{"title":"On the Decidability of Infix Inclusion Problem","authors":"","doi":"10.1007/s00224-023-10160-w","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>We introduce the infix inclusion problem of two languages <em>S</em> and <em>T</em> that decides whether or not <em>S</em> is a subset of the set of all infixes of <em>T</em>. This problem is motivated by the need for identifying malicious computation patterns according to their semantics, which are often disguised with additional sub-patterns surrounding information. In other words, malicious patterns are embedded as an infix of the whole pattern. We examine the infix inclusion problem for the case where a source <em>S</em> and a target <em>T</em> are finite, regular or context-free languages. We prove that the problem is 1) <span>co-NP-complete</span> when one of the languages is finite, 2) <span>PSPACE-complete</span> when both <em>S</em> and <em>T</em> are regular, 3) <span>EXPTIME-complete</span> when <em>S</em> is context-free and <em>T</em> is regular, 4) undecidable when <em>S</em> is either regular or context-free and <em>T</em> is context-free and 5) undecidable when one of <em>S</em> and <em>T</em> is in a language class where the emptiness of its languages is undecidable, even if the other is finite. We, furthermore, explore the infix inclusion problem for visibly pushdown languages, a subclass of context-free languages.</p>","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Computing Systems","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1007/s00224-023-10160-w","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce the infix inclusion problem of two languages S and T that decides whether or not S is a subset of the set of all infixes of T. This problem is motivated by the need for identifying malicious computation patterns according to their semantics, which are often disguised with additional sub-patterns surrounding information. In other words, malicious patterns are embedded as an infix of the whole pattern. We examine the infix inclusion problem for the case where a source S and a target T are finite, regular or context-free languages. We prove that the problem is 1) co-NP-complete when one of the languages is finite, 2) PSPACE-complete when both S and T are regular, 3) EXPTIME-complete when S is context-free and T is regular, 4) undecidable when S is either regular or context-free and T is context-free and 5) undecidable when one of S and T is in a language class where the emptiness of its languages is undecidable, even if the other is finite. We, furthermore, explore the infix inclusion problem for visibly pushdown languages, a subclass of context-free languages.
摘要 我们引入了两种语言 S 和 T 的后缀包含问题，该问题决定了 S 是否是 T 的所有后缀集合的子集。该问题的动机是根据恶意计算模式的语义识别恶意计算模式的需要，这些恶意计算模式通常用围绕信息的附加子模式进行伪装。换句话说，恶意模式是作为整个模式的下位数嵌入的。我们研究了源 S 和目标 T 均为有限、正则或无上下文语言情况下的下位包含问题。我们证明：1）当其中一种语言是有限语言时，该问题是 co-NP-complete 的；2）当 S 和 T 都是规则语言时，该问题是 PSPACE-complete 的；3）当 S 是无上下文且 T 是规则语言时，该问题是 EXPTIME-complete 的；4）当 S 是规则语言或无上下文且 T 是无上下文时，该问题是不可判定的；5）当 S 和 T 中的一种语言属于语言类时，即使另一种语言是有限语言，其语言的空性也是不可判定的。此外，我们还探讨了无上下文语言子类--明显推倒语言的下位包含问题。
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TOCS is devoted to publishing original research from all areas of theoretical computer science, ranging from foundational areas such as computational complexity, to fundamental areas such as algorithms and data structures, to focused areas such as parallel and distributed algorithms and architectures.
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