Bartocci & Radu Grosu

  3. Australian National University, Canberra, Australia

    Sergiy Bogomolov

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  1. Amit Gurung
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  2. Rajarshi Ray
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  3. Ezio Bartocci
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  4. Sergiy Bogomolov
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Correspondence to Amit Gurung.

Appendix

Appendix

Claim 1

A-GJH algorithm performs a BFS of a HA with the number of BFS levels \(=\) bound.

Proof

We show the correctness of the algorithm by the following loop invariant of the repeat–until loop of thealgorithm:

At the beginning of the \(j^{th}\) iteration of the repeat–until loop, the data structure R contains all the states of the HA reachable from Init with \(j-1\) levels of BFS.

We use a level of a BFS to signify the frontiers of states reachable from Init. For example, PostC(Init) denotes all states reachable up to a BFS level/frontier of 1 from Init, and PostC(PostD(PostC(Init))) denotes all states reachable up to a BFS level/frontier of 2 from Init and so on.

At initialization, R is assigned to Init. Therefore, the loop invariant is true at initialization, which says that at the beginning of the first iteration of the repeat–until loop, R contains all the states of the HA reachable from Init with no BFS.

We now show that the loop invariant is maintained. In lines 8 to 15, PostC operator is applied to every symbolic state in Wlist[t] and the result is included in R. The states reachable by PostD transitions are added to \(Wlist[1-t]\) for exploration in the next iteration. Therefore, at each iteration, the BFS frontier is increased by 1, maintaining the loop invariant. Parallel exploration causes no race condition and write contention on the shared data structure Wlist and R. The justifications are the same as in the G.J. Holzmann’s algorithm in the SPIN model checker [37].

The termination of the algorithm is evident from the termination condition of the repeat–until. The loop terminates either when there are no new symbolic states for further exploration in Wlist[t] or when the predetermined bound on the BFS levels is reached. It is clear that one of the condition must be eventually true, and hence, the algorithm must terminate.

At termination, the loop condition must be false, which means either \(level=bound\) or \(Wlist[t] = \emptyset \). In the former condition, the loop invariant at termination says that R contains all the states of the HA reachable from Init with bound levels of BFS, which is our claim. Termination due to the later condition implies that the fixed point has been found before BFS levels could reach the bound. In both cases, our claim holds. \(\square \)

Claim 2

TP-BFS algorithm performs a BFS of a HA with the number of BFS levels \(=\) bound.

Proof

The correctness of the algorithm can be proved using the same loop invariant used in the proof of claim 1. The arguments for the validity of the loop invariant are same except for the invariant maintenance. In lines 7 to 28, the PostC and PostD operations increase the BFS frontier/level by 1, maintaining the loop invariant. The PostC and PostD operations are split into atomic tasks and inserted into a tasks list data structure. Since the algorithm provides a partitioned access of the tasks in the tasks list data structure to the threads executing in parallel in the cores, the threads access exclusive portions in memory, with no read–write contention. Such an exclusive access to the tasks list data structure by the threads makes locking needless. The read–write switching of the data structure Wlist is the same as in the A-GJH algorithm, which makes the accesses to the symbolic states in the waiting list lock-free, during the BFS [37].

The termination proof of the algorithm is the same as the termination proof of A-GJH algorithm in claim 1. \(\square \)

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Gurung, A., Ray, R., Bartocci, E. et al. Parallel reachability analysis of hybrid systems in XSpeed. Int J Softw Tools Technol Transfer 21, 401–423 (2019). https://doi.org/10.1007/s10009-018-0485-6

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