Theoretical Computer Science最新文献

英文中文

Edge open packing: Complexity, algorithmic aspects, and bounds 边缘开包：复杂性、算法方面和界限

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-24 DOI: 10.1016/j.tcs.2024.114884

Boštjan Brešar , Babak Samadi

Given a graph G, two edges

e_{1}, e_{2} \in E (G)

are said to have a common edge

e \neq e_{1}, e_{2}

if e joins an endvertex of

e_{1}

to an endvertex of

e_{2}

. A subset

B \subseteq E (G)

is an edge open packing set in G if no two edges of B have a common edge in G, and the maximum cardinality of such a set in G is called the edge open packing number,

ρ_{e}^{o} (G)

, of G. In this paper, we prove that the decision version of the edge open packing number is NP-complete even when restricted to graphs with universal vertices, Eulerian bipartite graphs, and planar graphs with maximum degree 4, respectively. In contrast, we present a linear-time algorithm that computes the edge open packing number of a tree. We also resolve two problems posed in the seminal paper (Chelladurai et al. (2022) [5]). Notably, we characterize the graphs G that attain the upper bound

ρ_{e}^{o} (G) \leq | E (G) | / δ (G)

, and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.

给定一个图 G，如果 e 连接 e1 的一个末端顶点和 e2 的一个末端顶点，则称两条边 e1,e2∈E(G) 有一条公共边 e≠e1,e2。如果 B 中没有两条边在 G 中具有公共边，则子集 B⊆E(G)是 G 中的边开包集，这样的集在 G 中的最大心数称为 G 的边开包数 ρeo(G)。相比之下，我们提出了一种计算树的边开包数的线性时间算法。我们还解决了开创性论文（Chelladurai 等人 (2022) [5]）中提出的两个问题。值得注意的是，我们描述了达到上界ρeo(G)≤|E(G)|/δ(G)的图 G 的特征，并提供了图的删边子图的下界和上界，并建立了相应的实现结果。

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引用次数: 0

Geometrical Penrose tilings are characterized by their 1-atlas 彭罗斯几何倾斜的特点是其 1-atlas

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-23 DOI: 10.1016/j.tcs.2024.114883

Thomas Fernique , Victor Lutfalla

Penrose rhombus tilings are tilings of the plane by two decorated rhombi such that the decorations match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove the decorations, we get, by definition, a sofic tiling space that we here call geometrical Penrose tilings. Here, we show how to compute the patterns of a given size which appear in these tilings by three different methods: two based on the substitutive structure of the Penrose tilings and the last on their definition by the cut and projection method. We use this to prove that the geometrical Penrose tilings are characterized by a small set of patterns called vertex-atlas, i.e., they form a tiling space of finite type. Though considered as folklore, no complete proof of this result has been published, to our knowledge.

彭罗斯菱形平铺是由两个装饰菱形组成的平面平铺，在两个平铺之间的交界处（就像拼图一样），装饰菱形相匹配。用动力学术语来说，它们构成了一个有限类型的平铺空间。如果我们去掉装饰，根据定义，就会得到一个索菲克平铺空间，我们在这里称之为几何彭罗斯平铺空间。在这里，我们展示了如何通过三种不同的方法计算出现在这些平铺中的给定大小的图案：两种方法基于彭罗斯平铺的替代结构，最后一种方法基于切割和投影法的定义。我们以此证明，几何彭罗斯罗列的特征是一小部分称为顶点-阿特拉斯的图案，即它们构成了一个有限类型的罗列空间。据我们所知，这一结果虽然被视为民间传说，但还没有发表过完整的证明。

引用次数: 0

Arbitrary pattern formation on a continuous circle by oblivious robot swarm 遗忘机器人群在连续圆上形成任意图案

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-21 DOI: 10.1016/j.tcs.2024.114882

Brati Mondal, Pritam Goswami, Avisek Sharma, Buddhadeb Sau

In the field of distributed systems, the Arbitrary Pattern Formation (APF) problem is an extensively studied problem. The purpose of APF is to design an algorithm to move a swarm of robots to a particular position in an environment (discrete or continuous) such that the swarm can form a specific but arbitrary pattern given previously to every robot as an input. In this paper, the solvability of the APF problem on a continuous circle is discussed for a swarm of oblivious and silent robots without chirality under a semi-synchronous scheduler. Firstly a class of configurations (the initial placements of the robots on the circle) called Formable Configuration (FC) has been provided which is necessary to solve the APF problem on a continuous circle. Then considering the initial configuration to be an FC, a deterministic and distributed algorithm has been provided that solves the APF problem for n robots on a continuous circle of fixed radius within

O (n)

epochs without collision, where an epoch is considered to be a time interval in which all robots are activated at least once.

