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Parameter synthesis for hierarchical concurrent real-time systems

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Abstract

Modeling and verifying complex real-time systems, involving timing delays, are notoriously difficult problems. Checking the correctness of a system for one particular value for each delay does not give any information for other values. It is thus interesting to reason parametrically, by considering that the delays are parameters (unknown constants) and synthesizing a constraint guaranteeing a correct behavior. We present here Parametric Stateful Timed Communicating Sequential Processes, a language capable of specifying and verifying parametric hierarchical real-time systems with complex data structures. Although we prove that the synthesis is undecidable in general, we present several semi-algorithms for efficient parameter synthesis, which behave well in practice. This work has been implemented in a real-time model checker, PSyHCoS, and validated on a set of case studies.

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Notes

  1. For example, the PTA model of the SPSMALL memory (Chevallier et al. 2009) is made of 10 PTA in parallel, but only 31 symbolic states are reachable according to the semantics of Alur et al. (1993). Due to the static composition of PTA, HyTech crashes by memory overflow even before starting the actual exploration; in contrast, Imitator (André et al. 2012a) finishes the analysis within 0.079 s while using only 3.1 MiB of memory. Details are available in http://www.lsv.ens-cachan.fr/Software/imitator/hytech/.

  2. A semi-algorithm is a procedure that may not terminate but, if it does, then its result is correct.

  3. In the literature related to parametric timed systems, constants are either in the real or the rational domain. Here, to maintain consistency with STCSP (Sun et al.  2013), where constants are defined in \({\mathbb {R}}_{+}\), we choose reals.

  4. An alternative could be \(u\in (U\cup {\mathbb {R}}_{+})\). Our implementation actually allows the definition of either constants or parameters in the timed constructs, but defining \(u\in U\) simplifies the subsequent reasoning and proofs.

  5. For simplicity, in the following, we leave out general and internal choices from the classic CSP (Hoare 1985). The terminology is a little ambiguous in the literature: We assume that a general choice (\(P \ |\ Q\)) can be resolved by an occurrence of any event; an external choice (\(P \ {\square }\ Q\)) can be resolved only by visible events (not \(\tau \)); and an internal choice (\(P \ {\diamond }\ Q\)) is resolved “immediately”, hence cannot be delayed (which generates a \(\tau \)-transition). Although we only consider external choice in the following, all three constructions implemented in PSyHCoS, and used in our case studies.

  6. This example is a parametrization of the example from Sun et al. (2013, p. 3:5).

  7. In parametric timed automata (Jovanovic et al. 2013), the notion of reachable state is based on locations, viz., discrete control states. In PSTCSP, there are no such discrete control states; hence, we could define reachability based on a variable valuation, on a given process, or a combination of both. We choose here the first option to simplify the proof, but our results extend directly to the two other cases (see Remark 3).

  8. For timed systems, the state space is always infinite because of dense time. Here, we mean that the number of (symbolic) states \((V, P, C)\) is infinite too.

  9. The test \( Post _{\mathsf {M}}(S) \subseteq S\) is a classical fixpoint test given in an algorithmic manner. Here, one does not know yet whether \( Post _{\mathsf {M}}(S) \subseteq S\), since \( Post _{\mathsf {M}}(S)\) will be computed at the next iteration. In practice, this is handled using a set of “old” states (computed at previous iterations), and a set of “new” states (computed at the current iteration).

  10. In theory, nothing guarantees that the maximum number of clocks is the same for \( reachAll \), \( reachAll +\), \( IM \) and \( IM +\). Nevertheless, since it is always the same for all experiments, we factor it to save some space in the columns.

  11. Recall that \( IM \) does not need to maintain transitions. Hence, the transition number for \( IM \) and \( IM +\) is only an integer maintained within the program for statistics purpose.

  12. http://lipn.univ-paris13.fr/~andre/software/PSyHCoS/.

