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| Completeness of Hoare logic with inputs over the standard model | |
| Xu, ZW; Sui, YF; Zhang, WH | |
| 2016 | |
| Source | THEORETICAL COMPUTER SCIENCE
 |
| ISSN | 0304-3975 |
| Volume | 612Pages:23-28 |
| English Abstract | Hoare logic for the set of while-programs with the first-order logical language L and the first-order theory T subset of L is denoted by HL(T). Bergstra and Tucker have pointed out that the complete number theory Th(N) is the only extension T of Peano arithmetic PA for which HL(T) is logically complete. The completeness result is not satisfying, since it allows inputs to range over nonstandard models. The aim of this paper is to investigate under what circumstances HL(T) is logically complete when inputs range over the standard model N. PA(+) is defined by adding to PA all the unprovable Pi(1)-sentences that describe the nonterminating computations. It is shown that each computable function in N is uniformly Sigma(1)-definable in all models of PA(+), and that PA(+) is arithmetical. Finally, it is established, based on the reduction from HL(T) to T, that PA(+) is the minimal extension T of PA for which HL(T) is logically complete when inputs range over N. This completeness result has an advantage over Bergstra's and Tucker's one, in that PA(+) is arithmetical while Th(N) is not. (C) 2015 Elsevier B.V. All rights reserved.; Hoare logic for the set of while-programs with the first-order logical language L and the first-order theory T subset of L is denoted by HL(T). Bergstra and Tucker have pointed out that the complete number theory Th(N) is the only extension T of Peano arithmetic PA for which HL(T) is logically complete. The completeness result is not satisfying, since it allows inputs to range over nonstandard models. The aim of this paper is to investigate under what circumstances HL(T) is logically complete when inputs range over the standard model N. PA(+) is defined by adding to PA all the unprovable Pi(1)-sentences that describe the nonterminating computations. It is shown that each computable function in N is uniformly Sigma(1)-definable in all models of PA(+), and that PA(+) is arithmetical. Finally, it is established, based on the reduction from HL(T) to T, that PA(+) is the minimal extension T of PA for which HL(T) is logically complete when inputs range over N. This completeness result has an advantage over Bergstra's and Tucker's one, in that PA(+) is arithmetical while Th(N) is not. (C) 2015 Elsevier B.V. All rights reserved. |
| Indexed Type | SCI |
| Keyword | Hoare Logic Peano Arithmetic The Standard Model Computation Arithmetical Definability Logical Completeness |
| Department | Chinese Acad Sci, Inst Software, State Key Lab Comp Sci, Beijing, Peoples R China. Chinese Acad Sci, Inst Comp Technol, Key Lab Intelligent Informat Proc, Beijing, Peoples R China. |
| Language | 英语 |
| WOS ID | WOS:000369211200002 |
| Citation statistics | |
| Content Type | 期刊论文 |
| URI | http://ir.iscas.ac.cn/handle/311060/17412 |
| Collection | 中国科学院软件研究所 |
| Recommended Citation GB/T 7714 | Xu, ZW,Sui, YF,Zhang, WH. Completeness of Hoare logic with inputs over the standard model[J]. THEORETICAL COMPUTER SCIENCE,2016,612:23-28. |
| APA | Xu, ZW,Sui, YF,&Zhang, WH.(2016).Completeness of Hoare logic with inputs over the standard model.THEORETICAL COMPUTER SCIENCE,612,23-28. |
| MLA | Xu, ZW,et al."Completeness of Hoare logic with inputs over the standard model".THEORETICAL COMPUTER SCIENCE 612(2016):23-28. |
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