在分布式系统领域，任意模式形成（APF）问题是一个被广泛研究的问题。任意模式形成问题的目的是设计一种算法，将机器人群移动到环境（离散或连续）中的某个特定位置，使机器人群能够形成一个特定但任意的模式，并将该模式事先作为输入提供给每个机器人。本文讨论了在半同步调度程序下，无手性的遗忘沉默机器人群在连续圆上的 APF 问题的可解性。首先，本文提供了一类称为可形成配置（FC）的配置（机器人在圆圈上的初始位置），这是解决连续圆圈上的 APF 问题所必需的。然后，将初始配置视为 FC，提供了一种确定性分布式算法，可在 O(n) 个历时内解决半径固定的连续圆上 n 个机器人的 APF 问题，且不会发生碰撞，其中历时被视为所有机器人至少被激活一次的时间间隔。

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引用次数: 0

L(3,2,1)-labeling of certain planar graphs 某些平面图形的 L(3,2,1)- 标记

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-20 DOI: 10.1016/j.tcs.2024.114881

Tiziana Calamoneri

Given a graph

G = (V, E)

of maximum degree Δ, denoting by

d (x, y)

the distance in G between nodes

x, y \in V

, an

L (3, 2, 1)

-labeling of G is an assignment l from V to the set of non-negative integers such that

| l (x) - l (y) | \geq 3

if x and y are adjacent,

| l (x) - l (y) | \geq 2

d (x, y) = 2

, and

| l (x) - l (y) | \geq 1

d (x, y) = 3

, for all x and y in V. The

L (3, 2, 1)

-number

λ (G)

is the smallest positive integer such that G admits an

L (3, 2, 1)

-labeling with labels from

{0, 1, \dots, λ (G)}

In this paper, the

L (3, 2, 1)

-number of certain planar graphs is determined, proving that it is linear in Δ, although the general upper bound for the

L (3, 2, 1)

-number of planar graphs is quadratic in Δ.

给定一个最大度数为 Δ 的图 G=(V,E) ，用 d(x,y) 表示 G 中节点 x,y∈V 之间的距离，L(3,2、1)-labeling is an assignment l from V to the set of non-negative integers such that |l(x)-l(y)|≥3 if x and y are adjacent, |l(x)-l(y)|≥2 if d(x,y)=2, and |l(x)-l(y)|≥1 if d(x,y)=3, for all x and y in V.本文确定了某些平面图的 L(3,2,1)数，证明它在Δ中是线性的，尽管平面图的 L(3,2,1)数的一般上限在Δ中是二次方。

引用次数: 0

Complexity and enumeration in models of genome rearrangement 基因组重排模型的复杂性和枚举性

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-20 DOI: 10.1016/j.tcs.2024.114880

Lora Bailey , Heather Smith Blake , Garner Cochran , Nathan Fox , Michael Levet , Reem Mahmoud , Elizabeth Bailey Matson , Inne Singgih , Grace Stadnyk , Xinyi Wang , Alexander Wiedemann

In this paper, we examine the computational complexity of enumeration in certain genome rearrangement models. We first show that the Pairwise Rearrangement problem in the Single Cut-and-Join model (Bergeron et al., 2010 [8]) is

# P

-complete under polynomial-time Turing reductions. Next, we show that in the Single Cut or Join model (Feijão and Meidanis, 2011 [21]), the problem of enumerating all medians (

) is logspace-computable (

FL

), improving upon the previous polynomial-time (

FP

) bound of Miklós & Smith [41].

本文研究了某些基因组重排模型中枚举的计算复杂性。我们首先证明，在多项式时间图灵还原下，单切和连接模型（Bergeron 等，2010 [8]）中的配对重排问题是 #P-complete 的。接下来，我们证明了在单切或联接模型（Feijão 和 Meidanis，2011 [21]）中，枚举所有中值（）的问题是对数空间可计算的（FL），改进了 Miklós & Smith [41] 以前的多项式时间（FP）约束。

引用次数: 0

Probabilistic unifying relations for modelling epistemic and aleatoric uncertainty: Semantics and automated reasoning with theorem proving 为认识论和不确定性建模的概率统一关系：语义学和定理证明的自动推理

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-18 DOI: 10.1016/j.tcs.2024.114876

Kangfeng Ye, Jim Woodcock, Simon Foster

Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty. Probabilistic programs are ubiquitous, including having a significant impact on machine intelligence. While many probabilistic algorithms have been used in practice in different domains, their automated verification based on formal semantics is still a relatively new research area. In the last two decades, it has attracted much interest. Many challenges, however, remain. The work presented in this paper, probabilistic unifying relations (ProbURel), takes a step towards our vision to tackle these challenges.

