The Knaster-Tarski theorem, also known as Tarski's theorem, guarantees that every monotone function defined on a complete lattice has a fixed point. We analyze the query complexity of finding such a fixed point on the $k$-dimensional grid of side length $n$ under the $\leq$ relation. Specifically, there is an unknown monotone function $f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^k$ and an algorithm must query a vertex $v$ to learn $f(v)$. A key special case of interest is the Boolean hypercube $\{0,1\}^k$, which is isomorphic to the power set lattice -- the original setting of the Knaster-Tarski theorem. Our lower bound characterizes the randomized and deterministic query complexity of the Tarski search problem on the Boolean hypercube as $\Theta(k)$. More generally, we prove a randomized lower bound of $\Omega\left( k + \frac{k \cdot \log{n}}{\log{k}} \right)$ for the $k$-dimensional grid of side length $n$, which is asymptotically tight in high dimensions when $k$ is large relative to $n$.

翻译：克纳斯特-塔尔斯基定理（亦称塔尔斯基定理）保证了定义在完全格上的每个单调函数都存在不动点。我们分析了在边长为 $n$ 的 $k$ 维网格上，依据 $\leq$ 关系寻找此类不动点的查询复杂度。具体而言，存在一个未知的单调函数 $f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^k$，算法必须查询顶点 $v$ 以获知 $f(v)$。一个关键且受关注的特例是布尔超立方 $\{0,1\}^k$，它同构于幂集格——这正是克纳斯特-塔尔斯基定理的原始设定。我们的下界刻画了布尔超立方上塔尔斯基搜索问题的随机化与确定性查询复杂度均为 $\Theta(k)$。更一般地，我们证明了对于边长为 $n$ 的 $k$ 维网格，其随机化查询复杂度下界为 $\Omega\left( k + \frac{k \cdot \log{n}}{\log{k}} \right)$，当 $k$ 相对于 $n$ 较大时，该下界在高维情形下是渐近紧的。