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arXiv:2502.16679 · Leibniz international proceedings in informatics

Tarski Lower Bounds from Multi-Dimensional Herringbones

Simina Brânzei, Reed Phillips, Nicholas Recker

Published Jan 1, 2025• 0 citations
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Abstract

Tarski's theorem states that every monotone function from a complete lattice to itself 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. In this setting, 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 $f(v)$. The goal is to find a fixed point of $f$ using as few oracle queries as possible. We show that the randomized query complexity of this problem is $Ω\left( \frac{k \cdot \log^2{n}}{\log{k}} \right)$ for all $n,k \geq 2$. This unifies and improves upon two prior results: a lower bound of $Ω(\log^2{n})$ from [EPRY 2019] and a lower bound of $Ω\left( \frac{k \cdot \log{n}}{\log{k}}\right)$ from [BPR 2024], respectively.

DOI
10.4230/lipics.approx/random.2025.52

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Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We analyze the query complexity of finding such a fixed point on the kkk-dimensional grid of side length nnn under the ≤\leq≤ relation. In this setting, there is an unknown monotone function f:{0,1,…,n−1}k→{0,1,…,n−1}kf: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^kf:{0,1,…,n−1}k→{0,1,…,n−1}k and an algorithm must query a vertex vvv to learn f(v)f(v)f(v). The goal is to find a fixed point of fff using as few oracle queries as possible. We show that the randomized query complexity of this problem is Ω(k⋅log⁡2nlog⁡k)Ω\left( \frac{k \cdot \log^2{n}}{\log{k}} \right)Ω(logkk⋅log2n​) for all n,k≥2n,k \geq 2n,k≥2. This unifies and improves upon two prior results: a lower bound of Ω(log⁡2n)Ω(\log^2{n})Ω(log2n) from [EPRY 2019] and a lower bound of Ω(k⋅log⁡nlog⁡k)Ω\left( \frac{k \cdot \log{n}}{\log{k}}\right)Ω(logkk⋅logn​) from [BPR 2024], respectively.
The Tarski fixed-point problem asks for a fixed point of a monotone map on a kkk-dimensional grid lattice of width nnn and has applications in game theory, economics, and verification. Substantial gaps remain between the known upper and lower bounds on its quantum query complexity. We prove two hardness bootstrapping theorems for quantum query lower bounds. Starting from a hard instance family whose instances have pair-wise incomparable unique fixed points, and a lower bound established by the nonnegative adversary method, we bootstrap this lower bound by (i) a factor of Ω(t)Ω(t)Ω(t) when the dimension is multiplied by ttt and the width remains unchanged, and (ii) a factor of Ω(log⁡m)Ω(\log m)Ω(logm) when the width is multiplied by mmm and the dimension is doubled. Applying these theorems, we obtain three consequences: (i) an Ω(klog⁡2n)Ω(k\log^2 n)Ω(klog2n) lower bound for k≥4k\geq 4k≥4, improving previously best-known classical and quantum lower bounds; (ii) for every fixed constant α>0α>0α>0, there exists a constant δ(α)>0δ(α)>0δ(α)>0 such that Qε(Tarski(n,k))=Ω(knδ(α))Q_ε(\text{Tarski}(n,k))=Ω(kn^{δ(α)})Qε​(Tarski(n,k))=Ω(knδ(α)) whenever k>nαk>n^αk>nα, and in particular, an Ω(kn0.0072)Ω(kn^{0.0072})Ω(kn0.0072) lower bound for k>n2/203k>n^{2/203}k>n2/203, the first lower bound with simultaneous linear dependence on the dimension and polynomial dependence on the width; and (iii) a superpolynomial lower bound of (k/log⁡k)Ω(log⁡k)(k/\log k)^{Ω(\log k)}(k/logk)Ω(logk) for exponentially wide grids, where n=2Θ(k)n=2^{Θ(k)}n=2Θ(k). The last result rules out a route to polynomial-time algorithms for simple stochastic games and ARRIVAL through generic polynomial-query black-box Tarski solvers.
Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We specifically consider the two-dimensional lattice Ln2\mathcal{L}^2_nLn2​ on points {1,…,n}2\{1, \ldots, n\}^2{1,…,n}2 and where (x1,y1)≤(x2,y2)(x_1, y_1) \leq (x_2, y_2)(x1​,y1​)≤(x2​,y2​) if x1≤x2x_1 \leq x_2x1​≤x2​ and y1≤y2y_1 \leq y_2y1​≤y2​. We show that the quantum query complexity of finding a fixed point given query access to a monotone function on Ln2\mathcal{L}^2_nLn2​ is Ω((log⁡n)2)Ω((\log n)^2)Ω((logn)2), matching the classical deterministic upper bound. The proof consists of two main parts: a lower bound on the quantum query complexity of a composition of a class of functions including ordered search, and an extremely close relationship between finding Tarski fixed points and nested ordered search.
The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting. We prove an Ω(klog⁡n)Ω(k\log n)Ω(klogn) quantum query lower bound for finding a fixed point of a monotone function on [n]k[n]^k[n]k, using the nonnegative spectral adversary method. In the two extremal regimes n=2n = 2n=2 and k=1k = 1k=1, our quantum lower bound matches the previous classical lower bounds Ω(k)Ω(k)Ω(k) and Ω(log⁡n)Ω(\log n)Ω(logn), respectively. For n,k≥2n, k\geq 2n,k≥2, our bound improves the best previous classical lower bound when n<kn < kn<k and is within a factor of log⁡n/log⁡k\log n / \log klogn/logk compared to the known classical lower bound when n≥kn \geq kn≥k. To construct the adversary matrix, we develop the Tree--Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.
The Six-Dimensional Tarski Problem is Ω(log⁡3n)Ω(\log^3 n)Ω(log3n)-Hard
We study the deterministic query complexity of finding a fixed point of a monotone function f:[n]d→[n]df:[n]^d\to[n]^df:[n]d→[n]d, where querying xxx returns f(x)f(x)f(x). For fixed d≥2d\ge2d≥2, earlier work established a lower bound of Ω((log⁡n)2)Ω((\log n)^2)Ω((logn)2) and an upper bound of Od((log⁡n)⌈(d+2)/3⌉)O_d((\log n)^{\lceil(d+2)/3\rceil})Od​((logn)⌈(d+2)/3⌉) for the query complexity. Very recently, several papers obtained improved lower bounds using AI-assisted proofs. In this paper, we further improve these lower bounds to Tarski(n,d)=Ωd((log⁡n)2+⌊log⁡3(d/2)⌋)\text{Tarski}(n,d)=Ω_d((\log n)^{2+\lfloor\log_3(d/2)\rfloor})Tarski(n,d)=Ωd​((logn)2+⌊log3​(d/2)⌋). In particular, we obtain the optimal Θ((log⁡n)3)Θ((\log n)^3)Θ((logn)3) deterministic query complexity in six dimensions.
We consider the query complexity of finding a local minimum of a function defined on a graph. This abstract problem is fundamental to many optimization tasks, such as finding a local minimum of the loss function when training deep neural networks. In such applications, each query is an expensive loss evaluation, making it crucial to parallelize computations. This motivates our study of local search where at most k rounds of interaction (aka adaptivity) with the oracle are allowed. We focus on the d -dimensional grid \{1, 2, \ldots, n \}^{d} , where the dimension d \geq 2 is a constant. Our main contribution is to give algorithms and lower bounds that characterize the query complexity of finding a local minimum in k rounds, when k is constant and polynomial in n , respectively. Our proof technique for lower bounding the query complexity in rounds may be of independent interest as an alternative to the classical relational adversary method of Aaronson from the fully adaptive setting. The local search analysis also enables us to characterize the query complexity of computing a Brouwer fixed point in rounds.