Our work is based on Hehner's predicative probabilistic programming, but there are several obstacles to the broader adoption of his work. Our contributions here include (1) the formalisation of its syntax and semantics by introducing an Iverson bracket notation to separate relations from arithmetic; (2) the formalisation of relations using Unifying Theories of Programming (UTP) and probabilities outside the brackets using summation over the topological space of the real numbers; (3) the constructive semantics for probabilistic loops using Kleene's fixed-point theorem; (4) the enrichment of its semantics from distributions to subdistributions and superdistributions to deal with the constructive semantics; (5) the unique fixed-point theorem to simplify the reasoning about probabilistic loops; and (6) the mechanisation of our theory in Isabelle/UTP, an implementation of UTP in Isabelle/HOL, for automated reasoning using theorem proving.

We demonstrate our work with six examples, including problems in robot localisation, classification in machine learning, and the termination of probabilistic loops.

概率编程结合了通用计算机编程、统计推理和形式语义，可帮助系统在面临不确定性时做出决策。概率编程无处不在，对机器智能也有重大影响。虽然许多概率算法已在不同领域得到实际应用，但基于形式语义的自动验证仍是一个相对较新的研究领域。在过去的二十年里，它引起了广泛的关注。然而，许多挑战依然存在。本文介绍的工作--概率统一关系（ProbURel）--朝着我们应对这些挑战的愿景迈出了一步。我们的工作基于 Hehner 的谓词概率编程，但要更广泛地采用他的工作还存在一些障碍。我们在此的贡献包括：(1) 通过引入艾弗森括号符号将其语法和语义形式化，从而将关系与算术分开；(2) 使用统一编程理论（UTP）将关系形式化，并使用实数拓扑空间求和将括号外的概率形式化；(3) 使用克莱因定点定理为概率循环提供构造语义；(4) 丰富其语义，从分布到子分布和超分布，以处理构造语义；(5) 唯一定点定理，以简化概率循环的推理；以及 (6) 在 Isabelle/UTP 中机械化我们的理论，这是UTP 在 Isabelle/HOL 中的实现，用于使用定理证明进行自动推理。我们用六个例子演示了我们的工作，包括机器人定位、机器学习分类和概率循环终止等问题。

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引用次数: 0

Polynomial-time checking of generalized Sahlqvist syntactic shape 广义 Sahlqvist 句法形状的多项式时间检查

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-18 DOI: 10.1016/j.tcs.2024.114875

Krishna B. Manoorkar , Alessandra Palmigiano , Mattia Panettiere

The best known modal logics are axiomatized by Sahlqvist axioms, i.e., axioms of a syntactic shape which guarantees these formulas to have such excellent properties as canonicity and elementarity. Recently, the definition of Sahlqvist formulas has been generalized and extended from formulas in classical modal logic to inequalities (sequents) in a wide family of logics known as LE-logics. We introduce an algorithm which checks if a given inequality is generalized Sahlqvist in polynomial time.

最著名的模态逻辑是由萨克维斯特公理（Sahlqvist axioms）公理化的，即一种语法形式的公理，这种公理保证了这些公式具有完备性和元素性等优良性质。最近，Sahlqvist 公式的定义得到了推广，从经典模态逻辑中的公式扩展到了被称为 LE 逻辑的一大系列逻辑中的不等式（序列）。我们引入了一种算法，可以在多项式时间内检查给定不等式是否为广义 Sahlqvist。

引用次数: 0

Multiple-model polynomial regression and efficient algorithms for data analysis 多模型多项式回归和数据分析的高效算法

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-17 DOI: 10.1016/j.tcs.2024.114878

Bohan Lyu , Jianzhong Li

This paper newly proposes a data analysis method using multiple-model p-order polynomial regression (MMPR), which separates given datasets into subsets and constructs respective polynomial regression models for them. An approximate algorithm to construct MMPR models based on $(ϵ, δ)$ -estimator, and mathematical proofs of the correctness and efficiency of the algorithm are introduced. This paper empirically implements the method on both synthetic and real-world datasets, and it's shown to have comparable performance to existing regression methods in many cases, while it takes almost the shortest time to provide a regression model with high prediction accuracy.

