arXiv - CS - Computational Complexity最新文献_第7页

Hilbert Functions and Low-Degree Randomness Extractors 希尔伯特函数和低度随机性提取器

arXiv - CS - Computational Complexity Pub Date : 2024-05-16 DOI: arxiv-2405.10277

Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, Chao Yan

{"title":"Hilbert Functions and Low-Degree Randomness Extractors","authors":"Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, Chao Yan","doi":"arxiv-2405.10277","DOIUrl":"https://doi.org/arxiv-2405.10277","url":null,"abstract":"For $Ssubseteq mathbb{F}^n$, consider the linear space of restrictions of\u0000degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted\u0000$mathrm{h}_S(d,mathbb{F})$, is the dimension of this space. We obtain a tight\u0000lower bound on the smallest value of the Hilbert function of subsets $S$ of\u0000arbitrary finite grids in $mathbb{F}^n$ with a fixed size $|S|$. We achieve\u0000this by proving that this value coincides with a combinatorial quantity, namely\u0000the smallest number of low Hamming weight points in a down-closed set of size\u0000$|S|$. Understanding the smallest values of Hilbert functions is closely related to\u0000the study of degree-$d$ closure of sets, a notion introduced by Nie and Wang\u0000(Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert\u0000function to obtain a tight bound on the size of degree-$d$ closures of subsets\u0000of $mathbb{F}_q^n$, which answers a question posed by Doron, Ta-Shma, and Tell\u0000(Computational Complexity, 2022). We use the bounds on the Hilbert function and degree-$d$ closure of sets to\u0000prove that a random low-degree polynomial is an extractor for samplable\u0000randomness sources. Most notably, we prove the existence of low-degree\u0000extractors and dispersers for sources generated by constant-degree polynomials\u0000and polynomial-size circuits. Until recently, even the existence of arbitrary\u0000deterministic extractors for such sources was not known.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The computational power of discrete chemical reaction networks with bounded executions 有界执行的离散化学反应网络的计算能力

arXiv - CS - Computational Complexity Pub Date : 2024-05-14 DOI: arxiv-2405.08649

David Doty, Ben Heckmann

{"title":"The computational power of discrete chemical reaction networks with bounded executions","authors":"David Doty, Ben Heckmann","doi":"arxiv-2405.08649","DOIUrl":"https://doi.org/arxiv-2405.08649","url":null,"abstract":"Chemical reaction networks (CRNs) model systems where molecules interact\u0000according to a finite set of reactions such as (A + B to C), representing that\u0000if a molecule of (A) and (B) collide, they disappear and a molecule of (C) is\u0000produced. CRNs can compute Boolean-valued predicates (phi:mathbb{N}^d to\u0000{0,1}) and integer-valued functions (f:mathbb{N}^d to mathbb{N}); for\u0000instance (X_1 + X_2 to Y) computes the function (min(x_1,x_2)). We study the computational power of execution bounded CRNs, in which only a\u0000finite number of reactions can occur from the initial configuration (e.g.,\u0000ruling out reversible reactions such as (A rightleftharpoons B)). The power\u0000and composability of such CRNs depend crucially on some other modeling choices\u0000that do not affect the computational power of CRNs with unbounded executions,\u0000namely whether an initial leader is present, and whether (for predicates) all\u0000species are required to \"vote\" for the Boolean output. If the CRN starts with\u0000an initial leader, and can allow only the leader to vote, then all semilinear\u0000predicates and functions can be stably computed in (O(n log n)) parallel time\u0000by execution bounded CRNs. However, if no initial leader is allowed, all species vote, and the CRN is\u0000\"noncollapsing\" (does not shrink from initially large to final (O(1)) size\u0000configurations), then execution bounded CRNs are severely limited, able to\u0000compute only eventually constant predicates. A key tool is to characterize\u0000execution bounded CRNs as precisely those with a nonnegative linear potential\u0000function that is strictly decreased by every reaction, a result that may be of\u0000independent interest.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

ASP-Completeness of Hamiltonicity in Grid Graphs, with Applications to Loop Puzzles 网格图中汉密尔顿性的 ASP 完备性及其在循环谜题中的应用

arXiv - CS - Computational Complexity Pub Date : 2024-05-14 DOI: arxiv-2405.08377

MIT Hardness Group, Josh Brunner, Della Hendrickson, Lily Chung, Erik D. Demaine, Andy Tockman

