{"title":"On Interpretations of Presburger Arithmetic in Büchi Arithmetics","authors":"A. A. Zapryagaev","doi":"10.1134/S1064562423700655","DOIUrl":null,"url":null,"abstract":"<p>Büchi arithmetics <b>BA</b><sub><i>n</i></sub>, <span>\\(n \\geqslant 2\\)</span>, are extensions of Presburger arithmetic with an unary functional symbol <span>\\({{V}_{n}}(x)\\)</span> denoting Presburger arithmetic the largest power of <i>n</i> that divides <i>x</i>. Definability of a set in <b>BA</b><sub><i>n</i></sub> is equivalent to its recognizability by a finite automaton receiving numbers in their <i>n</i>-ary expansion. We consider the interpretations of in the standard model of <b>BA</b><sub><i>n</i></sub> and show that each such interpretation has an internal model isomorphic to the standard one. This answers a question by A. Visser on the interpretations of certain weak arithmetical theories in themselves.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"107 2","pages":"89 - 92"},"PeriodicalIF":0.5000,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Doklady Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S1064562423700655","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
Büchi arithmetics BAn, \(n \geqslant 2\), are extensions of Presburger arithmetic with an unary functional symbol \({{V}_{n}}(x)\) denoting Presburger arithmetic the largest power of n that divides x. Definability of a set in BAn is equivalent to its recognizability by a finite automaton receiving numbers in their n-ary expansion. We consider the interpretations of in the standard model of BAn and show that each such interpretation has an internal model isomorphic to the standard one. This answers a question by A. Visser on the interpretations of certain weak arithmetical theories in themselves.
期刊介绍：
Doklady Mathematics is a journal of the Presidium of the Russian Academy of Sciences. It contains English translations of papers published in Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences), which was founded in 1933 and is published 36 times a year. Doklady Mathematics includes the materials from the following areas: mathematics, mathematical physics, computer science, control theory, and computers. It publishes brief scientific reports on previously unpublished significant new research in mathematics and its applications. The main contributors to the journal are Members of the RAS, Corresponding Members of the RAS, and scientists from the former Soviet Union and other foreign countries. Among the contributors are the outstanding Russian mathematicians.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
