Cut-free sequent systems for temporal logic

Currently known sequent systems for temporal logics such as linear time temporal logic and computation tree logic either rely on a cut rule, an invariant rule, or an infinitary rule. The first and second violate the subformula property and the third has infinitely many premises. We present finitary cut-free invariant-free weakening-free and contraction-free sequent systems for both logics mentioned. In the case of linear time all rules are invertible. The systems are based on annotating fixpoint formulas with a history, an approach which has also been used in game-theoretic characterisations of these logics.     

Focusing and polarization in linear,intuitionistic, and classical logics

A focused proof system provides a normal form to cut-free proofs in which the application of invertible and non-invertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cut-free proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.     

Sequent calculi and decidability for intuitionistic hybrid logic

In this paper we study the proof theory of the first constructive version of hybrid logic called Intuitionistic Hybrid Logic (IHL) in order to prove its decidability. In this perspective we propose a sequent-style natural deduction system and then the first sequent calculus for this logic. We prove its main properties like soundness, completeness and also the cut-elimination property. Finally we provide, from our calculus, the first decision procedure for IHL and then prove its decidability.     

Specifying coalgebras with modal logic

We propose to use modal logic as a logic for coalgebras and discuss it in view of the work done on coalgebras as a semantics of object-oriented programming. Two approaches are taken: First, standard concepts of modal logic are applied to coalgebras. For a certain kind of functor it is shown that the logic exactly captures the notion of bisimulation and a complete calculus is given. Examples of verifications of object properties are given. Second, we discuss the relationship of this approach with the coalgebraic logic of Moss (Coalgebraic logic, Ann Pure Appl. Logic 96 (1999) 277–317.).     

A finite model construction for coalgebraic modal logic

In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatisation in rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy–Milner logic, graded modal logic and probabilistic modal logic.     

Generalized quantifiers and modal logic

We study several modal languages in which some (sets of) generalized quantifiers can be represented; the main language we consider is suitable for defining any first order definable quantifier, but we also consider a sublanguage thereof, as well as a language for dealing with the modal counterparts of some higher order quantifiers. These languages are studied both from a modal logic perspective and from a quantifier perspective. Thus the issues addressed include normal forms, expressive power, completeness both of modal systems and of systems in the quantifier tradition, complexity as well as syntactic characterizations of special semantic constraints. Throughout the paper several techniques current in the theory of generalized quantifiers are used to obtain results in modal logic, and conversely.This author was supported by the Foundation for Philosophical Research (SWON), which is subsidized by the Netherlands Organization for Scientific Research (NWO).     

The Global Properties of Valid Formulas in Modal Logic K

Global property is the necessary condition which must be satisfied by the provable formulas.It can help to find out some unprovable formula that does not satisfy some global property before proving it using formal automated reasoning systems,thus the efficiency of the whole system is improved.This paper presents some global properties of valid formulas in modal logic K.Such properties are structure characters of formulas,so they are simple and easy to check.At the same time,some global properties of K unsatisfiable formula set are also given.     

From modal logic to terminal coalgebras

We define a modal logic whose models are coalgebras of a polynomial functor. Bisimilarity turns out to be the same as logical equivalence. Ideas and concepts of modal logic are directly applied to the theory of coalgebras: we give an axiomatization and define canonical coalgebras. That leads to a completeness result. Each canonical coalgebra proves to be terminal in a certain class of coalgebras. The approach also yields a functional characterization of the terminal coalgebra of all coalgebras with respect to a given polynomial functor.     

启发式方法生成命题逻辑可读证明

探讨了自动生成命题逻辑系统R的可读证明.采用试探法和自然推理法分别从前推和后推模拟人类思维求证,试探法根据推理规则将待证公式反向分解,自然推理法从假设出发根据推理规则生成新的公式.两种方法都实现了相干命题逻辑系统R的可读证明,并结合实现了混合证明.试探法和自然推理法是生成可读证明的有效方法,前推和后推两种思维方法也适用于其他逻辑系统的自动证明.     

Constructive linear-time temporal logic: Proof systems and Kripke semantics

In this paper we study a version of constructive linear-time temporal logic (LTL) with the “next” temporal operator. The logic is originally due to Davies, who has shown that the proof system of the logic corresponds to a type system for binding-time analysis via the Curry-Howard isomorphism. However, he did not investigate the logic itself in detail; he has proved only that the logic augmented with negation and classical reasoning is equivalent to (the “next” fragment of) the standard formulation of classical linear-time temporal logic. We give natural deduction, sequent calculus and Hilbert-style proof systems for constructive LTL with conjunction, disjunction and falsehood, and show that the sequent calculus enjoys cut elimination. Moreover, we also consider Kripke semantics and prove soundness and completeness. One distinguishing feature of this logic is that distributivity of the “next” operator over disjunction “?(A∨B)⊃?A∨?B” is rejected in view of a type-theoretic interpretation.     

