Dimension Amplification for Tarski Fixed-Point Query Lower Bounds

2026-10-05

AI Summary

This study addresses the inherent difficulty of surpassing polylogarithmic upper bounds on the query complexity of Tarski fixed points in fixed dimensions. To overcome this barrier, the authors propose a dimension amplification technique that elevates the dimension from d to 4d+1 via combinatorial constructions, introducing an additional logarithmic factor while preserving solution uniqueness. By integrating monotone mapping theory with deterministic query analysis, they iteratively derive tighter complexity lower bounds for higher dimensions. The primary contribution is a proof that the query complexity in any fixed dimension admits no finite polylogarithmic upper bound. Furthermore, the work establishes a tight Ω((log N)³) lower bound for nine-dimensional grids and derives a general lower bound formula applicable to arbitrary high-dimensional settings.

Abstract

We prove that finding a fixed point of a monotone map on the nine-dimensional grid $[N]^9$ requires $\Omega((\log N)^3)$ deterministic queries, even when the fixed point is unique and each query returns the entire function value. The proof gives a construction that raises the dimension from $d$ to $4d+1$, preserves uniqueness, and adds a logarithmic factor to the lower bound. Iteration gives $\Omega((\log N)^{r+2})$ queries in dimension $(7\cdot4^r-1)/3$, for every fixed nonnegative integer $r$, with an implicit constant that may depend on $r$. Consequently, no finite logarithmic exponent bounds the query complexity in all fixed dimensions.

Problem

• Tarski fixed-point
• query complexity
• lower bounds
• monotone map
• dimension amplification

Innovation

• Tarski fixed-point
• query complexity
• dimension amplification
• lower bounds
• monotone map