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International SPIN Workshop on Model Checking of Software

SPIN 2015: Model Checking Software pp 84–101Cite as

Practical Stutter-Invariance Checks for \(\omega \)-Regular Languages

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9232))

Abstract

An \(\omega \)-regular language is stutter-invariant if it is closed by the operation that duplicates some letter in a word or that removes some duplicate letter. Model checkers can use powerful reduction techniques when the specification is stutter-invariant.

We propose several automata-based constructions that check whether a specification is stutter-invariant. These constructions assume that a specification and its negation can be translated into Büchi automata, but aside from that, they are independent of the specification formalism. These transformations were inspired by a construction due to Holzmann and Kupferman, but that we broke down into two operations that can have different realizations, and that can be combined in different ways. As it turns out, implementing only one of these operations is needed to obtain a functional stutter-invariant check.

Finally we have implemented these techniques in a tool so that users can easily check whether an LTL or PSL formula is stutter-invariant.

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Notes

  1. 1.

    While testing our implementation we found Lemma 2 of [9] to be incorrect w.r.t. the \(\cap \) operator. A counterexample is the SERE \(r=a\cap (a;a)\) since \(L_\sharp (r)=\emptyset \) but \(L_\sharp (\kappa (r))=\{a\}\). Also Lemma 4 is incorrect w.r.t. the operator; a counterexample is the PSL formula a b which gets rewritten as \(a^+\) b: two stutter-invariant formulas with different languages. We are in contact with the authors. (Note that these lemmas are numbered 4 and 9 in the authors’ copy.).

  2. 2.

    \(A_1\), \(A_2\), and \(A_3\) are equivalent automata. The only reason we used two acceptance sets in \(A_1\) was to demonstrate how \(\mathsf {cl}\) deals with multiple acceptance sets.

  3. 3.

    Unlike automata-based constructions such as \(\mathsf {cl}(\mathsf {sl}(A))\), the formula \(\tau '(\varphi )\) is not necessarily an over-approximation of \(\varphi \), so the equivalence check between \(\varphi \) and \(\tau '(\varphi )\) cannot be replaced by a simple inclusion check.

  4. 4.

    The only actual implementation of a construction similar to the one of Holzmann and Kupferman [19] that we know about is in ltl2dstar[20], but it decides only stutter-invariance for one letter at a time, is used to improve Safra’s construction, and is not directly accessible to the user.

  5. 5.

    Measurements were done on a dedicated Intel Xeon E5-2620 2 GHz, running Debian GNU/Linux, with the memory limited to 32 GB (out of the 64 GB installed).

  6. 6.

    A better implementation of this check would be to construct the automaton for \(\varphi \leftrightarrow \tau '(\varphi )\) on-the-fly during its emptiness check, as done in dedicated satifiability checkers [27]. Alas, the implementation of our algorithm for translating LTL/PSL formulas into TGBA is not implemented in a way that would allow an on-the-fly construction. So this experiment should not be read as a dismissal of the idea of testing whether \(\mathscr {L}(A_{\lnot (\varphi \leftrightarrow \tau '(\varphi ))})=\emptyset \) but simply as a justification of why we used \(\mathscr {L}(A_{\tau '(\varphi )}\otimes A_{\lnot \varphi })=\emptyset \wedge \mathscr {L}(A_{\lnot \tau '(\varphi )}\otimes A_{\varphi })=\emptyset \) in our former implementation.

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Acknowledgments

The authors are indebted to Joachim Klein and Akim Demaille for some influencing comments on the first drafts of this article, and to Etienne Renault, Souheib Baarir and the anonymous reviewers of ICALP’15 and SPIN’15 from some valuable feedback on earlier versions.

Author information

Authors and Affiliations

  1. LRDE, EPITA, Le Kremlin-Bicêtre, France

    Thibaud Michaud & Alexandre Duret-Lutz

Authors
  1. Thibaud Michaud
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  2. Alexandre Duret-Lutz
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Corresponding author

Correspondence to Alexandre Duret-Lutz .

Editor information

Editors and Affiliations

  1. Stellenbosch University, Matieland, South Africa

    Bernd Fischer

  2. Stellenbosch University, Matieland, South Africa

    Jaco Geldenhuys

A Tool Support

A Tool Support

All the checks described in this article are implemented in Spot 1.99.1 which can be obtained from https://spot.lrde.epita.fr/.

Stutter-invariance of LTL or PSL formulas can be tested on-line without installing anything:

  1. 1.

    Load https://spot.lrde.epita.fr/trans.html.

  2. 2.

    Type an LTL or PSL formula.

  3. 3.

    Select “Desired Output: Formula” and then “property information”.

  4. 4.

    Scan the resulting properties for “syntactic stutter invariant” (this means the formula belongs to LTL\(\setminus \mathsf {X}\)or siPSL), “stutter invariant” or “stutter sensitive”. In the latter two cases, the automata-based check had to be performed.

If Spot is installed, the tool ltlfilt can be used from the command-line to make the same decision. For instance ltlfilt -f  ’\(\varphi \)’ --stutter-invariant will print \(\varphi \) back iff \(\varphi \) is stutter-invariant.

Similarly the tool autfilt can be used to apply the operations \(\mathsf {cl}\), \(\mathsf {sl}\), and \(\mathsf {sl}_2\) to any automaton (with any acceptance condition). The corresponding options are --destut, --instut, and --instut=2 respectively.

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Michaud, T., Duret-Lutz, A. (2015). Practical Stutter-Invariance Checks for \(\omega \)-Regular Languages. In: Fischer, B., Geldenhuys, J. (eds) Model Checking Software. SPIN 2015. Lecture Notes in Computer Science(), vol 9232. Springer, Cham. https://doi.org/10.1007/978-3-319-23404-5_7

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  • DOI: https://doi.org/10.1007/978-3-319-23404-5_7

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  • Online ISBN: 978-3-319-23404-5

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