{"title":"非标准分析的有状态实现器","authors":"Bruno Dinis, Étienne Miquey","doi":"10.46298/lmcs-19(2:7)2023","DOIUrl":null,"url":null,"abstract":"In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of (semi)intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the $\\lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $\\cal{M}^{\\mathbb{N}}$ the computation is being done. We pay attention to the nonstandard principles (and their computational content) obtainable in this setting. In particular, we give non-trivial realizers to Idealization and a non-standard version of the LLPO principle. We then discuss how to quotient this product to mimic the Lightstone-Robinson construction.","PeriodicalId":49904,"journal":{"name":"Logical Methods in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stateful Realizers for Nonstandard Analysis\",\"authors\":\"Bruno Dinis, Étienne Miquey\",\"doi\":\"10.46298/lmcs-19(2:7)2023\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of (semi)intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the $\\\\lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $\\\\cal{M}^{\\\\mathbb{N}}$ the computation is being done. We pay attention to the nonstandard principles (and their computational content) obtainable in this setting. In particular, we give non-trivial realizers to Idealization and a non-standard version of the LLPO principle. We then discuss how to quotient this product to mimic the Lightstone-Robinson construction.\",\"PeriodicalId\":49904,\"journal\":{\"name\":\"Logical Methods in Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-04-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Logical Methods in Computer Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.46298/lmcs-19(2:7)2023\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Logical Methods in Computer Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.46298/lmcs-19(2:7)2023","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of (semi)intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the $\lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $\cal{M}^{\mathbb{N}}$ the computation is being done. We pay attention to the nonstandard principles (and their computational content) obtainable in this setting. In particular, we give non-trivial realizers to Idealization and a non-standard version of the LLPO principle. We then discuss how to quotient this product to mimic the Lightstone-Robinson construction.
期刊介绍:
Logical Methods in Computer Science is a fully refereed, open access, free, electronic journal. It welcomes papers on theoretical and practical areas in computer science involving logical methods, taken in a broad sense; some particular areas within its scope are listed below. Papers are refereed in the traditional way, with two or more referees per paper. Copyright is retained by the author.
Topics of Logical Methods in Computer Science:
Algebraic methods
Automata and logic
Automated deduction
Categorical models and logic
Coalgebraic methods
Computability and Logic
Computer-aided verification
Concurrency theory
Constraint programming
Cyber-physical systems
Database theory
Defeasible reasoning
Domain theory
Emerging topics: Computational systems in biology
Emerging topics: Quantum computation and logic
Finite model theory
Formalized mathematics
Functional programming and lambda calculus
Inductive logic and learning
Interactive proof checking
Logic and algorithms
Logic and complexity
Logic and games
Logic and probability
Logic for knowledge representation
Logic programming
Logics of programs
Modal and temporal logics
Program analysis and type checking
Program development and specification
Proof complexity
Real time and hybrid systems
Reasoning about actions and planning
Satisfiability
Security
Semantics of programming languages
Term rewriting and equational logic
Type theory and constructive mathematics.