On the (High) Undecidability of Distributed Synthesis Problems

• Authors: David Janin
• Venue: SOFSEM '07 Proceedings of the 33rd conference on Current Trends in Theory and Practice of Computer Science
• Year: 2007

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