建构型理论中一阶逻辑的材料对话:扩展版

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-11-03 DOI:10.1017/s0960129523000348
Dominik Wehr, Dominik Kirst
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引用次数: 0

摘要

抽象对话是一种回合制游戏,它模拟了关于逻辑公式满足度的辩论。一阶结构上的一种新变体产生了一阶满足的概念。在归纳构造演算的构造背景下,研究了经典和直觉一阶逻辑的归纳有效性概念。我们证明了经典一阶逻辑的这种物质对话语义具有构造健全性和完备性证明,将其与一阶逻辑的标准模型论语义区分开来。此外,我们证明了关于直觉主义材料对话的完备性在建设性和古典背景下都失败了。作为替代方案,我们建议在Kripke结构上播放材料对话。这些克里普克材料对话在限制于否定片段时表现出建设性的完整性。利用Coq相互作用定理证明,对经典材料对话的结果进行了机械化处理。
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Material dialogues for first-order logic in constructive type theory: extended version
Abstract Dialogues are turn-taking games which model debates about the satisfaction of logical formulas. A novel variant played over first-order structures gives rise to a notion of first-order satisfaction. We study the induced notion of validity for classical and intuitionistic first-order logic in the constructive setting of the calculus of inductive constructions. We prove that such material dialogue semantics for classical first-order logic admits constructive soundness and completeness proofs, setting it apart from standard model-theoretic semantics of first-order logic. Furthermore, we prove that completeness with regard to intuitionistic material dialogues fails in both constructive and classical settings. As an alternative, we propose material dialogues played over Kripke structures. These Kripke material dialogues exhibit constructive completeness when restricting to the negative fragment. The results concerning classical material dialogues have been mechanized using the Coq interactive theorem prover.
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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