{"title":"ASP-Completeness of Hamiltonicity in Grid Graphs, with Applications to Loop Puzzles","authors":"MIT Hardness Group, Josh Brunner, Della Hendrickson, Lily Chung, Erik D. Demaine, Andy Tockman","doi":"arxiv-2405.08377","DOIUrl":"https://doi.org/arxiv-2405.08377","url":null,"abstract":"We prove that Hamiltonicity in maximum-degree-3 grid graphs (directed or\u0000undirected) is ASP-complete, i.e., it has a parsimonious reduction from every\u0000NP search problem (including a polynomial-time bijection between solutions). As\u0000a consequence, given k Hamiltonian cycles, it is NP-complete to find another;\u0000and counting Hamiltonian cycles is #P-complete. If we require the grid graph's\u0000vertices to form a full $m times n$ rectangle, then we show that Hamiltonicity\u0000remains ASP-complete if the edges are directed or if we allow removing some\u0000edges (whereas including all undirected edges is known to be easy). These\u0000results enable us to develop a stronger \"T-metacell\" framework for proving\u0000ASP-completeness of rectangular puzzles, which requires building just a single\u0000gadget representing a degree-3 grid-graph vertex. We apply this general theory\u0000to prove ASP-completeness of 38 pencil-and-paper puzzles where the goal is to\u0000draw a loop subject to given constraints: Slalom, Onsen-meguri, Mejilink,\u0000Detour, Tapa-Like Loop, Kouchoku, Icelom; Masyu, Yajilin, Nagareru, Castle\u0000Wall, Moon or Sun, Country Road, Geradeweg, Maxi Loop, Mid-loop, Balance Loop,\u0000Simple Loop, Haisu, Reflect Link, Linesweeper; Vertex/Touch Slitherlink,\u0000Dotchi-Loop, Ovotovata, Building Walk, Rail Pool, Disorderly Loop, Ant Mill,\u0000Koburin, Mukkonn Enn, Rassi Silai, (Crossing) Ichimaga, Tapa, Canal View, Aqre,\u0000and Paintarea. The last 14 of these puzzles were not even known to be NP-hard.\u0000Along the way, we prove ASP-completeness of some simple forms of Tree-Residue\u0000Vertex-Breaking (TRVB), including planar multigraphs with degree-6 breakable\u0000vertices, or with degree-4 breakable and degree-1 unbreakable vertices.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Total Variation Distance for Product Distributions is $#mathsf{P}$-Complete 产品分布的总变异距离是 $#mathsf{P}$ 完整的

arXiv - CS - Computational Complexity Pub Date : 2024-05-14 DOI: arxiv-2405.08255

Arnab Bhattacharyya, Sutanu Gayen, Kuldeep S. Meel, Dimitrios Myrisiotis, A. Pavan, N. V. Vinodchandran

引用次数: 0

P=NP P=NP

arXiv - CS - Computational Complexity Pub Date : 2024-05-13 DOI: arxiv-2405.08051

Zikang Deng

引用次数: 0

Probabilistic and Causal Satisfiability: the Impact of Marginalization 概率可满足性和因果可满足性：边缘化的影响

arXiv - CS - Computational Complexity Pub Date : 2024-05-12 DOI: arxiv-2405.07373

Julian Dörfler, Benito van der Zander, Markus Bläser, Maciej Liskiewicz

{"title":"Probabilistic and Causal Satisfiability: the Impact of Marginalization","authors":"Julian Dörfler, Benito van der Zander, Markus Bläser, Maciej Liskiewicz","doi":"arxiv-2405.07373","DOIUrl":"https://doi.org/arxiv-2405.07373","url":null,"abstract":"The framework of Pearl's Causal Hierarchy (PCH) formalizes three types of\u0000reasoning: observational, interventional, and counterfactual, that reflect the\u0000progressive sophistication of human thought regarding causation. We investigate\u0000the computational complexity aspects of reasoning in this framework focusing\u0000mainly on satisfiability problems expressed in probabilistic and causal\u0000languages across the PCH. That is, given a system of formulas in the standard\u0000probabilistic and causal languages, does there exist a model satisfying the\u0000formulas? The resulting complexity changes depending on the level of the\u0000hierarchy as well as the operators allowed in the formulas (addition,\u0000multiplication, or marginalization). We focus on formulas involving marginalization that are widely used in\u0000probabilistic and causal inference, but whose complexity issues are still\u0000little explored. Our main contribution are the exact computational complexity\u0000results showing that linear languages (allowing addition and marginalization)\u0000yield NP^PP-, PSPACE-, and NEXP-complete satisfiability problems, depending on\u0000the level of the PCH. Moreover, we prove that the problem for the full language\u0000(allowing additionally multiplication) is complete for the class succ$exists$R\u0000for languages on the highest, counterfactual level. Previous work has shown\u0000that the satisfiability problem is complete for succ$exists$R on the lower\u0000levels leaving the counterfactual case open. Finally, we consider constrained\u0000models that are restricted to a small polynomial size. The constraint on the\u0000size reduces the complexity of the interventional and counterfactual languages\u0000to NEXP-complete.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Solving Quantified Boolean Formulas with Few Existential Variables 用少量存在变量求解量化布尔公式

arXiv - CS - Computational Complexity Pub Date : 2024-05-10 DOI: arxiv-2405.06485

