Randomized Lower Bounds for Tarski Fixed Points in High Dimensions

2024-09-05

arXiv.org

AI Summary

This paper investigates the randomized query complexity of computing Tarski fixed points of monotone functions on high-dimensional lattices. The problem is formalized as finding a point (x) satisfying (f(x) = x) via function evaluations on either the (k)-dimensional Boolean hypercube or the (k)-dimensional (n imes cdots imes n) grid. Employing techniques from combinatorial game theory, information-theoretic lower bounds, and lattice theory, the authors establish the first tight (Theta(k)) lower bounds—both randomized and deterministic—for the Boolean hypercube. They further prove an (Omega(k + k log n / log k)) randomized lower bound for the (n)-sided (k)-dimensional grid, showing asymptotic optimality when (k gg log n). These results provide a unified characterization of how dimensionality (k) and domain size (n) jointly govern query complexity, significantly improving prior lower bounds that applied only to low-dimensional or highly structured settings.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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 $f: {0,1,ldots, n-1}^k o {0,1,ldots, n-1}^k$ 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 $Omegaleft( k + frac{k cdot log{n}}{log{k}} ight)$ for the $k$-dimensional grid of side length $n$, which is asymptotically tight in high dimensions when $k$ is large relative to $n$.

Problem

• Analyze query complexity of finding Tarski fixed points
• Focus on Boolean hypercube and k-dimensional grid
• Prove lower bounds for randomized and deterministic queries

Innovation

• Randomized query complexity analysis method
• Lower bound characterization for Boolean hypercube
• Asymptotically optimal randomized lower bound