公式分层的谓词模态逻辑

由于必然模态词□的引入,谓词模态逻辑的公式在一个可能世界中的真假值可能依赖于其可达的可能世界.在谓词模态逻辑中存在个体跨可能世界相等问题.针对这一问题,Lewis提出了对应物理论,并且在对应物理论中用对应物关系来表示个体跨可能世界相等.但是,当一个对象具有一个以上的对应物时,谓词模态逻辑中的跨可能世界相等关系无法与对应物关系建立一一对应.通过限制谓词模态逻辑中全称量词∀的范围,给出了一种公式分层的谓词模态逻辑.它是谓词模态逻辑的一个子逻辑,并且其语言与谓词模态逻辑的语言是相同的.但其公式是分层定义的,使得∀可以出现在□的范围内,并且□不能出现在∀的范围内.由于任意形如∀x□φ（x）的表达式都不是该逻辑的公式,以量词开头的公式在一个可能世界w中的真假值只依赖于w,该逻辑避免了个体跨可能世界相等问题.给出了该逻辑的语言、语法和语义,并证明了该逻辑是可靠的和完备的.     

谓词模态逻辑到对应物理论的翻译

对应物理论(counterpart theory)是一阶逻辑的一种理论.Lewis利用谓词模态逻辑到对应物理论的翻译来研究谓词模态逻辑的性质,但是Lewis的翻译存在把不可满足的公式翻译为可满足公式的情况针对这个问题,提出了一种扩展语义的谓词模态逻辑,建立了扩展语义后谓词模态逻辑模型与对应物理论模型的一一对应关系,并在此基础上建立了谓词模态逻辑到对应物理论的语义忠实语义满翻译(faithful and full translation),其可确保将谓词模态逻辑的可满足公式和不可满足公式分别翻译为对应物理论的可满足公式和不可满足公式.由对应物理论是可靠的、完备的一阶逻辑的理论且语义忠实语义满翻译保持可靠性和完备性,进一步证明了扩展语义的谓词模态逻辑也是可靠和完备的.     

模态逻辑中的(n)真度理论与和谐定理

首次在模态逻辑中通过有限模型建立了模态公式的(n)真度理论,得到了当模态词不出现时(n)真度与经典二值命题逻辑中的真度保持一致的和谐定理.研究了时态逻辑中命题的(n)真度随n变化的性态.提出了模态公式间的(n)相似度理论,并由此在全体公式之集中建立了(n)伪距离.得出了(n)模态逻辑度量空间,该空间以经典逻辑度量空间为子空间,从而可将经典命题逻辑中的近似推理理论推广到模态逻辑之中.     

多值模态逻辑的计量化方法

在多值模态逻辑中构建了n-值模态模型及相应的语义理论,并指出这种语义是经典模态逻辑语义的推广.定义了W,R n-型框架的概念,并在该框架下用归纳的方法构建了由模态公式诱导的局部化映射,给出公式的局部化真度的概念,并指出任意模态公式的局部化真度都可以转化为另一个不含模态词的公式在同一可能世界处的局部化真度.定义了模态公式的全局真度,并证明了当某模态公式不含模态词时,其全局真度与其在一般命题逻辑中的真度一致.     

命题模态逻辑的模型构造和完备性证明

模态逻辑是研究必然、可能及其相关概念的逻辑。模态公式的可满足性问题和证明系统的完备性问题是模态逻辑中的两个经典的问题。为了解决这两个问题,提出一个构造模态公式的canonical model的方法。通过这个方法,对于给定模态公式φ,如果φ是可满足的,可以得到φ的一个canonical model;如果φ是不可满足的,可以得到φ的证明。此外,还给出命题模态逻辑完备性的一个构造性证明方法。     

模态逻辑系统S5中相容理论的构造方法

在模态逻辑系统S5中借助公式的状态描述等概念，提出了一种从最简单的原子公式对应的相容理论开始，对于较复杂的公式形式，寻找恰当的状态描述，再将对应的存在状态加入最初的相容理论后仍相容的方法。     

