Convergence Refinement

Refinement tools such as compilers do not necessarily preserve fault-tolerance. That is, given a fault-tolerant program in a high-level language as input, the output of a compiler in a lower-level language will not necessarily be faulttolerant. In this paper, we identify a type of refinement, namely “convergence refinement”, that preserves the faulttolerance property of stabilization. We illustrate the use of convergence refinement by presenting the first formal design of Dijkstra’s little-understood 3-state stabilizing token-ring system. Our designs begin with simple, abstract token-ring systems that are not stabilizing, and then add an abstract “wrapper” to the systems so as to achieve stabilization. The system and the wrapper are then refined to obtain a concrete token-ring system, while preserving stabilization. In fact, the two are refined independently, which demonstrates that convergence refinement is amenable for “graybox” design of stabilizing implementations, i.e., design of system stabilization based solely on system specification and without knowledge of system implementation details.