This paper shows, in the framework of the logical method,the unsolvability of $k$-set agreement task by devising a suitable formula of epistemic logic. The unsolvability of $k$-set agreement task is a well-known fact, which is a direct consequence of Sperner's lemma, a classic result from combinatorial topology. However, Sperner's lemma does not provide a good intuition for the unsolvability,hiding it behind the elegance of its combinatorial statement. The logical method has a merit that it can account for the reason of unsolvability by a concrete formula, but no epistemic formula for the general unsolvability result for $k$-set agreement task has been presented so far. We employ a variant of epistemic $\mu$-calculus, which extends the standard epistemic logic with distributed knowledge operators and propositional fixpoints, as the formal language of logic. With these extensions, we can provide an epistemic $\mu$-calculus formula that mentions higher-dimensional connectivity, which is essential in the original proof of Sperner's lemma, and thereby show that $k$-set agreement tasks are not solvable even by multi-round protocols. Furthermore, we also show that the same formula applies to establish the unsolvability for $k$-concurrency, a submodel of the 2-round protocol.
翻译:本文在逻辑方法的框架内显示, $k$- set 协议任务无法解析, 其方法是设计一个合适的缩写逻辑公式。 $k$- set 协议任务无法解解析, 是一个众所周知的事实, 这是Sperner lemma 的典型结果, 这是组合式表层学的经典结果 。 然而, Sperner 的 Lemma 没有为无法解析提供一种良好的直觉, 它隐藏在组合式语句的优雅背后。 这个逻辑方法有其优点, 它可以用一个具体公式来解释无法解析的原因, 但对于 $k$- set 协议任务的一般不可解析结果, 却没有提出如此远的缩写公式。 我们使用一个缩略式 $mum- calulus 的缩略图变量, 将标准的缩略图逻辑逻辑逻辑与分布式知识操作员和方略图固定点相扩展为同一语言。 通过这些扩展, 我们可以用一个缩写 $mall- $- colulual- 公式的缩略性公式来证明我们无法解 $- sal- fal- falal- commall