Topological Characterization of Stabilizing Consensus

TL;DR

It is proved that semi-open decision sets and semi-continuous decision functions as introduced by Levin (AMM, 1963) are the appropriate means for this characterization of the solvability/impossibility of deterministic stabilizing consensus in any computing model with benign process and communication faults using point-set topology.

Abstract

We provide a complete characterization of the solvability/impossibility of deterministic stabilizing consensus in any computing model with benign process and communication faults using point-set topology. Relying on the topologies for infinite executions introduced by Nowak, Schmid and Winkler (JACM, 2024) for terminating consensus, we prove that semi-open decision sets and semi-continuous decision functions as introduced by Levin (AMM, 1963) are the appropriate means for this characterization: Unlike the decision functions for terminating consensus, which are continuous, semi-continuous functions do not require the inverse image of an open set to be open and hence allow to map a connected space to a disconnected one. We also show that multi-valued stabilizing consensus with weak and strong validity are equivalent, as is the case for terminating consensus. By applying our results to (variants of) all the known possibilities/impossibilities for stabilizing consensus, we easily provide a topological explanation of these results.

Figures (5)

• Figure 1: Examples of two connected components of the decision sets $\Sigma_0=\Sigma_{\gamma_0}\cup \Sigma_{\gamma_0'}$ and $\Sigma_1=\Sigma_{\gamma_1}\cup \Sigma_{\gamma_1'}$. Common limit points (like for $\Sigma_{\gamma_0}$ and $\Sigma_{\gamma_1}$, marked by $\times$) must be excluded by \ref{['cor:consensusseparation']}.
• Figure 2: Illustration of the two cases in the proof of \ref{['thm:Schar:nonunif']}, depicting the four different cases for the decision function $\Delta_p(C^{t_i})$. The blue, green, yellow, and red cloud represents $D_p(\gamma,t^i)$ for Case (a), (b), (c), and (d), respectively. The included boundaries $\mathop{\mathrm{\partial in}}\nolimits \Sigma_v$ resp. $\mathop{\mathrm{\partial in}}\nolimits \Sigma_w$ are formed by the limit points (marked by a cross) that lie on the blue resp. red dotted lines. The thick limit point at the center of the yellow cloud lies in the intersection of $\mathop{\mathrm{\partial}}\nolimits \mathop{\mathrm{\partial in}}\nolimits \Sigma_v$ and $\mathop{\mathrm{\partial}}\nolimits \mathop{\mathrm{\partial in}}\nolimits \Sigma_w$; we assume here w.l.o.g. that it belongs to $\mathop{\mathrm{\partial in}}\nolimits \Sigma_v$ (blue).
• Figure 3: Prefix order for the 1-prefixes and 2-prefixes in the LL model.
• Figure 4: Complete representation of 1-prefixes in the LL model, for all input assignments.
• Figure 5: Representation of all 1-prefixes and 2-prefixes in the DLL model, for a fixed input assignment.

Theorems & Definitions (46)

• Definition 2.1: Non-uniform and uniform terminating consensus
• Definition 2.2: Stabilizing consensus
• Lemma 3.1
• Lemma 3.2
• Definition 3.3: Semi-open sets Lev63
• Theorem 3.4: Lev63
• Theorem 3.5: Lev63
• Lemma 3.6: Lev63
• Definition 3.7: Included boundary points
• Theorem 3.8: Lev63