On the (high) undecidability of distributed synthesis problems
Résumé
The distributed synthesis problem [11] is known to be undecidable. Our purpose here is to study further this undecidability. For this, we consider distributed games [8], an infinite variant of Peterson and Reif multiplayer games with partial information [10], in which Pnueli and Rosner?s distributed synthesis problem can be encoded and, when decidable [11,6,7], uniformly solved [8]. We first prove that even the simple problem of solving 2-process distributed game with reachability conditions is undecidable ( $\Sigma^0_1$ -complete). This decision problem, equivalent to two process distributed synthesis with fairly restricted FO-specification was left open [8]. We prove then that the safety case is $\Pi^0_1$ -complete. More generally, we establish a correspondence between 2-process distributed game with Mostowski?s weak parity conditions [9] and levels of the arithmetical hierarchy. finally, distributed games with general ?-regular infinitary conditions are shown to be highly undecidable ( $\Sigma^1_1$ -complete)
Domaines
Autre [cs.OH]
Origine : Fichiers produits par l'(les) auteur(s)