Alexander Golovnev, Zeyu Guo, Pooya Hatami, Satyajeet Nagargoje, Chao Yan
For $Ssubseteq mathbb{F}^n$, consider the linear space of restrictions of degree-$d$ polynomials to $S$. The Hilbert function of $S$, denoted $mathrm{h}_S(d,mathbb{F})$, is the dimension of this space. We obtain a tight lower bound on the smallest value of the Hilbert function of subsets $S$ of arbitrary finite grids in $mathbb{F}^n$ with a fixed size $|S|$. We achieve this by proving that this value coincides with a combinatorial quantity, namely the smallest number of low Hamming weight points in a down-closed set of size $|S|$. Understanding the smallest values of Hilbert functions is closely related to the study of degree-$d$ closure of sets, a notion introduced by Nie and Wang (Journal of Combinatorial Theory, Series A, 2015). We use bounds on the Hilbert function to obtain a tight bound on the size of degree-$d$ closures of subsets of $mathbb{F}_q^n$, which answers a question posed by Doron, Ta-Shma, and Tell (Computational Complexity, 2022). We use the bounds on the Hilbert function and degree-$d$ closure of sets to prove that a random low-degree polynomial is an extractor for samplable randomness sources. Most notably, we prove the existence of low-degree extractors and dispersers for sources generated by constant-degree polynomials and polynomial-size circuits. Until recently, even the existence of arbitrary deterministic extractors for such sources was not known.
{"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":null,"pages":null},"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}
Chemical reaction networks (CRNs) model systems where molecules interact according to a finite set of reactions such as (A + B to C), representing that if a molecule of (A) and (B) collide, they disappear and a molecule of (C) is produced. CRNs can compute Boolean-valued predicates (phi:mathbb{N}^d to {0,1}) and integer-valued functions (f:mathbb{N}^d to mathbb{N}); for instance (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 finite number of reactions can occur from the initial configuration (e.g., ruling out reversible reactions such as (A rightleftharpoons B)). The power and composability of such CRNs depend crucially on some other modeling choices that do not affect the computational power of CRNs with unbounded executions, namely whether an initial leader is present, and whether (for predicates) all species are required to "vote" for the Boolean output. If the CRN starts with an initial leader, and can allow only the leader to vote, then all semilinear predicates and functions can be stably computed in (O(n log n)) parallel time by execution bounded CRNs. However, if no initial leader is allowed, all species vote, and the CRN is "noncollapsing" (does not shrink from initially large to final (O(1)) size configurations), then execution bounded CRNs are severely limited, able to compute only eventually constant predicates. A key tool is to characterize execution bounded CRNs as precisely those with a nonnegative linear potential function that is strictly decreased by every reaction, a result that may be of independent interest.
{"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":null,"pages":null},"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}
MIT Hardness Group, Josh Brunner, Della Hendrickson, Lily Chung, Erik D. Demaine, Andy Tockman
We prove that Hamiltonicity in maximum-degree-3 grid graphs (directed or undirected) is ASP-complete, i.e., it has a parsimonious reduction from every NP search problem (including a polynomial-time bijection between solutions). As a consequence, given k Hamiltonian cycles, it is NP-complete to find another; and counting Hamiltonian cycles is #P-complete. If we require the grid graph's vertices to form a full $m times n$ rectangle, then we show that Hamiltonicity remains ASP-complete if the edges are directed or if we allow removing some edges (whereas including all undirected edges is known to be easy). These results enable us to develop a stronger "T-metacell" framework for proving ASP-completeness of rectangular puzzles, which requires building just a single gadget representing a degree-3 grid-graph vertex. We apply this general theory to prove ASP-completeness of 38 pencil-and-paper puzzles where the goal is to draw a loop subject to given constraints: Slalom, Onsen-meguri, Mejilink, Detour, Tapa-Like Loop, Kouchoku, Icelom; Masyu, Yajilin, Nagareru, Castle Wall, Moon or Sun, Country Road, Geradeweg, Maxi Loop, Mid-loop, Balance Loop, Simple Loop, Haisu, Reflect Link, Linesweeper; Vertex/Touch Slitherlink, Dotchi-Loop, Ovotovata, Building Walk, Rail Pool, Disorderly Loop, Ant Mill, Koburin, Mukkonn Enn, Rassi Silai, (Crossing) Ichimaga, Tapa, Canal View, Aqre, and Paintarea. The last 14 of these puzzles were not even known to be NP-hard. Along the way, we prove ASP-completeness of some simple forms of Tree-Residue Vertex-Breaking (TRVB), including planar multigraphs with degree-6 breakable vertices, or with degree-4 breakable and degree-1 unbreakable vertices.