一阶模态逻辑归结推理

1.引言模态逻辑自动推理的归结方法始于Farinas-del-Cerro的工作,近几年,又得到了进一步的研究和发展。一阶模态逻辑是在命题模态逻辑系统上增加了全称量词和存在量词,由于模态算子与量词之间存在着相互作用,因而使得模态公式的Skolem化、模态替换更为复杂。1986年,Cialdea和Farinas-del-Cerro将经典逻辑的Herbrand定理推广到一阶模态逻辑,并在此基础上,Cialdea于1991年建立起了一阶模态逻辑的归结推理方法,有效地克服了模态算子与量词之间复杂的相互作用。     

基于模态逻辑D公理系统的Conformant规划方法

2006年,conformant规划问题成为国际规划竞赛不确定性问题域中的标准测试问题,得到研究人员的广泛关注.目前,conformant规划系统都是将其看成信念状态空间上的启发式搜索问题予以求解.通过分析conformant规划问题的语法和语义,提出新的基于模态逻辑的规划框架.将其转换为模态逻辑D公理系统的一系列定理证明问题.提出2种基于模态逻辑的编码方式.构造相应的公理与推理规则形成模态公式集,保证对于D系统的定理证明过程等同于原问题的规划过程.并通过问题实例验证该方法的有效性.继基于SAT、CSP、线性规划、模型检测等求解技术的规划方法后,该规划框架是基于转换的规划方法的一种新的尝试.     

模态逻辑公式的真度理论

在模态逻辑系统中,对可能世界进行了深入的分析,首次提出了完整模型的概念,并且在这个完整模型的框架下定义了模态公式的真度概念,建立了公式的真度理论。并且证明了：若模态公式[φ]不含任何模态词,即经典逻辑公式,它对应的模态真度[τ(φ)]就由区间退化为一个点,并且这个点就是该公式的Borel型真度值。     

谓词模态逻辑中变量的严格指派问题

目前,命题模态逻辑已成为人工智能以及计算机科学等其他领域的有效工具,但谓词模态逻辑却不是.关于谓词模态逻辑还存在着很多的争议和问题.表面上看,在命题模态逻辑中添加相应的量词而得到谓词模态逻辑似乎是一件很简单的事情,但是这些新添加的量词却带来了许多的问题和麻烦,例如,变量的严格指派问题、不变论域与可变论域问题、跨界相等问题等.主要研究谓词模态逻辑中变量的严格指派问题.首先,介绍什么是变量的严格指派问题;其次,讨论对于该问题现有的解决办法及所存在的不足;最后,分析造成该问题的深层次原因,并简要介绍该问题的解决办法.     

Theory of （n） truth degrees of formulas in modal logic and a consistency theorem

The theory of (n) truth degrees of formulas is proposed in modal logic for the first time. A consistency theorem is obtained which says that the (n) truth degree of a modality-free formula equals the truth degree of the formula in two-valued propositional logic. Variations of (n) truth degrees of formulas w.r.t. n in temporal logic is investigated. Moreover, the theory of (n) similarity degrees among modal formulas is proposed and the (n) modal logic metric space is derived therefrom which contains the classical logic metric space as a subspace. Finally, a kind of approximate reasoning theory is proposed in modal logic. Supported by the National Natural Science Foundation of China (Grant Nos. 10331010 and 10771129), and the Foundation of 211 Construction of Shaanxi Normal University     

