On the (High) Undecidability of Distributed Synthesis Problems

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 i¾?-regular infinitary conditions are shown to be highly undecidable ($\Sigma^1_1$-complete).