{"title":"Decidability bounds for Presburger arithmetic extended by sine","authors":"","doi":"10.1016/j.apal.2024.103487","DOIUrl":null,"url":null,"abstract":"<div><p>We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic (<strong>sin-PA</strong>), and systematically study decision problems for sets of sentences in <strong>sin-PA</strong>. In particular, we detail a decision algorithm for existential sin-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of sin-PA sentences. To do so, we explicitly interpret the weak monadic second-order theory of the grid, which is undecidable, in <strong>sin-PA</strong>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103487"},"PeriodicalIF":0.6000,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000915/pdfft?md5=8ec66c0193137f3b2153f14d4d1e4bed&pid=1-s2.0-S0168007224000915-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pure and Applied Logic","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168007224000915","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
Abstract
We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic (sin-PA), and systematically study decision problems for sets of sentences in sin-PA. In particular, we detail a decision algorithm for existential sin-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of sin-PA sentences. To do so, we explicitly interpret the weak monadic second-order theory of the grid, which is undecidable, in sin-PA.
期刊介绍：
The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.
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