{"title":"Short Presburger Arithmetic Is Hard","authors":"Danny Nguyen, I. Pak","doi":"10.1109/FOCS.2017.13","DOIUrl":null,"url":null,"abstract":"We study the computational complexity of short sentences in Presburger arithmetic (SHORT-PA). Here by short we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of SHORT-PA sentences with m+2 alternating quantifiers is SigmaP_m-complete or PiP_m-complete, when the first quantifier is exists or forall, respectively. Counting versions and restricted systems are also analyzed.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"124 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2017-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"13","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/FOCS.2017.13","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 13
Abstract
We study the computational complexity of short sentences in Presburger arithmetic (SHORT-PA). Here by short we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of SHORT-PA sentences with m+2 alternating quantifiers is SigmaP_m-complete or PiP_m-complete, when the first quantifier is exists or forall, respectively. Counting versions and restricted systems are also analyzed.
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