Undecidable Verification Problems for Programs with Unreliable Channels

We consider the veriication of a particular class of innnite-state systems, namely systems consisting of nite-state processes that communicate via unbounded lossy FIFO channels. This class is able to model e.g. link protocols such as the Alternating Bit Protocol and HDLC. In an earlier paper, we showed that several interesting veriication problems are decidable for this class of systems, namely (1) the reachability problem: is a set of states reachable from some other state of the system, (2) safety property over traces formulated as regular sets of allowed nite traces, and (3) eventuality properties: do all computations of a system eventually reach a given set of states. In this paper, we show that the following problems are undecidable, namely The model checking problem in propositional temporal logics such as Propositional Linear Time Logic (PTL) and Computation Tree Logic (CTL). The problem of deciding eventuality properties with fair channels: do all computations eventually reach a given set of states if the unreliable channels are fair in the sense that they deliver innnitely many messages if innnitely many messages are transmitted. This problem can model the question of whether a link protocol, such as HDLC, will eventually reliably transfer messages across a medium that is not permanently broken. The results are obtained through a reduction from a variant of Post's Correspondence Problem.