{"title":"PTIME中的逻辑层次结构","authors":"L. Hella","doi":"10.1109/LICS.1992.185548","DOIUrl":null,"url":null,"abstract":"A generalized quantifier is n-ary if it binds any finite number of formulas, but at most n variables in each formula. It is proved that for each integer n, there is a property of finite models which is expressible in fixpoint logic, or even in DATALOG, but not in the extension of first-order logic by any set of n-ary quantifiers. It follows that no extension of first-order logic by a finite set of quantifiers captures all DATALOG-definable properties. Furthermore, it is proved that for each integer n, there is a LOGSPACE-computable property of finite models which is not definable in any extension of fixpoint logic by n-ary quantifiers. Hence, the expressive power of LOGSPACE, and a fortiori, that of PTIME, cannot be captured by adding to fixpoint logic any set of quantifiers of bounded arity.<<ETX>>","PeriodicalId":6412,"journal":{"name":"[1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science","volume":"124 21 1","pages":"360-368"},"PeriodicalIF":0.0000,"publicationDate":"1992-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"168","resultStr":"{\"title\":\"Logical hierarchies in PTIME\",\"authors\":\"L. Hella\",\"doi\":\"10.1109/LICS.1992.185548\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A generalized quantifier is n-ary if it binds any finite number of formulas, but at most n variables in each formula. It is proved that for each integer n, there is a property of finite models which is expressible in fixpoint logic, or even in DATALOG, but not in the extension of first-order logic by any set of n-ary quantifiers. It follows that no extension of first-order logic by a finite set of quantifiers captures all DATALOG-definable properties. Furthermore, it is proved that for each integer n, there is a LOGSPACE-computable property of finite models which is not definable in any extension of fixpoint logic by n-ary quantifiers. Hence, the expressive power of LOGSPACE, and a fortiori, that of PTIME, cannot be captured by adding to fixpoint logic any set of quantifiers of bounded arity.<<ETX>>\",\"PeriodicalId\":6412,\"journal\":{\"name\":\"[1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science\",\"volume\":\"124 21 1\",\"pages\":\"360-368\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1992-06-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"168\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"[1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/LICS.1992.185548\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"[1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/LICS.1992.185548","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A generalized quantifier is n-ary if it binds any finite number of formulas, but at most n variables in each formula. It is proved that for each integer n, there is a property of finite models which is expressible in fixpoint logic, or even in DATALOG, but not in the extension of first-order logic by any set of n-ary quantifiers. It follows that no extension of first-order logic by a finite set of quantifiers captures all DATALOG-definable properties. Furthermore, it is proved that for each integer n, there is a LOGSPACE-computable property of finite models which is not definable in any extension of fixpoint logic by n-ary quantifiers. Hence, the expressive power of LOGSPACE, and a fortiori, that of PTIME, cannot be captured by adding to fixpoint logic any set of quantifiers of bounded arity.<>