Properties of graphs specified by a regular language

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Acta Informatica Pub Date : 2022-08-12 DOI:10.1007/s00236-022-00427-z
Volker Diekert, Henning Fernau, Petra Wolf
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引用次数: 3

Abstract

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property \(\varPhi \). What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying \(\varPhi \) in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language L if a certain torsion condition is satisfied. This condition holds trivially if L is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet \(\varSigma \), and we define a regular set \(\mathbb {G}\subseteq \varSigma ^*\) such that every nonempty word \(w\in \mathbb {G}\) defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over \(\varSigma \). Then, we ask whether the automaton \(\mathcal {A}\) specifies some graph satisfying a certain property \(\varPhi \). Our structural results show that we can answer this question for all “typical” graph properties. In order to show our results, we split L into a finite union of subsets and every subset of this union defines in a natural way a single finite graph F where some edges and vertices are marked. The marked graph in turn defines an infinite graph \(F^\infty \) and therefore the family of finite subgraphs of \(F^\infty \) where F appears as an induced subgraph. This yields a geometric description of all graphs specified by L based on splitting L into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece.

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由常规语言指定的图的属性
传统上,图算法得到一个单一的图作为输入,然后他们应该决定这个图是否满足某个属性\(\varPhi \)。如果这个问题被修改了,我们得到一个可能无限的图族作为输入,问题是是否有一个图族中满足\(\varPhi \) ?我们通过使用形式语言来指定图族,特别是正则词集来解决这个问题。我们证明了如果满足一定的扭转条件,可以通过研究规范语言L的语法单群来确定某些图的性质。如果L是正则的,这个条件一般成立。更具体地说,我们在二进制字母表\(\varSigma \)上使用有限图的自然二进制编码,并定义一个正则集\(\mathbb {G}\subseteq \varSigma ^*\),使得每个非空单词\(w\in \mathbb {G}\)定义一个有限和非空图。此外,图形属性可以在语法上定义为\(\varSigma \)上的语言。然后,我们问自动机\(\mathcal {A}\)是否指定满足某一属性的某个图\(\varPhi \)。我们的结构结果表明,我们可以对所有“典型”图属性回答这个问题。为了证明我们的结果,我们将L分割成一个子集的有限并集,这个并集的每个子集以自然的方式定义了一个有限图F,其中一些边和顶点被标记。标记的图又定义了一个无限图\(F^\infty \),因此定义了\(F^\infty \)的有限子图族,其中F作为诱导子图出现。这产生了L所指定的所有图的几何描述,基于将L分成有限多块;然后利用图可收回的概念,我们得到了每个图块中易于理解的图的描述。
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来源期刊
Acta Informatica
Acta Informatica 工程技术-计算机:信息系统
CiteScore
2.40
自引率
16.70%
发文量
24
审稿时长
>12 weeks
期刊介绍: Acta Informatica provides international dissemination of articles on formal methods for the design and analysis of programs, computing systems and information structures, as well as related fields of Theoretical Computer Science such as Automata Theory, Logic in Computer Science, and Algorithmics. Topics of interest include: • semantics of programming languages • models and modeling languages for concurrent, distributed, reactive and mobile systems • models and modeling languages for timed, hybrid and probabilistic systems • specification, program analysis and verification • model checking and theorem proving • modal, temporal, first- and higher-order logics, and their variants • constraint logic, SAT/SMT-solving techniques • theoretical aspects of databases, semi-structured data and finite model theory • theoretical aspects of artificial intelligence, knowledge representation, description logic • automata theory, formal languages, term and graph rewriting • game-based models, synthesis • type theory, typed calculi • algebraic, coalgebraic and categorical methods • formal aspects of performance, dependability and reliability analysis • foundations of information and network security • parallel, distributed and randomized algorithms • design and analysis of algorithms • foundations of network and communication protocols.
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