Abstract
We study the reactive synthesis problem (\(\mathsf {RS}\)) for specifications given in Metric Interval Temporal Logic (\(\mathsf {MITL}\)). \(\mathsf {RS}\) is known to be undecidable in a very general setting, but on infinite words only; and only the very restrictive \(\mathsf {BResRS}\) subcase is known to be decidable (see D’Souza et al. and Bouyer et al.). In this paper, we sharpen the decidability border of \(\mathsf {MITL}\) synthesis. We show \(\mathsf {RS}\) is undecidable on finite words too, and present a landscape of restrictions (both on the logic and on the possible controllers) that are still undecidable. On the positive side, we revisit \(\mathsf {BResRS}\) and introduce an efficient on-the-fly algorithm to solve it.
More technical details and proofs can be found in the full version of this paper [8]. This work has been supported by The European Union Seventh Framework Programme under Grant Agreement 601148 (Cassting) and by the FRS/F.N.R.S. PDR grant SyVeRLo.
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Notes
- 1.
In order to keep the discussion focused and concise, we give the formal definitions for finite words only. It is straightforward to adapt them to the infinite words case.
- 2.
We assume that for every location q and every valuation \(\nu \), there exists a timed action \((t,\sigma )\in \mathbb {R}^{+}\times \varSigma \) and a transition \((q,(\sigma ,g,R),q')\in \delta _\mathcal P\) such that \(\nu +t\models g\).
- 3.
- 4.
Empty word \(\varepsilon \) is added for convenience, in case it is not already in \(\mathcal L(\varphi )\).
- 5.
Observe that the proof does not require any plant (or uses the trivial plant accepting \(T\varSigma ^\star \)). This entails undecidability of the ‘realisability problem’, which is more restrictive than \(\mathsf {RS} _d^\star \) and another difference with respect to the proof in [4].
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Brihaye, T., Estiévenart, M., Geeraerts, G., Ho, HM., Monmege, B., Sznajder, N. (2016). Real-Time Synthesis is Hard!. In: Fränzle, M., Markey, N. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2016. Lecture Notes in Computer Science(), vol 9884. Springer, Cham. https://doi.org/10.1007/978-3-319-44878-7_7
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