{"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":null,"pages":null},"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}
Arnab Bhattacharyya, Sutanu Gayen, Kuldeep S. Meel, Dimitrios Myrisiotis, A. Pavan, N. V. Vinodchandran
We show that computing the total variation distance between two product distributions is $#mathsf{P}$-complete. This is in stark contrast with other distance measures such as Kullback-Leibler, Chi-square, and Hellinger, which tensorize over the marginals leading to efficient algorithms.
{"title":"Total Variation Distance for Product Distributions is $#mathsf{P}$-Complete","authors":"Arnab Bhattacharyya, Sutanu Gayen, Kuldeep S. Meel, Dimitrios Myrisiotis, A. Pavan, N. V. Vinodchandran","doi":"arxiv-2405.08255","DOIUrl":"https://doi.org/arxiv-2405.08255","url":null,"abstract":"We show that computing the total variation distance between two product\u0000distributions is $#mathsf{P}$-complete. This is in stark contrast with other\u0000distance measures such as Kullback-Leibler, Chi-square, and Hellinger, which\u0000tensorize over the marginals leading to efficient algorithms.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062605","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}
This paper investigates an extremely classic NP-complete problem: How to determine if a graph G, where each vertex has a degree of at most 4, can be 3-colorable(The research in this paper focuses on graphs G that satisfy the condition where the degree of each vertex does not exceed 4. To conserve space, it is assumed throughout the paper that graph G meets this condition by default.). The author has meticulously observed the relationship between the coloring problem and semidefinite programming, and has creatively constructed the corresponding semidefinite programming problem R(G) for a given graph G. The construction method of R(G) refers to Theorem 1.1 in the paper. I have obtained and proven the conclusion: A graph G is 3-colorable if and only if the objective function of its corresponding optimization problem R(G) is bounded, and when the objective function is bounded, its minimum value is 0.
本文研究了一个极其经典的 NP-完备问题：如何确定一个每个顶点的度最多为 4 的图 G 是否可以 3-着色（本文的研究重点是满足每个顶点的度不超过 4 的条件的图 G。为了节省篇幅，本文假设图 G 默认满足这个条件）。笔者仔细观察了着色问题与半定式编程之间的关系，并针对给定的图 G 创造性地构造了相应的半定式编程问题 R(G)，R(G)的构造方法参考了论文中的定理 1.1。我已经得到并证明了结论：当且仅当相应优化问题 R(G) 的目标函数有界，且目标函数有界时，其最小值为 0 时，图 G 是可 3 色的。
{"title":"P=NP","authors":"Zikang Deng","doi":"arxiv-2405.08051","DOIUrl":"https://doi.org/arxiv-2405.08051","url":null,"abstract":"This paper investigates an extremely classic NP-complete problem: How to\u0000determine if a graph G, where each vertex has a degree of at most 4, can be\u00003-colorable(The research in this paper focuses on graphs G that satisfy the\u0000condition where the degree of each vertex does not exceed 4. To conserve space,\u0000it is assumed throughout the paper that graph G meets this condition by\u0000default.). The author has meticulously observed the relationship between the\u0000coloring problem and semidefinite programming, and has creatively constructed\u0000the corresponding semidefinite programming problem R(G) for a given graph G.\u0000The construction method of R(G) refers to Theorem 1.1 in the paper. I have\u0000obtained and proven the conclusion: A graph G is 3-colorable if and only if the\u0000objective function of its corresponding optimization problem R(G) is bounded,\u0000and when the objective function is bounded, its minimum value is 0.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062565","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}
Julian Dörfler, Benito van der Zander, Markus Bläser, Maciej Liskiewicz
The framework of Pearl's Causal Hierarchy (PCH) formalizes three types of reasoning: observational, interventional, and counterfactual, that reflect the progressive sophistication of human thought regarding causation. We investigate the computational complexity aspects of reasoning in this framework focusing mainly on satisfiability problems expressed in probabilistic and causal languages across the PCH. That is, given a system of formulas in the standard probabilistic and causal languages, does there exist a model satisfying the formulas? The resulting complexity changes depending on the level of the hierarchy as well as the operators allowed in the formulas (addition, multiplication, or marginalization). We focus on formulas involving marginalization that are widely used in probabilistic and causal inference, but whose complexity issues are still