Modal logic as metalogic

The goal of this paper is to show how modal logic may be conceived as recording the derived rules of a logical system in the system itself. This conception of modal logic was propounded by Dana Scott in the early seventies. Here, similar ideas are pursued in a context less classical than Scott's.First a family of propositional logical systems is considered, which is obtained by gradually adding structural rules to a variant of the nonassociative Lambek calculus. In this family one finds systems that correspond to the associative Lambek calculus, linear logic, relevant logics, BCK logic and intuitionistic logic. Above these basic systems, sequent systems parallel to the basic systems are constructed, which formalize various notions of derived rules for the basic systems. The deduction theorem is provable for the basic systems if, and only if, they are at least as strong as systems corresponding to linear logic, or BCK logic, depending on the language, and their deductive metalogic is not stronger than they are.However, though we do not always have the deduction theorem, we may always obtain a modal analogue of the deduction theorem for conservative modal extensions of the basic systems. Modal postulates which are necessary and sufficient for that are postulates of S4 plus modal postulates which mimic structural rules. For example, the modal postulates which Girard has recently considered in linear logic are necessary and sufficient for the modal analogue of the deduction theorem.All this may lead towards results about functional completeness in categories. When functional completeness, which is analogous to the deduction theorem, fails, we may perhaps envisage a modal analogue of functional completeness in a modal category, of which our original category is a full subcategory.     

Bounded Model Generation for Isabelle/HOL

A translation from higher-order logic (on top of the simply typed λ-calculus) to propositional logic is presented, such that the resulting propositional formula is satisfiable iff the HOL formula has a model of a given finite size. A standard SAT solver can then be used to search for a satisfying assignment, and such an assignment can be transformed back into a model for the HOL formula. The algorithm has been implemented in the interactive theorem prover Isabelle/HOL, where it is used to automatically generate countermodels for non-theorems.     

描述逻辑FL0概念及术语公理集的表达能力刻画

表达能力和推理复杂性是一个逻辑的两个重要特征,也是一对相互制约的关系。解释之间的互模拟关系是从语义的角度刻画逻辑表达能力的一个有效途径,其代表性的结果是命题模态逻辑表达能力的刻画定理-van Benthem刻画定理。文中给出了描述逻辑FL0(含构造子:原子概念、顶概念、概念交、全称量词约束)的模拟关系,建立了FL0中概念和术语公理集的表达能力刻画定理,即一阶逻辑公式与FL0概念和术语公理集等价的充分必要条件。上述结果为寻求表达能力与推理复杂性之间的最佳平衡提供了有效的支持。     

Modal Predicates and Coequations

We show how coalgebras can be presented by operations and equations. This is a special case of Linton's approach to algebras over a general base category , namely where is taken as the dual of sets. Since the resulting equations generalise coalgebraic coequations to situations without cofree coalgebras, we call them coequations. We prove a general co-Birkhoff theorem describing covarieties of coalgebras by means of coequations. We argue that the resulting coequational logic generalises modal logic.     

Labelled Modal Logics: Quantifiers

In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4.2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logics with varying, increasing, decreasing, or constant domains. The result is modular with respect to both properties of the accessibility relation in the Kripke frame and the way domains of individuals change between worlds. Our approach has a modular metatheory too; soundness, completeness and normalization are proved uniformly for every logic in our class. Finally, our work leads to a simple implementation of a modal logic theorem prover in a standard logical framework.     

A note on assumptions about Skolem functions

Skolemization is not an equivalence preserving transformation. For the purposes of refutational theorem proving it is sufficient that skolemization preserves satisfiability and unsatisfiability. Therefore there is sometimes some freedom in interpreting Skolem functions in a particular way. We show that in certain cases it is possible to exploit this freedom for simplifying formulae considerably. Examples for cases where this occurs systematically are the relational translation from modal logics to predicate logic and the relativization of first-order logics with sorts.     

Stratified institutions and elementary homomorphisms

For conventional logic institutions, when one extends the sentences to contain open sentences, their satisfaction is then parameterized. For instance, in the first-order logic, the satisfaction is parameterized by the valuation of unbound variables, while in modal logics it is further by possible worlds. This paper proposes a uniform treatment of such parameterization of the satisfaction relation within the abstract setting of logics as institutions, by defining the new notion of stratified institutions. In this new framework, the notion of elementary model homomorphisms is defined independently of an internal stratification or elementary diagrams. At this level of abstraction, a general Tarski style study of connectives is developed. This is an abstract unified approach to the usual Boolean connectives, to quantifiers, and to modal connectives. A general theorem subsuming Tarski's elementary chain theorem is then proved for stratified institutions with this new notion of connectives.