纤维逻辑

形式逻辑已经从简单命题逻辑发展到比较复杂的模态逻辑系列。但是在主体环境下,已有逻辑的复杂性仍然不能有效刻画主体复杂的心智。有一些人工智能研究者根据主体心智的多重性,在模态逻辑中引入多种模态算子,并借此对主体加以刻画。但是原来的可能世界语义却难以容纳如此复杂的语法,出现了很多不合理的地方。本文首先介绍了新近出现的纤维逻辑（fibring logics）,然后归纳了目前将此理论应用在主体BDI建模的研究现状,最后分析纤维逻辑的不足之处,讨论了其他可能的应用,并对今后的工作做了展望。     

Multi-modal nonmonotonic logics of minimal knowledge

In this paper we introduce multi-modal logics of minimal knowledge. Such a family of logics constitutes the first proposal in the field of epistemic nonmonotonic logic in which the three following aspects are simultaneously addressed: (1) the possibility of formalizing multiple agents through multiple modal operators; (2) the possibility of using first-order quantification in the modal language; (3) the possibility of formalizing nonmonotonic reasoning abilities for the agents modeled, based on the principle of minimal knowledge. We illustrate the expressive capabilities of multi-modal logics of minimal knowledge to provide a formal semantics to peer-to-peer data integration systems, which constitute one of the most recent and complex architectures for distributed information systems.      

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 survey on temporal logics for specifying and verifying real-time systems

Over the last two decades, there has been an extensive study of logical formalisms on specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for formal specification of real-time and complex systems, an up to date survey of these logics does not exist in the literature. In this paper we analyse various temporal formalisms introduced for specification, including propositional/first-order linear temporal logics, branching temporal logics, interval temporal logics, real-time temporal logics and probabilistic temporal logics. We give decidability, axiomatizability, expressiveness, model checking results for each logic analysed. We also provide a comparison of features of the temporal logics discussed.     

A Framework for Automated Reasoning in Multiple-Valued Logics

The language of signed formulas offers a first-order classical logic framework for automated reasoning in multiple-valued logics. It is sufficiently general to include both annotated logics and fuzzy operator logics. Signed resolution unifies the two inference rules of annotated logics, thus enabling the development of an SLD-style proof procedure for annotated logic programs. Signed resolution also captures fuzzy resolution. The logic of signed formulas offers a means of adapting most classical inference techniques to multiple-valued logics.     

Reasoning about Minimal Knowledge in Nonmonotonic Modal Logics

We study the problem of embedding Halpern and Moses's modal logic of minimal knowledge states into two families of modal formalism for nonmonotonic reasoning, McDermott and Doyle's nonmonotonic modal logics and ground nonmonotonic modal logics. First, we prove that Halpern and Moses's logic can be embedded into all ground logics; moreover, the translation employed allows for establishing a lower bound (3p) for the problem of skeptical reasoning in all ground logics. Then, we show a translation of Halpern and Moses's logic into a significant subset of McDermott and Doyle's formalisms. Such a translation both indicates the ability of Halpern and Moses's logic of expressing minimal knowledge states in a more compact way than McDermott and Doyle's logics, and allows for a comparison of the epistemological properties of such nonmonotonic modal formalisms.     

Eliminability of cut in hypersequent calculi for some modal logics of linear frames

Hypersequent calculi, introduced independently by Pottinger and Avron, provide a powerful generalization of ordinary sequent calculi. In the paper we present a proof of eliminability of cut in hypersequent calculi for three modal logics of linear frames: K4.3, KD4.3 and S4.3. Our cut-free calculus is based on Avron's HC formalization for Gödel–Dummett's logic. The presented proof of eliminability of cut is purely syntactical and based on Ciabattoni, Metcalfe, Montagna's proof of eliminability of cut for hypersequent calculi for some fuzzy logics with modalities.     

TABLEAUX: A general theorem prover for modal logics

We present a general theorem proving system for propositional modal logics, called TABLEAUX. The main feature of the system is its generality, since it provides an unified environment for various kinds of modal operators and for a wide class of modal logics, including usual temporal, epistemic or dynamic logics. We survey the modal languages covered by TABLEAUX, which range from the basic one L(, ) through a complex multimodal language including several families of operators with their transitive-closure and converse. The decision procedure we use is basically a semantic tableaux method, but with slight modifications compared to the traditional one. We emphasize the advantages of such semantical proof methods for modal logics, since we believe that the models construction they provide represents perhaps the most attractive feature of these logics for possible applications in computer science and AI. The system has been implemented in Prolog, and appears to be of reasonable efficiency for most current examples. Experimental results are given in the paper, with two lists of test examples.A preliminary version of this paper appeared in the Proceedings of the International Computer Science Conference (ICSC'88), Hong-Kong, December 19–21, 1988.     

Hybridizing a Logical Framework

Logical connectives familiar from the study of hybrid logic can be added to the logical framework LF, a constructive type theory of dependent functions. This extension turns out to be an attractively simple one, and maintains all the usual theoretical and algorithmic properties, for example decidability of type-checking. Moreover it results in a rich metalanguage for encoding and reasoning about a range of resource-sensitive substructural logics, analagous to the use of LF as a metalanguage for more ordinary logics.This family of applications of the language, contrary perhaps to expectations of how hybridized systems are typically used, does not require the usual modal connectives box and diamond, nor any internalization of a Kripke accessibility relation. It does, however, make essential use of distinctively hybrid connectives: universal quantifiation over worlds, truth of a proposition at a named world, and local binding of the current world. This supports the claim that the innovations of hybrid logic have independent value even apart from their traditional relationship to temporal and alethic modal logics.     

