In our previous papers [7,11,23], we presented advanced type systems for the -calculus, which can guarantee deadlock-freedom in the sense that certain communications will eventually succeed unless the whole process diverges. Although such guarantee is quite useful for reasoning about the behavior of concurrent programs, there still remains a weakness that the success of a communication cannot be completely guaranteed due to the problem of divergence. For example, while a server process that has received a request message cannot discard the request, it is allowed to infinitely delegate the request to other processes, causing a livelock. In this paper, we show that we can guarantee not only deadlock-freedom but also livelock-freedom, by modifying our previous type systems for deadlock-freedom. The resulting type system guarantees that certain communications will eventually succeed under fair scheduling, no matter whether processes diverge. Moreover, it can also guarantee that some of those communications will succeed within a certain amount of time.
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