{"title":"关于在正则ILP实例集中找到正ILP实例的判定性","authors":"Petra Wolf","doi":"10.1007/s00236-022-00429-x","DOIUrl":null,"url":null,"abstract":"<div><p>The regular intersection emptiness problem for a decision problem <i>P</i> (<span>\\({{\\textit{int}}_{{\\mathrm {Reg}}}}\\)</span>(<i>P</i>)) is to decide whether a potentially infinite regular set of encoded <i>P</i>-instances contains a positive one. Since <span>\\({{\\textit{int}}_{{\\mathrm {Reg}}}}\\)</span>(<i>P</i>) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the <span>\\({{\\textit{int}}_{{\\mathrm {Reg}}}}\\)</span>-problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the <span>\\({{\\textit{int}}_{{\\mathrm {Reg}}}}\\)</span>-problem for the well-known NP-complete problem <span>Integer Linear Programming</span> (<span>ILP</span>). It is shown that any DFA that describes a set of <span>ILP</span>-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of <span>\\({{\\textit{int}}_{{\\mathrm {Reg}}}}\\)</span>(<span>ILP</span>).</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00236-022-00429-x.pdf","citationCount":"0","resultStr":"{\"title\":\"On the decidability of finding a positive ILP-instance in a regular set of ILP-instances\",\"authors\":\"Petra Wolf\",\"doi\":\"10.1007/s00236-022-00429-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The regular intersection emptiness problem for a decision problem <i>P</i> (<span>\\\\({{\\\\textit{int}}_{{\\\\mathrm {Reg}}}}\\\\)</span>(<i>P</i>)) is to decide whether a potentially infinite regular set of encoded <i>P</i>-instances contains a positive one. Since <span>\\\\({{\\\\textit{int}}_{{\\\\mathrm {Reg}}}}\\\\)</span>(<i>P</i>) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the <span>\\\\({{\\\\textit{int}}_{{\\\\mathrm {Reg}}}}\\\\)</span>-problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the <span>\\\\({{\\\\textit{int}}_{{\\\\mathrm {Reg}}}}\\\\)</span>-problem for the well-known NP-complete problem <span>Integer Linear Programming</span> (<span>ILP</span>). It is shown that any DFA that describes a set of <span>ILP</span>-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of <span>\\\\({{\\\\textit{int}}_{{\\\\mathrm {Reg}}}}\\\\)</span>(<span>ILP</span>).</p></div>\",\"PeriodicalId\":7189,\"journal\":{\"name\":\"Acta Informatica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-08-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00236-022-00429-x.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Informatica\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00236-022-00429-x\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INFORMATION SYSTEMS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Informatica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00236-022-00429-x","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
On the decidability of finding a positive ILP-instance in a regular set of ILP-instances
The regular intersection emptiness problem for a decision problem P (\({{\textit{int}}_{{\mathrm {Reg}}}}\)(P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since \({{\textit{int}}_{{\mathrm {Reg}}}}\)(P) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the \({{\textit{int}}_{{\mathrm {Reg}}}}\)-problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the \({{\textit{int}}_{{\mathrm {Reg}}}}\)-problem for the well-known NP-complete problem Integer Linear Programming (ILP). It is shown that any DFA that describes a set of ILP-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of \({{\textit{int}}_{{\mathrm {Reg}}}}\)(ILP).
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.