突破Ω(n)空间壁垒：群体协议可判定双指数阈值
Breaking through the $Ω(n)$-space barrier: Population Protocols Decide Double-exponential Thresholds
AI总结：
本研究突破群体协议的Ω(n)空间壁垒，构造出状态数仅为O(log|φₙ|)的双指数阈值判定协议，匹配已知下界，且为首个非1感知、几乎自稳定的阈值协议构造。
摘要：
群体协议是一种分布式计算模型，其中有限状态智能体以两两随机交互的方式运行。协议可判定任意初始配置是否满足某一固定性质，该性质被定义为配置集合上的谓词。若一族用于判定谓词$φ_n$的协议使用$\boldsymbol{\textit{O}}(|φ_n|)$个状态，则称其为简洁的，其中$φ_n$被编码为系数以二进制表示的无量词Presburger公式（所有可由群体协议判定的谓词都能以此方式编码）。尽管已知所有谓词都存在简洁协议，但对于**任意**谓词族$φ_n$，是否存在状态数为$o(|φ_n|)$的协议仍是开放问题。我们通过为一类阈值谓词$φ_n(x)\boldsymbol{\textit{⇔}} x\boldsymbol{\textit{≥}} k_n$（其中$k_1,k_2,...\boldsymbol{\textit{∈}}\boldsymbol{\textit{ℕ}}$）构造状态数为$\boldsymbol{\textit{O}}(\boldsymbol{\textit{log}}|φ_n|)$的协议，给出了肯定答案（换言之，使用$\boldsymbol{\textit{O}}(n)$个状态的协议可判定$k\boldsymbol{\textit{≥}} 2^{2^n}$的$x\boldsymbol{\textit{≥}} k$问题）。该结果与已知下界相匹配。此外，我们针对阈值谓词的构造是首个不具备1感知性的构造，且几乎是自稳定的。
英文摘要：
Population protocols are a model of distributed computation in which finite-state agents interact randomly in pairs. A protocol decides for any initial configuration whether it satisfies a fixed property, specified as a predicate on the set of configurations. A family of protocols deciding predicates $φ_n$ is succinct if it uses $\mathcal{O}(|φ_n|)$ states, where $φ_n$ is encoded as quantifier-free Presburger formula with coefficients in binary. (All predicates decidable by population protocols can be encoded in this manner.) While it is known that succinct protocols exist for all predicates, it is open whether protocols with $o(|φ_n|)$ states exist for \emph{any} family of predicates $φ_n$. We answer this affirmatively, by constructing protocols with $\mathcal{O}(\log|φ_n|)$ states for some family of threshold predicates $φ_n(x)\Leftrightarrow x\ge k_n$, with $k_1,k_2,...\in\mathbb{N}$. (In other words, protocols with $\mathcal{O}(n)$ states that decide $x\ge k$ for a $k\ge 2^{2^n}$.) This matches a known lower bound. Moreover, our construction for threshold predicates is the first that is not $1$-aware, and it is almost self-stabilising.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
