命题逻辑公式的压缩表示及其相应的形式系统

提出一种将命题逻辑公式压缩表示的方法--公式的压缩图,给出相应的形式系统,并证明该系统的证明效率比传统相继式演算系统Gentzen\{cut}有指数级的提高,从而为命题逻辑提供了一种新的有效的推理系统.     

Translations from natural deduction to sequent calculus

Gentzen's “Untersuchungen” [1] gave a translation from natural deduction to sequent calculus with the property that normal derivations may translate into derivations with cuts. Prawitz in [8] gave a translation that instead produced cut‐free derivations. It is shown that by writing all elimination rules in the manner of disjunction elimination, with an arbitrary consequence, an isomorphic translation between normal derivations and cut‐free derivations is achieved. The standard elimination rules do not permit a full normal form, which explains the cuts in Gentzen's translation. Likewise, it is shown that Prawitz' translation contains an implicit process of cut elimination.     

扰动模糊命题逻辑系统中的广义重言式

通过定义二维R0-蕴涵算子，将王国俊教授在逻辑系统形中的广义重言式理论推广并应用到二维赋值的扰动模糊命题逻辑系统西中，证明了这一系统中（μ，δ）-重言式就是某个（λ，1-λ）-重言式，最终获得与一维线性赋值格上完全相应的广义重言式分类。     

A Sequent Calculus for Propositional Dynamic Logic for Agents

We consider propositional dynamic logic for agents. For this logic, we present a sequent calculus with a restricted cut rule and prove the soundness and completeness for the calculus.     

A Sequent Calculus for Propositional Dynamic Logic for Agents with Interactions

We consider a propositional dynamic logic for agents with interactions such as known commitment, no learning, and perfect recall. For this logic, we present a sequent calculus with a restricted cut rule and prove the soundness and completeness for the calculus.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 261–269, April–June, 2005.     

公式真度的大小之比较

在不同的逻辑系统中,同一公式的真度往往差别很大.因此,本文系统探讨了六个n值命题逻辑系统中公式真度的分布情况.首先,计算了六个n值命题逻辑系统中一个典型公式的真度;然后通过列表比较了该公式在这六个逻辑系统中真度的大小.最后,分析了该公式在每个逻辑系统中的真度随n变化的情况.     

经典命题逻辑中的近似推理与强近似推理

基于命题逻辑公式之间相似度的概念,在经典命题逻辑系统中提出了两种近似推理模式,得到了它们的一些基本性质。讨论了这两种推理模式与经典推理模式之间的关系。此外,当理论Γ有限时,本文分别给出Γ在α程度上近似推出公式A和理论Σ的充要条件。最后,我们给出公式A与有限理论Γ的结论集D(Γ)之间距离的计算公式。     

Uniform interpolation and the existence of sequent calculi

This paper presents a uniform and modular method to prove uniform interpolation for several intermediate and intuitionistic modal logics. The proof-theoretic method uses sequent calculi that are extensions of the terminating sequent calculus $\mathsf{G4ip}$ for intuitionistic propositional logic. It is shown that whenever the rules in a calculus satisfy certain structural properties, the corresponding logic has uniform interpolation. It follows that the intuitionistic versions of $\mathsf{K}$ and $\mathsf{KD}$ (without the diamond operator) have uniform interpolation. It also follows that no intermediate or intuitionistic modal logic without uniform interpolation has a sequent calculus satisfying those structural properties, thereby establishing that except for the seven intermediate logics that have uniform interpolation, no intermediate logic has such a sequent calculus.     

A coding method for a sequent calculus of propositional logic

The paper deals with a coding method for a sequent calculus of the propositional logic. The method is based on the sequent calculus. It allows us to determine if a formula is derivable in the calculus without constructing a derivation tree. The main advantage of the coding method is its compactness in comparison with derivation trees of the sequent calculus. The coding method can be used as a decision procedure for the propositional logic.     

