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Parameterized synthesis of self-stabilizing protocols in symmetric networks

Abstract

Self-stabilization in distributed systems is a technique to guarantee convergence to a set of legitimate states without external intervention when a transient fault or bad initialization occurs. Recently, there has been a surge of efforts in designing techniques for automated synthesis of self-stabilizing algorithms that are correct by construction. Most of these techniques, however, are not parameterized, meaning that they can only synthesize a solution for a fixed and predetermined number of processes. In this paper, we report a breakthrough in parameterized synthesis of self-stabilizing algorithms in symmetric networks, including ring, line, mesh, and torus. First, we develop cutoffs that guarantee (1) closure in legitimate states, and (2) deadlock-freedom outside the legitimate states. We also develop a sufficient condition for convergence in self-stabilizing systems. Since some of our cutoffs grow with the size of the local state space of processes, scalability of the synthesis procedure is still a problem. We address this problem by introducing a novel SMT-based technique for counterexample-guided synthesis of self-stabilizing algorithms in symmetric networks. We have fully implemented our technique and successfully synthesized solutions to maximal matching, three coloring, and maximal independent set problems for ring and line topologies.

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Notes

  1. 1.

    Note that a computation may be terminating in a state in \(\textit{LS}\).

  2. 2.

    One can see that it is possible to find such a sequence of length \(l^2\) by an induction on l.

  3. 3.

    Note that this property can be easily transformed to a property without the universal quantifier by introducing new variables that can take arbitrary values at the initialization and then keep their values.

  4. 4.

    This approach is inspired by similar abstraction-based methods for the verification and synthesis of systems with many processes [3, 10].

  5. 5.

    For example, for a topology with \(l=3\) in a torus with \(|m|=3\), the cutoff for the first case will be 2187.

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Correspondence to Fathiyeh Faghih.

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Mirzaie, N., Faghih, F., Jacobs, S. et al. Parameterized synthesis of self-stabilizing protocols in symmetric networks. Acta Informatica (2019). https://doi.org/10.1007/s00236-019-00361-7

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