本文新近提出了一种使用多模型 p 阶多项式回归（MMPR）的数据分析方法，该方法将给定数据集分成若干子集，并为其构建相应的多项式回归模型。本文介绍了基于（ϵ,δ）估计器构建 MMPR 模型的近似算法，并对算法的正确性和效率进行了数学证明。本文在合成数据集和实际数据集上对该方法进行了实证，结果表明该方法在很多情况下与现有的回归方法性能相当，而且几乎用最短的时间就能提供预测准确率很高的回归模型。

引用次数: 0

Generating Java code pairing with ChatGPT 生成与 ChatGPT 配对的 Java 代码

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-17 DOI: 10.1016/j.tcs.2024.114879

Zelong Zhao, Nan Zhang, Bin Yu, Zhenhua Duan

The Large Language Models (LLMs) like ChatGPT 3.5 have created a new era of automatic code generation. However, the existing research primarily focuses on generating simple code based on datasets (such as HumanEval, etc.). Most of approaches pay less attention to complex and practical code generation. Therefore, in this paper, we propose a new approach called “Xd-CodeGen” which can be used to generate large scale Java code. This approach is composed of four phases: requirement analysis, modeling, code generation, and code verification. In the requirement analysis phase, ChatGPT 3.5 is utilized to decompose and restate user requirements. To do so, a knowledge graph is developed to describe entities and their relationship in detail. Further, Propositional Projection Temporal Logic (PPTL) formulas are employed to define the properties of requirements. In the modeling phase, we use knowledge graphs to enhance prompts and generate UML class and activity diagrams for each sub-requirement using ChatGPT 3.5. In the code generation phase, based on established UML models, we make use of prompt engineering and knowledge graph to generate Java code. In the code verification phase, a runtime verification at code level approach is employed to verify generated Java code. Finally, we apply the proposed approach to develop a practical Java web project.

ChatGPT 3.5 等大型语言模型（LLM）开创了自动代码生成的新时代。然而，现有的研究主要集中在基于数据集（如 HumanEval 等）生成简单代码。大多数方法较少关注复杂和实用的代码生成。因此，在本文中，我们提出了一种名为 "Xd-CodeGen "的新方法，可用于生成大规模 Java 代码。该方法由四个阶段组成：需求分析、建模、代码生成和代码验证。在需求分析阶段，利用 ChatGPT 3.5 对用户需求进行分解和重述。为此，开发了一个知识图谱来详细描述实体及其关系。此外，我们还使用命题投影时态逻辑（PPTL）公式来定义需求的属性。在建模阶段，我们使用知识图谱来增强提示，并使用 ChatGPT 3.5 为每个子需求生成 UML 类图和活动图。在代码生成阶段，基于已建立的 UML 模型，我们利用提示工程和知识图谱生成 Java 代码。在代码验证阶段，我们采用代码级运行时验证方法来验证生成的 Java 代码。最后，我们应用所提出的方法开发了一个实用的 Java 网络项目。

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引用次数: 0

Counting vanishing matrix-vector products 计算消失的矩阵向量积

IF 0.9 4区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS

Theoretical Computer Science

Pub Date : 2024-09-16 DOI: 10.1016/j.tcs.2024.114877

Cornelius Brand , Viktoriia Korchemna , Kirill Simonov , Michael Skotnica

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces. Let $v \in Q^{d}$ be a rational vector, $(T_{1}, T_{2} \dots, T_{m})$ a list of $d \times d$ rational matrices, $S \in Q^{h \times d}$ a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices $T_{i_{1}}, T_{i_{2}}, \dots, T_{i_{k}}$ from the list such that $S T_{i_{k}} \dots T_{i_{1}} v = 0 \in Q^{h}$ .

In this paper, we show that this problem is $# W [2]$ -hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for $d > 3$ is $# W [2]$ -hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show $W [1] / W [2]$ -hardness. This is in contrast to the parameterized k-sum problem, which is only $W [1]$ -hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.

考虑下面这个经典子集和问题的参数化计数变化，它主要出现在拓扑空间的高同调群中。设 v∈Qd 是一个有理向量，(T1,T2...,Tm) 是一个 d×d 有理矩阵列表，S∈Qh×d 是一个不一定是正方形的有理矩阵，k 是一个参数。我们的目标是从列表中选择 k 个矩阵 Ti1,Ti2,...,Tik，使得 STik⋯Ti1v=0∈Qh 的方法数。因此，计算 d>3 的 d 维 1-connected 拓扑空间的 k-th 同调群对于参数 k 是 #W[2]-困难的。我们还讨论了该问题的决策版本及其若干修改，并证明了其 W[1]/W[2]-hardness 性。这与参数化 k 和问题形成鲜明对比，后者只有 W[1]-hardness (Abboud-Lewi-Williams, ESA'14)。此外，我们还证明了该问题的无参数决策版本是一个无法判定的问题，并给出了一种针对有限域上有界矩阵的固定参数可控算法，该算法以矩阵维数和域的阶数为参数。

引用次数: 0

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