A context model for fuzzy concept analysis based upon modal logic

In this paper we present interesting relationships between the context model, modal logic and fuzzy concept analysis. It has been shown that the context model proposed by Gebhardt and Kruse [Int. J. Approx. Reason. 9 (1993) 283] can be semantically extended and considered as a data model for fuzzy concept analysis within the framework of the meta-theory developed by Resconi et al. in 1990s. Consequently, the context model provides a practical framework for constructing membership functions of fuzzy concepts and gives the basis for a theoretical justification of suitably use of well-known t-norm based connectives such as min–max and product–sum rules in applications. Furthermore, an interpretation of mass assignments of fuzzy concepts within the context model is also established.     

Expressivity of coalgebraic modal logic: The limits and beyond

Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modelled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioural equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviourally equivalent depend on the existence of separating sets of predicate liftings for the signature functor at hand. Here, we provide a classification result for predicate liftings which leads to an easy criterion for the existence of such separating sets, and we give simple examples of functors that fail to admit expressive normal or monotone modal logics, respectively, or in fact an expressive (unary) modal logic at all. We then move on to polyadic modal logic, where modal operators may take more than one argument formula. We show that every accessible functor admits an expressive polyadic modal logic. Moreover, expressive polyadic modal logics are, unlike unary modal logics, compositional.     

A framework for checking proofs naturally

We propose a logical framework, called Natural Framework (NF), which supports formal reasoning about computation and logic (CAL) on a computer. NF is based on a theory of Judgments and Derivations. NF is designed by observing how working mathematical theories are created and developed. Our observation is that the notions of judgments and derivations are the two fundamental notions used in any mathematical activity. We have therefore developed a theory of judgments and derivations and designed a framework in which the theory provides a uniform and common play ground on which various mathematical theories can be defined as derivation games and can be played, namely, can write and check proofs. NF is equipped with a higher-order intuitionistic logic and derivations (proofs) are described following Gentzen’s natural deduction style. NF is part of an interactive computer environment CAL and it is also referred to as NF/CAL. CAL is written in Emacs Lisp and it is run within a special buffer of the Emacs editor. CAL consists of user interface, a general purpose parser and a checker for checking proofs of NF derivation games. NF/CAL system has been successfully used as an education system for teaching CAL for undergraduate students for about 8 years. We will give an overview of the NF/CAL system both from theoretical and practical sides.     

Eigenvariables, bracketing and the decidability of positive minimal predicate logic

We give a new proof of a theorem of Mints that the positive fragment of minimal predicate logic is decidable. The idea of the proof is to replace the eigenvariable condition of sequent calculus by an appropriate scoping mechanism. The algorithm given by this proof seems to be more practical than that given by the original proof. A naive implementation is given at the end of the paper. Another contribution is to show that this result extends to a large class of theories, including simple type theory (higher-order logic) and second-order propositional logic. We obtain this way a new proof of the decidability of the inhabitation problem for positive types in system F.     

A proof procedure for normal default theories

Recent research by Delgrande [6] and Geffner and Pearl [10] suggests two different semantic interpretations for normal defaults with one single representation as conditional sentences. However, they both need additional formal mechanisms for handling irrelevant information when their approaches are applied to formalising default reasoning. Delgrande in [5, 6] suggests two meta-strategies which he considers to be adequately strong to handle the orderings of defaults, and he claims they are equivalent. Furthermore, each of Delgrande's strategies is defined in terms of all sentences of the object language. In this paper, we shall prove that Delgrande's claim that his meta-strategies are equivalent is incorrect and that one of his meta-strategies can be reformulated within the framework of First Order Predicate Calculus (FOPC) and without having to consider every sentence of the object language. One advantage of such a reformalisation is its computational simplicity: to give an extension of a default theory there is only a need to consider those sentences which occur in the default theory under consideration rather than every sentence in the object language; furthermore, to provide a proof procedure for Delgrande's system as based on the meta-strategy we have formalised, one need only employ a FOPC proof procedure, rather than a conditional one.     

Compositionality of Hennessy–Milner logic by structural operational semantics

This paper presents a method for the decomposition of HML formulas. It can be used to decide whether a process algebra term satisfies a HML formula, by checking whether subterms satisfy certain formulas, obtained by decomposing the original formula. The method uses the structural operational semantics of the process algebra. The main contribution of this paper is the extension of an earlier decomposition method for the De Simone format from the Ph.D. thesis of Larsen in 1986, to more general formats.