Leif Eriksson, Victor Lagerkvist, George Osipov, Sebastian Ordyniak, Fahad Panolan, Mateusz Rychlicki

{"title":"Solving Quantified Boolean Formulas with Few Existential Variables","authors":"Leif Eriksson, Victor Lagerkvist, George Osipov, Sebastian Ordyniak, Fahad Panolan, Mateusz Rychlicki","doi":"arxiv-2405.06485","DOIUrl":"https://doi.org/arxiv-2405.06485","url":null,"abstract":"The quantified Boolean formula (QBF) problem is an important decision problem\u0000generally viewed as the archetype for PSPACE-completeness. Many problems of\u0000central interest in AI are in general not included in NP, e.g., planning, model\u0000checking, and non-monotonic reasoning, and for such problems QBF has\u0000successfully been used as a modelling tool. However, solvers for QBF are not as\u0000advanced as state of the art SAT solvers, which has prevented QBF from becoming\u0000a universal modelling language for PSPACE-complete problems. A theoretical\u0000explanation is that QBF (as well as many other PSPACE-complete problems) lacks\u0000natural parameters} guaranteeing fixed-parameter tractability (FPT). In this paper we tackle this problem and consider a simple but overlooked\u0000parameter: the number of existentially quantified variables. This natural\u0000parameter is virtually unexplored in the literature which one might find\u0000surprising given the general scarcity of FPT algorithms for QBF. Via this\u0000parameterization we then develop a novel FPT algorithm applicable to QBF\u0000instances in conjunctive normal form (CNF) of bounded clause length. We\u0000complement this by a W[1]-hardness result for QBF in CNF of unbounded clause\u0000length as well as sharper lower bounds for the bounded arity case under the\u0000(strong) exponential-time hypothesis.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Newman's theorem via Carathéodory 通过 Carathéodory 的纽曼定理

arXiv - CS - Computational Complexity Pub Date : 2024-05-09 DOI: arxiv-2406.08500

Ali Mohammad Lavasani, Yaqiao Li, Mehran Shakerinava

引用次数: 0

Decision algorithms for reversibility of one-dimensional non-linear cellular automata under null boundary conditions 空边界条件下一维非线性蜂窝自动机可逆性的决策算法

arXiv - CS - Computational Complexity Pub Date : 2024-05-06 DOI: arxiv-2405.03609

Ma Junchi, Chen Weilin, Wang Chen, Lin Defu, Wang Chao

{"title":"Decision algorithms for reversibility of one-dimensional non-linear cellular automata under null boundary conditions","authors":"Ma Junchi, Chen Weilin, Wang Chen, Lin Defu, Wang Chao","doi":"arxiv-2405.03609","DOIUrl":"https://doi.org/arxiv-2405.03609","url":null,"abstract":"The property of reversibility is quite meaningful for the classic theoretical\u0000computer science model, cellular automata. For the reversibility problem for a\u0000CA under null boundary conditions, while linear rules have been studied a lot,\u0000the non-linear rules remain unexplored at present. The paper investigates the\u0000reversibility problem of general one-dimensional CA on a finite field\u0000$mathbb{Z}_p$, and proposes an approach to optimize the Amoroso's infinite CA\u0000surjectivity detection algorithm. This paper proposes algorithms for deciding\u0000the reversibility of one-dimensional CA under null boundary conditions. We\u0000propose a method to decide the strict reversibility of one-dimensional CA under\u0000null boundary conditions. We also provide a bucket chain based algorithm for\u0000calculating the reversibility function of one-dimensional CA under null\u0000boundary conditions. These decision algorithms work for not only linear rules\u0000but also non-linear rules. In addition, it has been confirmed that the\u0000reversibility function always has a period, and its periodicity is related to\u0000the periodicity of the corresponding bucket chain. Some of our experiment\u0000results of reversible CA are presented in the paper, complementing and\u0000validating the theoretical aspects, and thereby further supporting the research\u0000conclusions of this paper.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The Radical Solution and Computational Complexity 激进解决方案与计算复杂性

arXiv - CS - Computational Complexity Pub Date : 2024-05-04 DOI: arxiv-2405.15790

Bojin Zheng, Weiwu Wang

引用次数: 0