This paper contains supplementary material to the IEEE TPDS paper entitled "On necessary and sufficient conditions for deadlock-free routing in wormhole networks". In Section 1, we prove that deciding deadlock freedom of wormhole networks is co-NP-complete. In Section 2, we provide a counter example to a polynomial algorithm for this decision problem.

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Abstract-Deadlock freedom is a key challenge in the design of communication networks. Wormhole switching is a popular switching technique, which is also prone to deadlocks. Deadlock analysis of routing functions is a manual and complex task. We propose an algorithm that automatically proves routing functions deadlock-free or outputs a minimal counter-example explaining the source of the deadlock. Our algorithm is the first to automatically check a necessary and sufficient condition for deadlock-free routing. We illustrate its efficiency in a complex adaptive routing function for torus topologies. Results are encouraging. Deciding deadlock freedom is co-NP-Complete for wormhole networks. Nevertheless, our tool proves a 13 Â 13 torus deadlock-free within seconds. Finding minimal deadlocks is more difficult. Our tool needs four minutes to find a minimal deadlock in a 11 Â 11 torus while it needs nine hours for a 12 Â 12 network.

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Avoiding deadlock is crucial to interconnection networks. In '87, Dally and Seitz proposed a necessary and sufficient condition for deadlock-free routing. This condition states that a routing function is deadlock-free if and only if its channel dependency graph is acyclic. We formally define and prove a slightly different condition from which the original condition of Dally and Seitz can be derived. Dally and Seitz prove that a deadlock situation induces cyclic dependencies by reductio ad absurdum. In contrast we introduce the notion of a waiting graph from which we explicitly construct a cyclic dependency from a deadlock situation. Moreover, our proof is structured in such a way that it only depends on a small set of proof obligations associated to arbitrary routing functions and switching policies. Discharging these proof obligations is sufficient to instantiate our condition for deadlock-free routing on particular networks. Our condition and its proof have been formalized using the ACL2 theorem proving system.

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