二值逻辑中命题的条件真度理论

基于条件概率的思想,在二值经典命题逻辑中引入条件真度的概念。在二值逻辑系统中初步给出了在信息Σ下的近似推理理论。     

R_0-代数上的距离结构及其在命题逻辑中的应用

设Ω是全体从R0-代数M到R0单位区间[0,1]的同态之集,μ是Ω上的一概率测度。引进M上的元素的尺寸和元素对间的相似度,然后在M上建立了伪距离。作为应用,将距离R0-代数理论应用到命题逻辑的近似推理理论。     

二值命题逻辑中基于前提信息的近似推理理论

以真度为基础,给出二值命题逻辑系统中基于前提信息的相似度和伪距离的概念以及伪距离的真度表示式,对二值命题逻辑中具有前提信息的近似推理问题进行讨论.     

A sequent calculus for logic of knowledge and past time: Completeness and decidability

We consider logic of knowledge and past time. This logic involves the discrete-time linear temporal operators next, until, weak yesterday, and since. In addition, it contains an indexed set of unary modal operators agent i knows.We consider the semantic constraint of the unique initial states for this logic. For the logic, we present a sequent calculus with a restricted cut rule. We prove the soundness and completeness of the sequent calculus presented. We prove the decidability of provability in the considered calculus as well. So, this calculus can be used as a basis for automated theorem proving. The proof method for the completeness can be used to construct complete sequent calculi with a restricted cut rule for this logic with other semantical constraints as well. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 427–437, July–September, 2006.     

Sequent calculi for propositional star-free likelihood logic

We consider classical, multisuccedent intuitionistic, and intuitionistic sequent calculi for propositional likelihood logic. We prove the admissibility of structural rules and cut rule, invertibility of rules, correctness of the calculi, and completeness of the classical calculus with respect to given semantics.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 3–21, January–March, 2005.     

Dual‐Context Sequent Calculus and Strict Implication

We introduce a dual‐context style sequent calculus which is complete with respectto Kripke semantics where implication is interpreted as strict implication in the modal logic K. The cut‐elimination theorem for this calculus is proved by a variant of Gentzen's method.     

二值命题逻辑中有限理论带误差结论集的结构

二值命题逻辑系统中理论Г的带误差结论集是近似推理研究的基本对象,对其结构进行分析是近似推理研究中需要解决的问题。通过公式是有限理论Г的带误差结论的充要条件,利用集合划分方法,对有限理论Г的带误差结论集分别基于真度相等关系和逻辑等价关系进行分类,得到了基于两类等价关系的包含等价类个数和代表元表示形式的分类定理,进一步体现了二值命题逻辑系统近似推理研究中理论Г的带误差结论集的特征。     

基于形式概念的对象粒的属性逻辑公式描述

基于形式概念在属性集上建立逻辑语言系统,证明基于形式概念的基本对象粒描述定理,讨论合取原子属性逻辑公式所描述对象粒的性质,提出一个求解描述对象粒的属性逻辑公式的算法。     

Sequent systems for compact bilinear logic

Compact Bilinear Logic (CBL), introduced by Lambek [14], arises from the multiplicative fragment of Noncommutative Linear Logic of Abrusci [1] (also called Bilinear Logic in [13]) by identifying times with par and 0 with 1. In this paper, we present two sequent systems for CBL and prove the cut‐elimination theorem for them. We also discuss a connection between cut‐elimination for CBL and the Switching Lemma from [14].     

Labeled sequent calculus for justification logics

Justification logics are modal-like logics that provide a framework for reasoning about justifications. This paper introduces labeled sequent calculi for justification logics, as well as for combined modal-justification logics. Using a method due to Sara Negri, we internalize the Kripke-style semantics of justification and modal-justification logics, known as Fitting models, within the syntax of the sequent calculus to produce labeled sequent calculi. We show that all rules of these systems are invertible and the structural rules (weakening and contraction) and the cut rule are admissible. Soundness and completeness are established as well. The analyticity for some of our labeled sequent calculi are shown by proving that they enjoy the subformula, sublabel and subterm properties. We also present an analytic labeled sequent calculus for $\mathsf{S4LPN}$ based on Artemov–Fitting models.     

Glivenko Classes of Sequents for Temporal Logic with Time Gaps

Let LB be a sequent calculus of the first-order classical temporal logic TB with time gaps. Let, further, LBJ be the intuitionistic counterpart of LB. In this paper, we consider conditions under which a sequent is derivable in the calculus LBJ if and only if it is derivable in the calculus LB. Such conditions are defined for sequents with one formula in the succedent (purely Glivenko -classes) and for sequents with the empty succedent (Glivenko -classes).