Mechanising first-order temporal resolution

First-order temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics has identified important enumerable and even decidable fragments. Although a complete and correct resolution-style calculus has already been suggested for this specific fragment, this calculus involves constructions too complex to be of practical value. In this paper, we develop a machine-oriented clausal resolution method which features radically simplified proof search. We first define a normal form for monodic formulae and then introduce a novel resolution calculus that can be applied to formulae in this normal form. By careful encoding, parts of the calculus can be implemented using classical first-order resolution and can, thus, be efficiently implemented. We prove correctness and completeness results for the calculus and illustrate it on a comprehensive example. An implementation of the method is briefly discussed.     

Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning

In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatio-temporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4_u, which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with multi-dimensional formalisms and discuss possible directions for further research.     

First-Order Temporal Verification in Practice

First-order temporal logic, the extension of first-order logic with operators dealing with time, is a powerful and expressive formalism with many potential applications. This expressive logic can be viewed as a framework in which to investigate problems specified in other logics. The monodic fragment of first-order temporal logic is a useful fragment that possesses good computational properties such as completeness and sometimes even decidability. Temporal logics of knowledge are useful for dealing with situations where the knowledge of agents in a system is involved. In this paper we present a translation from temporal logics of knowledge into the monodic fragment of first-order temporal logic. We can then use a theorem prover for monodic first-order temporal logic to prove properties of the translated formulas. This allows problems specified in temporal logics of knowledge to be verified automatically without needing a specialized theorem prover for temporal logics of knowledge. We present the translation, its correctness, and examples of its use. Partially supported by EPSRC project: Analysis and Mechanisation of Decidable First-Order Temporal Logics (GR/R45376/01).     

Deciding Global Partial-Order Properties

Model checking of asynchronous systems is traditionally based on the interleaving model, where an execution is modeled by a total order between atomic events. Recently, the use of partial order semantics, representing the causal order between events, is becoming popular. This paper considers the model checking problem for partial-order temporal logics. Solutions to this problem exist for partial order logics over local states. For the more general global logics that are interpreted over global states, only undecidability results have been proved. In this paper, we present a decision procedure for a partial order temporal logic over global states. We also sharpen the undecidability results by showing that a single until operator is sufficient for undecidability.A preliminary version of this paper appears in Proceedings of the 25th International Colloquium on Automata, Languages, and Programming (ICALP98), LNCS 1443, pp. 41–52, 1998.     

Disjunction property and complexity of substructural logics

We systematically identify a large class of substructural logics that satisfy the disjunction property (DP), and show that every consistent substructural logic with the DP is PSPACE-hard. Our results are obtained by using algebraic techniques. PSPACE-completeness for many of these logics is furthermore established by proof theoretic arguments.     

Fibring Non-Truth-Functional Logics: Completeness Preservation

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This completeness transfer result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established.     

Fibred semantics for feature-based grammar logic

This paper gives a simple method for providing categorial brands of feature-based unification grammars with a model-theoretic semantics. The key idea is to apply the paradigm of fibred semantics (or layered logics, see Gabbay (1990)) in order to combine the two components of a feature-based grammar logic. We demonstrate the method for the augmentation of Lambek categorial grammar with Kasper/Rounds-style feature logic. These are combined by replacing (or annotating) atomic formulas of the first logic, i.e. the basic syntactic types, by formulas of the second. Modelling such a combined logic is less trivial than one might expect. The direct application of the fibred semantics method where a combined atomic formula like np (num: sg & pers: 3rd) denotes those strings which have the indicated property and the categorial operators denote the usual left- and right-residuals of these string sets, does not match the intuitive, unification-based proof theory. Unification implements a global bookkeeping with respect to a proof whereas the direct fibring method restricts its view to the atoms of the grammar logic. The solution is to interpret the (embedded) feature terms as global feature constraints while maintaining the same kind of fibred structures. For this adjusted semantics, the anticipated proof system is sound and complete.     

Deciding Regular Grammar Logics with Converse Through First-Order Logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF², which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. The translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. It is practically relevant because it makes it possible to use a decision procedure for the guarded fragment in order to decide regular grammar logics with converse. The class of regular grammar logics includes numerous logics from various application domains. A consequence of the translation is that the general satisfiability problem for every regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Other logics that can be translated into GF² include nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF² without extra machinery such as fixed-point operators.     

Model checking for action-based logics

A model checker is described that supports proving logical properties of concurrent systems. The logical properties can be described in different action-based logics (variants of Hennessy-Milner logic). The tools is based on the EMC model checker for the logic CTL. It therefore employs a set of translation functions from the considered logics to CTL, as well as a model translation function from labeled transition systems (models of the action-based logics) to Kripke structures (models for CTL). The obtained tool performs model checking in linear time complexity, and its correctness is guaranteed by the proof that the set of translation functions, coupled with the model translation function, preserves satisfiability of logical formulae.