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Acknowledgments

We are grateful to the anonymous reviewers for their very constructive comments, and to Zhu Huiquan for solving several implementation issues in our model checking tool PSyHCoS. This manuscript also benefited from discussions with Didier Lime. We thank Emmanuelle Encrenaz and Laurent Fribourg for discussions regarding the comparison between Imitator and HyTech. Yang Liu is supported by ”Formal Verification on Cloud” project under Grant No: M4081155.020 and ”Verification of Security Protocol Implementations” project under Grant No: M4080996.020. Jun Sun is supported by research grant “IDD11100102 / IDG31100105” from Singapore University of Technology and Design. Jin-Song Dong is supported by MOE T2 Project “Advanced Model Checking Systems”. All four authors are supported by STIC Asie project “CATS (Compositional Analysis of Timed Systems)”. Étienne André is partially supported by the ANR national research program PACS (ANR-2014).

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Authors and Affiliations

  1. Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS UMR 7030, Villetaneuse, France

    Étienne André

  2. School of Computer Engineering, Nanyang Technological University, Singapore, Singapore

    Yang Liu

  3. Singapore University of Technology and Design, Singapore, Singapore

    Jun Sun

  4. School of Computing, National University of Singapore, Singapore, Singapore

    Jin-Song Dong

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  1. Étienne André
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  2. Yang Liu
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Appendix: A firing rules for PSTCSP

Appendix: A firing rules for PSTCSP

Given a program \( program \) and a valuation \(V\), the valuation obtained by executing \( program \) with \(V\) is denoted as \( program (V)\). Let \( active (V,P)\) be the set of enabled events given \(P\) and \(V\), i.e., the set of events that can be fired at the current state (and which lead to states with satisfiable constraints). We give below all firing rules for PSTCSP.

