Synthesizing Self-Stabilizing Parameterized Protocols with Unbounded Variables

The focus of this paper is on the synthesis of unidirectional symmetric ring protocols that are self-stabilizing. Such protocols have an unbounded number of processes and unbounded variable domains, yet they ensure recovery to a set of legitimate states from any state. This is a significant problem as many distributed systems should preserve their fault tolerance properties when they scale. While previous work addresses this problem for constant-space protocols where domain size of variables are fixed regardless of the ring size, this work tackles the synthesis problem assuming that both variable domains and the number of processes in the ring are unbounded (but finite). We present a sufficient condition for synthesis and develop a sound algorithm that takes a conjunctive state predicate representing legitimate states, and generates the parameterized actions of a protocol that is self-stabilizing to legitimate states. We characterize the unbounded nature of protocols as semilinear sets, and show that such characterization simplifies synthesis. The proposed method addresses a longstanding problem because recovery is required from any state in an unbounded state space. For the first time, we synthesize some self-stabilizing unbounded protocols, including a near agreement and a parity protocol.