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Symbolic bounded synthesis

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Abstract

Synthesizing finite-state systems from full linear-time temporal logic (LTL) is an ambitious way to tackle the challenge of constructing correct-by-construction systems. One particularly promising approach in this context is bounded synthesis, originally proposed by Schewe and Finkbeiner, which in turn builds upon Safraless synthesis, as described by Kupferman and Vardi. Previous implementations of these approaches performed the computation either in an explicit way or used symbolic data structures other than binary decision diagrams (BDDs). In this paper, we reconsider BDDs as state space representation and use it as data structure for bounded synthesis. The key to this construction is the application of two novel optimisation techniques that decrease the number of state bits in such a representation significantly. The first technique uses signalling bits to connect sub-games representing the safety- and non-safety parts of the specification. The second technique is based on a closer analysis of the step of building a safety game from a universal automaton and uses a sufficient condition to remove some so-called counters from the state space of the game.

We evaluate our approach on several benchmark suites and show that the new approach leads to a computation time improvement of several orders of magnitude.

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Notes

  1. While this representation of the winning condition is rather uncommon in the literature, it will allow us later to discuss the composition of games in a simplified way.

  2. In other words, whether a play is winning for player 1 can be determined by evaluating which of the sets of \(\mathcal{S}(\mathcal{F})\) contain a position that is visited along the play, and then substituting all sets in \(\mathcal{F}\) for which some position is visited along the play by true and the other ones by false. If the resulting Boolean formula evaluates to true, the play is winning for player 1. While this definition of the acceptance condition is uncommon for obligation games in the literature, we use it here as it is very helpful to describe the game compositions in the following chapters in a simple and intuitive way.

  3. The solution process described here is related to solving so-called games with a weak winning condition (see [13] for a definition and details) and a reformulation of a procedure for solving games with a weak transition structure (with uniform treatment of all game positions in a strongly connected component, see [19] for a definition and details).

  4. The same problem also occurs in the context of generalized reactivity (1) synthesis [22, 29], an approach that trades the full expressivity of LTL against the possibility to use a simpler and more efficient algorithm solving the synthesis problem.

  5. A similar idea was also pursued by Henzinger et al. [20] for simplifying the process of automaton determinisation.

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Authors and Affiliations

  1. Reactive Systems Group, Fachrichtung Informatik, Universität des Saarlandes, Campus, Gebäude E1.3, 5. OG, 66123, Saarbrücken, Germany

    Rüdiger Ehlers

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  1. Rüdiger Ehlers
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Correspondence to Rüdiger Ehlers.

Additional information

This is an extended version of the paper “Symbolic Bounded Synthesis”, presented at the 22nd International Conference on Computer Aided Verification (CAV 2011).

This work was supported by the German Research Foundation (DFG) within the program “Performance Guarantees for Computer Systems” and the Transregional Collaborative Research Center “Automatic Verification and Analysis of Complex Systems” (SFB/TR 14 AVACS).

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Ehlers, R. Symbolic bounded synthesis. Form Methods Syst Des 40, 232–262 (2012). https://doi.org/10.1007/s10703-011-0137-x

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