The Mystery Deepens: On the Query Complexity of Tarski Fixed Points

2026-03-31

AI Summary

This work investigates the query complexity of finding Tarski fixed points on high-dimensional lattices. We introduce the first algorithmic framework based on secure partial information functions and directly leverage this function to construct an efficient fixed-point search algorithm. On the four-dimensional lattice $[n]^4$, our algorithm achieves a query complexity of $O(\log^2 n)$, matching the known lower bound. For the general $k$-dimensional case, we improve the best-known upper bound to $O(\log^{\lceil (k-1)/3 \rceil + 1} n)$, substantially reducing the number of required queries.

Abstract

We give an $O(\log^2 n)$-query algorithm for finding a Tarski fixed point over the $4$-dimensional lattice $[n]^4$, matching the $Ω(\log^2 n)$ lower bound of [EPRY20]. Additionally, our algorithm yields an ${O(\log^{\lceil (k-1)/3\rceil+1} n)}$-query algorithm for any constant $k$, improving the previous best upper bound ${O(\log^{\lceil (k-1)/2\rceil+1} n)}$ of [CL22]. Our algorithm uses a new framework based on \emph{safe partial-information} functions. The latter were introduced in [CLY23] to give a reduction from the Tarski problem to its promised version with a unique fixed point. This is the first time they are directly used to design new algorithms for Tarski fixed points.

Problem

• Tarski fixed point
• query complexity
• lattice
• algorithm
• computational complexity

Innovation

• Tarski fixed point
• query complexity
• safe partial-information functions
• lattice algorithms
• upper bound improvement