Abstract
In this paper, we study \({\textsc {ltl}}_f\) synthesis under environment specifications for arbitrary reachability and safety properties. We consider both kinds of properties for both agent tasks and environment specifications, providing a complete landscape of synthesis algorithms. For each case, we devise a specific algorithm (optimal wrt complexity of the problem) and prove its correctness. The algorithms combine common building blocks in different ways. While some cases are already studied in literature others are studied here for the first time.
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Notes
- 1.
In fact, this algorithm can solve all cases, but it’s much more involved compared to the direct algorithms we provide for each case.
- 2.
For pure-past ltl, obtaining the dfa from a pure-past ltl formula is single exponential [11], and indeed the problems and all our algorithms become EXPTIME-complete.
- 3.
Technically, the properties should be Borel, which all our properties are.
- 4.
In [9] is shown that the case of \({\textsc {ltl}}_f\) synthesis under safety and reachability properties can be solved by reducing to games on infinite-word automata. This certain case is covered in our paper, nevertheless, we provide a direct approach that only involves games on finite-word automata.
- 5.
Since strategies can depend on the history, and thus on the starting state in particular, there is always a strategy that wins from every state in \(\textsf{Win}_p\).
- 6.
We remark that (i) when we restrict the transition system of a da \((\mathcal {D},\alpha )\) we may need to revise the winning-condition \(\alpha \) to express whether reaching sink is good for the protagonist or not (although many times it is not, e.g., when restricting to the winning-region for a safety condition); (ii) in one case, in Algorithm 7, we will add two sink states.
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Aminof, B., De Giacomo, G., Di Stasio, A., Francon, H., Rubin, S., Zhu, S. (2023). \({\textsc {ltl}}_f\) Synthesis Under Environment Specifications for Reachability and Safety Properties. In: Malvone, V., Murano, A. (eds) Multi-Agent Systems. EUMAS 2023. Lecture Notes in Computer Science(), vol 14282. Springer, Cham. https://doi.org/10.1007/978-3-031-43264-4_17
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