Cut-elimination theorems for some logics associated with double Stone algebras

IF 3.2 3区计算机科学 Q2 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

International Journal of Approximate Reasoning Pub Date : 2025-07-09 DOI:10.1016/j.ijar.2025.109526

Martín Figallo, Juan S. Slagter

引用次数: 0

Abstract

A double Stone algebra is a Stone algebra whose dual lattice is also a Stone algebra. Logics that may be associated with double Stone algebras are based on bounded distributive lattices which are endowed with two negations: a Heyting negation (the pseudocomplement) and a Brouwer negation (the dual pseudocomplement) possibly satisfying some constraints. Different authors have studied the order-preserving logic associated with double Stone algebras. Recently, the four-valued character of this logic was exploited by providing a rough set semantics for it.

In this paper, we explore the proof-theoretical aspect of two logics associated with double Stone algebras, namely, the truth-preserving and the order-preserving logic, respectively. We provide sequent systems sound and complete for these logics and prove the cut-elimination theorem for both systems.

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双斯通代数相关逻辑的切消定理

双斯通代数是一种斯通代数，它的对偶格也是一种斯通代数。可能与双斯通代数相关的逻辑是基于有界分配格的，这些有界分配格被赋予两个否定：一个Heyting否定（伪补）和一个browwer否定（对偶伪补），可能满足某些约束。不同的作者研究了与双斯通代数相关的保序逻辑。最近，通过为该逻辑提供粗糙集语义，利用了该逻辑的四值特性。在本文中，我们分别探讨了与双石代数相关的两种逻辑的证明理论方面，即保真逻辑和保序逻辑。我们为这些逻辑提供了完备的序系统，并证明了它们的切消定理。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Approximate Reasoning 工程技术-计算机：人工智能

CiteScore

6.90

自引率

12.80%

发文量

170

审稿时长

67 days

期刊介绍： The International Journal of Approximate Reasoning is intended to serve as a forum for the treatment of imprecision and uncertainty in Artificial and Computational Intelligence, covering both the foundations of uncertainty theories, and the design of intelligent systems for scientific and engineering applications. It publishes high-quality research papers describing theoretical developments or innovative applications, as well as review articles on topics of general interest. Relevant topics include, but are not limited to, probabilistic reasoning and Bayesian networks, imprecise probabilities, random sets, belief functions (Dempster-Shafer theory), possibility theory, fuzzy sets, rough sets, decision theory, non-additive measures and integrals, qualitative reasoning about uncertainty, comparative probability orderings, game-theoretic probability, default reasoning, nonstandard logics, argumentation systems, inconsistency tolerant reasoning, elicitation techniques, philosophical foundations and psychological models of uncertain reasoning. Domains of application for uncertain reasoning systems include risk analysis and assessment, information retrieval and database design, information fusion, machine learning, data and web mining, computer vision, image and signal processing, intelligent data analysis, statistics, multi-agent systems, etc.