little explored. Our main contribution are the exact computational complexity results showing that linear languages (allowing addition and marginalization) yield NP^PP-, PSPACE-, and NEXP-complete satisfiability problems, depending on the level of the PCH. Moreover, we prove that the problem for the full language (allowing additionally multiplication) is complete for the class succ$exists$R for languages on the highest, counterfactual level. Previous work has shown that the satisfiability problem is complete for succ$exists$R on the lower levels leaving the counterfactual case open. Finally, we consider constrained models that are restricted to a small polynomial size. The constraint on the size reduces the complexity of the interventional and counterfactual languages to NEXP-complete.
{"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":null,"pages":null},"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}
Leif Eriksson, Victor Lagerkvist, George Osipov, Sebastian Ordyniak, Fahad Panolan, Mateusz Rychlicki
The quantified Boolean formula (QBF) problem is an important decision problem generally viewed as the archetype for PSPACE-completeness. Many problems of central interest in AI are in general not included in NP, e.g., planning, model checking, and non-monotonic reasoning, and for such problems QBF has successfully been used as a modelling tool. However, solvers for QBF are not as advanced as state of the art SAT solvers, which has prevented QBF from becoming a universal modelling language for PSPACE-complete problems. A theoretical explanation is that QBF (as well as many other PSPACE-complete problems) lacks natural parameters} guaranteeing fixed-parameter tractability (FPT). In this paper we tackle this problem and consider a simple but overlooked parameter: the number of existentially quantified variables. This natural parameter is virtually unexplored in the literature which one might find surprising given the general scarcity of FPT algorithms for QBF. Via this parameterization we then develop a novel FPT algorithm applicable to QBF instances in conjunctive normal form (CNF) of bounded clause length. We complement this by a W[1]-hardness result for QBF in CNF of unbounded clause length as well as sharper lower bounds for the bounded arity case under the (strong) exponential-time hypothesis.
{"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":null,"pages":null},"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}
Ali Mohammad Lavasani, Yaqiao Li, Mehran Shakerinava
We give a streamlined short proof of Newman's theorem in communication complexity by applying the classical and the approximate Carath'eodory's theorems.
{"title":"Newman's theorem via Carathéodory","authors":"Ali Mohammad Lavasani, Yaqiao Li, Mehran Shakerinava","doi":"arxiv-2406.08500","DOIUrl":"https://doi.org/arxiv-2406.08500","url":null,"abstract":"We give a streamlined short proof of Newman's theorem in communication\u0000complexity by applying the classical and the approximate Carath'eodory's\u0000theorems.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141518334","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}
Ma Junchi, Chen Weilin, Wang Chen, Lin Defu, Wang Chao
The property of reversibility is quite meaningful for the classic theoretical computer science model, cellular automata. For the reversibility problem for a CA under null boundary conditions, while linear rules have been studied a lot, the non-linear rules remain unexplored at present. The paper investigates the reversibility problem of general one-dimensional CA on a finite field $mathbb{Z}_p$, and proposes an approach to optimize the Amoroso's infinite CA surjectivity detection algorithm. This paper proposes algorithms for deciding the reversibility of one-dimensional CA under null boundary conditions. We propose a method to decide the strict reversibility of one-dimensional CA under null boundary conditions. We also provide a bucket chain based algorithm for calculating the reversibility function of one-dimensional CA under null boundary conditions. These decision algorithms work for not only linear rules but also non-linear rules. In addition, it has been confirmed that the reversibility function always has a period, and its periodicity is related to the periodicity of the corresponding bucket chain. Some of our experiment results of reversible CA are presented in the paper, complementing and validating the theoretical aspects, and thereby further supporting the research conclusions of this paper.