$$\begin{aligned} \dfrac{}{(V, \mathtt {Skip}, C) \mathop {\rightsquigarrow }\limits ^{\checkmark }(V, \mathtt {Stop}, {C}^{\uparrow })}(aki) \end{aligned}$$
$$\begin{aligned} \dfrac{}{(V, e \rightarrow P, C) \mathop {\rightsquigarrow }\limits ^{e} (V, P, {C}^{\uparrow })}(aev) \end{aligned}$$
$$\begin{aligned} \dfrac{}{(V, a\{ program \} \rightarrow P, C) \mathop {\rightsquigarrow }\limits ^{a} ( program (V), P, {C}^{\uparrow })}(aac) \end{aligned}$$
$$\begin{aligned} \dfrac{V \vDash b}{(V, \mathtt {if}~b~\mathtt {then}~\{ P \}~\mathtt {else}~\{ Q \}, C) \mathop {\rightsquigarrow }\limits ^{\tau } (V, P, {C}^{\uparrow })}(co2) \end{aligned}$$
$$\begin{aligned} \dfrac{V \not \vDash b}{(V, \mathtt {if}~b~\mathtt {then}~\{ P \}~\mathtt {else}~\{ Q \}, C) \mathop {\rightsquigarrow }\limits ^{\tau } (V, Q, {C}^{\uparrow })}(co3) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{e} (V', P', C')}{(V, P \ {\square }\ Q, C) \mathop {\rightsquigarrow }\limits ^{e} (V', P', C' \wedge idle (Q))}(aex1) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, Q, C) \mathop {\rightsquigarrow }\limits ^{e} (V', Q', C)}{(V, P \ {\square }\ Q, C) \mathop {\rightsquigarrow }\limits ^{e} (V', Q', C' \wedge idle (P))}(aex2) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{a} (V', Q', C')}{(V, P \ {\backslash } \, E, C) \mathop {\rightsquigarrow }\limits ^{a} (V', Q', C')}(ahi1) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{a} (V', Q', C') \ , \ \ active (V, P, C) \cap E\ne \emptyset \ , \ \ a \notin E}{(V, P \ {\backslash } \, E, C) \mathop {\rightsquigarrow }\limits ^{a} (V', Q', C' \wedge C)}(ahi2) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{a} (V', Q', C'), active (V, P, C) \cap E\ne \emptyset \ , \ \ a \in E}{(V, P \ {\backslash } \, E, C) \mathop {\rightsquigarrow }\limits ^{\tau } (V', Q', C' \wedge C)}(ahi3) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{a} (V', P', C') \ , \ \ \checkmark \notin active (V, P, C)}{(V, P; Q, C) \mathop {\rightsquigarrow }\limits ^{a} (V', P'; Q, C')}(ase1) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{\checkmark } (V', P', C')}{(V, P; Q, C) \mathop {\rightsquigarrow }\limits ^{\tau } (V, Q, C \wedge C')}(ase2) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{a} (V', P', C') \ , \ \ a \notin E}{(V, P \ [\![E ]\!]\ Q, C) \mathop {\rightsquigarrow }\limits ^{a} (V', P' \ [\![E ]\!]\ Q, C' \wedge idle (Q))}(apa1) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, Q, C) \mathop {\rightsquigarrow }\limits ^{a} (V', Q', C') \ , \ \ a \notin E}{(V, P \ [\![E ]\!]\ Q, C) \mathop {\rightsquigarrow }\limits ^{a} (V', P \ [\![E ]\!]\ Q', C' \wedge idle (P))}(apa2) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{e} (V, P', C') \ , \ \ (V, Q, C) \mathop {\rightsquigarrow }\limits ^{e} (V, Q', C'') \ , \ \ e \in E}{(V, P \ [\![E ]\!]\ Q, C) \mathop {\rightsquigarrow }\limits ^{e} (V, P' \ [\![E ]\!]\ Q', C' \wedge C'')}(apa3) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, Q, C) \mathop {\rightsquigarrow }\limits ^{a} (V', Q', C') \ , \ \ P \doteq Q}{(V, P, C) \mathop {\rightsquigarrow }\limits ^{a} (V', Q', C')}(ade) \end{aligned}$$
$$\begin{aligned} \dfrac{}{(V, \mathtt {Wait}{}[u]_x, C) \mathop {\rightsquigarrow }\limits ^{\tau }(V, \mathtt {Skip}, {C}^{\uparrow } \wedge x= u)}(await) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{\tau }(V', P', C')}{(V, P~\mathtt {timeout} [u]_x~ Q, C) \mathop {\rightsquigarrow }\limits ^{\tau }(V', P' ~\mathtt {timeout} [u]_x~ Q, C' \wedge x\le u)}(ato1) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{e}(V', P', C')}{(V, P ~\mathtt {timeout} [u]_x~ Q, C) \mathop {\rightsquigarrow }\limits ^{e}(V', P', C' \wedge x\le u)}(ato2) \end{aligned}$$
$$\begin{aligned} \dfrac{}{(V, P~\mathtt {timeout} [u]_x~ Q, C) \mathop {\rightsquigarrow }\limits ^{\tau }(V, Q, {C}^{\uparrow } \wedge x= u\wedge idle (P) ) }(ato3) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{a}(V', P', C')}{(V, P~\mathtt {interrupt}[u]_x~ Q, C) \mathop {\rightsquigarrow }\limits ^{a}(V', P' ~\mathtt {interrupt}[u]_x~ Q, C' \wedge x\le u)}(ait1) \end{aligned}$$
$$\begin{aligned} \dfrac{}{(V, P~\mathtt {interrupt}[u]_x~ Q, C) \mathop {\rightsquigarrow }\limits ^{\tau }(V, Q, {C}^{\uparrow } \wedge x= u\wedge idle (P))}(ait2) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{\tau }(V', P', C')}{(V, P~\mathtt {within}[u]_x, C) \mathop {\rightsquigarrow }\limits ^{\tau }(V', P'~\mathtt {within}[u]_x, C' \wedge x\le u)}(awi1) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{e}(V', P', C')}{(V, P~\mathtt {within}[u]_x, C) \mathop {\rightsquigarrow }\limits ^{e}(V', P', C' \wedge x\le u)}(awi2) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{a}(V', P', C') \ , \ \ a \ne \checkmark }{(V, P~\mathtt {deadline} [u]_x, C) \mathop {\rightsquigarrow }\limits ^{a}(V', P'~\mathtt {deadline} [u]_x, C' \wedge x\le u)}(adl1) \end{aligned}$$
$$\begin{aligned} \dfrac{(V, P, C) \mathop {\rightsquigarrow }\limits ^{\checkmark }(V', P', C')}{(V, P~\mathtt {deadline} [u]_x, C) \mathop {\rightsquigarrow }\limits ^{\checkmark }(V', P', C' \wedge x\le u)}(adl2) \end{aligned}$$

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André, É., Liu, Y., Sun, J. et al. Parameter synthesis for hierarchical concurrent real-time systems. Real-Time Syst 50, 620–679 (2014). https://doi.org/10.1007/s11241-014-9208-6

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