{"title":"Effective interpolation and preservation in guarded logics","authors":"Michael Benedikt, Balder ten Cate, M. V. Boom","doi":"10.1145/2603088.2603108","DOIUrl":null,"url":null,"abstract":"Desirable properties of a logic include decidability, and a model theory that inherits properties of first-order logic, such as interpolation and preservation theorems. It is known that the Guarded Fragment (GF) of first-order logic is decidable and satisfies some preservation properties from first-order model theory; however, it fails to have Craig interpolation. The Guarded Negation Fragment (GNF), a recently-defined extension, is known to be decidable and to have Craig interpolation. Here we give the first results on effective interpolation for extensions of GF. We provide an interpolation procedure for GNF whose complexity matches the doubly exponential upper bound for satisfiability of GNF. We show that the same construction gives not only Craig interpolation, but Lyndon interpolation and Relativized interpolation, which can be used to provide effective proofs of some preservation theorems. We provide upper bounds on the size of GNF interpolants for both GNF and GF input, and complement this with matching lower bounds.","PeriodicalId":20649,"journal":{"name":"Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","volume":"30 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2014-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2603088.2603108","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2

Abstract

Desirable properties of a logic include decidability, and a model theory that inherits properties of first-order logic, such as interpolation and preservation theorems. It is known that the Guarded Fragment (GF) of first-order logic is decidable and satisfies some preservation properties from first-order model theory; however, it fails to have Craig interpolation. The Guarded Negation Fragment (GNF), a recently-defined extension, is known to be decidable and to have Craig interpolation. Here we give the first results on effective interpolation for extensions of GF. We provide an interpolation procedure for GNF whose complexity matches the doubly exponential upper bound for satisfiability of GNF. We show that the same construction gives not only Craig interpolation, but Lyndon interpolation and Relativized interpolation, which can be used to provide effective proofs of some preservation theorems. We provide upper bounds on the size of GNF interpolants for both GNF and GF input, and complement this with matching lower bounds.
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保护逻辑中的有效插值和保存
逻辑的理想性质包括可判定性和继承一阶逻辑性质的模型理论,例如插值定理和保存定理。已知一阶逻辑的保护片段(GF)是可判定的,并且满足一阶模型理论的一些保存性质;然而,它没有克雷格插值。守护否定片段(GNF)是最近定义的一个扩展,已知是可决定的并且具有克雷格插值。本文首次给出了GF扩展的有效插值结果。我们提供了一种复杂度匹配GNF可满足性的双指数上界的GNF插值方法。我们证明了同样的构造不仅可以给出Craig插值,还可以给出Lyndon插值和相对化插值,这可以用来提供一些保存定理的有效证明。我们为GNF和GF输入提供了GNF插值大小的上界,并用匹配的下界进行补充。
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