可逆性这一特性对于经典的计算机科学理论模型--蜂窝自动机--来说意义重大。对于空边界条件下的细胞自动机的可逆性问题，虽然线性规则已被大量研究，但非线性规则目前仍未被探索。本文研究了有限域$mathbb{Z}_p$上一般一维CA的可逆性问题，并提出了一种优化阿莫罗索无限CA可射性检测算法的方法。本文提出了在空边界条件下判定一维 CA 可逆性的算法。我们提出了一种在空边界条件下判定一维 CA 严格可逆性的方法。我们还提供了一种基于桶链的算法，用于计算空边界条件下一维 CA 的可逆性函数。这些决策算法不仅适用于线性规则，也适用于非线性规则。此外，还证实了可逆性函数总是有周期的，而且其周期性与相应桶链的周期性有关。本文介绍了我们对可逆 CA 的一些实验结果，对理论方面进行了补充和验证，从而进一步支持了本文的研究结论。
{"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":null,"pages":null},"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}
The radical solution of polynomials with rational coefficients is a famous solved problem. This paper found that it is a $mathbb{NP}$ problem. Furthermore, this paper found that arbitrary $ mathscr{P} in mathbb{P}$ shall have a one-way running graph $G$, and have a corresponding $mathscr{Q} in mathbb{NP}$ which have a two-way running graph $G'$, $G$ and $G'$ is isomorphic, i.e., $G'$ is combined by $G$ and its reverse $G^{-1}$. When $mathscr{P}$ is an algorithm for solving polynomials, $G^{-1}$ is the radical formula. According to Galois' Theory, a general radical formula does not exist. Therefore, there exists an $mathbb{NP}$, which does not have a general, deterministic and polynomial time-complexity algorithm, i.e., $mathbb{P} neq mathbb{NP}$. Moreover, this paper pointed out that this theorem actually is an impossible trinity.
{"title":"The Radical Solution and Computational Complexity","authors":"Bojin Zheng, Weiwu Wang","doi":"arxiv-2405.15790","DOIUrl":"https://doi.org/arxiv-2405.15790","url":null,"abstract":"The radical solution of polynomials with rational coefficients is a famous\u0000solved problem. This paper found that it is a $mathbb{NP}$ problem.\u0000Furthermore, this paper found that arbitrary $ mathscr{P} in mathbb{P}$\u0000shall have a one-way running graph $G$, and have a corresponding $mathscr{Q}\u0000in mathbb{NP}$ which have a two-way running graph $G'$, $G$ and $G'$ is\u0000isomorphic, i.e., $G'$ is combined by $G$ and its reverse $G^{-1}$. When\u0000$mathscr{P}$ is an algorithm for solving polynomials, $G^{-1}$ is the radical\u0000formula. According to Galois' Theory, a general radical formula does not exist.\u0000Therefore, there exists an $mathbb{NP}$, which does not have a general,\u0000deterministic and polynomial time-complexity algorithm, i.e., $mathbb{P} neq\u0000mathbb{NP}$. Moreover, this paper pointed out that this theorem actually is an\u0000impossible trinity.","PeriodicalId":501024,"journal":{"name":"arXiv - CS - Computational Complexity","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165499","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}
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