We consider search by mobile agents for a hidden, idle target, placed on the infinite line. Feasible solutions are agent trajectories in which all agents reach the target sooner or later. A special feature of our problem is that the agents are $p$-faulty, meaning that every attempt to change direction is an independent Bernoulli trial with known probability $p$, where $p$ is the probability that a turn fails. We are looking for agent trajectories that minimize the worst-case expected termination time, relative to competitive analysis. First, we study linear search with one deterministic $p$-faulty agent, i.e., with no access to random oracles, $p\in (0,1/2)$. For this problem, we provide trajectories that leverage the probabilistic faults into an algorithmic advantage. Our strongest result pertains to a search algorithm (deterministic, aside from the adversarial probabilistic faults) which, as $p\to 0$, has optimal performance $4.59112+\epsilon$, up to the additive term $\epsilon$ that can be arbitrarily small. Additionally, it has performance less than $9$ for $p\leq 0.390388$. When $p\to 1/2$, our algorithm has performance $\Theta(1/(1-2p))$, which we also show is optimal up to a constant factor. Second, we consider linear search with two $p$-faulty agents, $p\in (0,1/2)$, for which we provide three algorithms of different advantages, all with a bounded competitive ratio even as $p\rightarrow 1/2$. Indeed, for this problem, we show how the agents can simulate the trajectory of any $0$-faulty agent (deterministic or randomized), independently of the underlying communication model. As a result, searching with two agents allows for a solution with a competitive ratio of $9+\epsilon$, or a competitive ratio of $4.59112+\epsilon$. Our final contribution is a novel algorithm for searching with two $p$-faulty agents that achieves a competitive ratio $3+4\sqrt{p(1-p)}$.

翻译：我们考虑移动智能体在无限直线上搜索一个隐藏的、静止的目标。可行的解决方案是智能体的轨迹，使得所有智能体最终都能到达目标。我们问题的一个特殊之处在于智能体是 $p$ 故障的，这意味着每次尝试改变方向都是一个独立的伯努利试验，已知概率为 $p$，其中 $p$ 是转向失败的概率。我们寻求能够最小化最坏情况下预期终止时间的智能体轨迹，基于竞争分析。首先，我们研究一个确定性 $p$ 故障智能体的线性搜索，即无法访问随机预言，$p\in (0,1/2)$。针对该问题，我们提供的轨迹将概率性故障利用为算法优势。我们最强的结果涉及一种搜索算法（确定性算法，除对抗性概率故障外），当 $p\to 0$ 时，其性能最优为 $4.59112+\epsilon$，其中加性项 $\epsilon$ 可任意小。此外，当 $p\leq 0.390388$ 时，其性能小于 $9$。当 $p\to 1/2$ 时，我们的算法性能为 $\Theta(1/(1-2p))$，并且我们证明该性能在常数因子内是最优的。其次，我们考虑两个 $p$ 故障智能体的线性搜索，$p\in (0,1/2)$，对此我们提供了三种具有不同优势的算法，即使在 $p\rightarrow 1/2$ 时，所有算法都具有有界竞争比。实际上，针对该问题，我们展示了智能体如何独立于底层通信模型，模拟任意 $0$ 故障智能体（确定性或随机性）的轨迹。因此，使用两个智能体搜索可实现竞争比为 $9+\epsilon$ 或 $4.59112+\epsilon$ 的解决方案。我们最后的贡献是一种新颖的算法，用于两个 $p$ 故障智能体的搜索，实现了竞争比 $3+4\sqrt{p(1-p)}$。