This paper investigates a swarm of autonomous mobile robots in the Euclidean plane, under the semi-synchronous ($\cal SSYNC$) scheduler. Each robot has a target function to determine a destination point from the robots' positions. All robots in the swarm take the same target function conventionally. We allow the robots to take different target functions, and investigate the effects of the number of distinct target functions on the problem-solving ability, regarding target function as a resource to solve a problem like time. Specifically, we are interested in how many distinct target functions are necessary and sufficient to solve a problem $\Pi$. The number of distinct target functions necessary and sufficient to solve $\Pi$ is called the minimum algorithm size (MAS) for $\Pi$. The MAS is defined to be $\infty$, if $\Pi$ is unsolvable even for the robots with unique target functions. We show that the problems form an infinite hierarchy with respect to their MASs; for each integer $c > 0$ and $\infty$, the set of problems whose MAS is $c$ is not empty, which implies that target function is a resource irreplaceable, e.g., with time. We propose MAS as a natural measure to measure the complexity of a problem. We establish the MASs for solving the gathering and related problems from any initial configuration, i.e., in a self-stabilizing manner. For example, the MAS for the gathering problem is 2. It is 3, for the problem of gathering {\bf all non-faulty} robots at a single point, regardless of the number $(< n)$ of crash failures. It is however $\infty$, for the problem of gathering all robots at a single point, in the presence of at most one crash failure.

翻译：本文研究欧几里得平面中半同步 ($\cal SSYNC$)调度器下的自主移动机器人群体。每个机器人拥有一个目标函数，用于根据机器人的位置确定目标点。传统上，群体中所有机器人采用相同的目标函数。我们允许机器人采用不同的目标函数，并研究不同目标函数数量对问题求解能力的影响，将目标函数视为类似于时间的一种解决问题的资源。具体而言，我们关注解决问题 $\Pi$ 所需且足够的不同目标函数数量。解决 $\Pi$ 所需且足够的不同目标函数数量称为 $\Pi$ 的极小算法规模 (MAS)。如果即使对于具有唯一目标函数的机器人也无法解决 $\Pi$，则定义 MAS 为 $\infty$。我们证明这些问题根据其 MAS 形成一个无限层次结构；对于每个整数 $c > 0$ 以及 $\infty$，MAS 为 $c$ 的问题集合非空，这意味着目标函数是一种不可替代的资源，例如无法用时间替代。我们提出 MAS 作为衡量问题复杂性的自然度量。我们建立了从任意初始配置（即以自稳定方式）解决聚集及相关问题的 MAS。例如，聚集问题的 MAS 为 2。无论崩溃故障的数量 $(< n)$ 是多少，将所有 {\bf 非故障} 机器人聚集于单一点的问题的 MAS 为 3。然而，在最多存在一个崩溃故障的情况下，将所有机器人聚集于单一点的问题的 MAS 为 $\infty$。