{"title":"短普雷斯伯格算术很难","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":"{\"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}","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}
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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