Undivided and Indistinguishable Histories in Branching-Time Logics

In the tree-like representation of Time, two histories are undivided at a moment t whenever they share a common moment in the future of t. In the present paper, it will first be proved that Ockhamist and Peircean branching-time logics are unable to express some important sentences in which the notion of undividedness is involved. Then, a new semantics for branching-time logic will be presented. The new semantics is based on trees endowed with an indistinguishability function, a generalization of the notion of undividedness. It will be shown that Ockhamist and Peircean semantics can be viewed as limit cases of the semantics developed in this paper.     

Step semantics for “true” concurrency with recursion

We present a variety of denotational linear time semantics for a language with recursion and true concurrency in a form of synchronous co-operation, which in the literature is known as step semantics. We show that this can be done by a generalization of known results for interleaving semantics. A general method is presented to define semantical operators and denotational semantics in the Smyth powerdomain of streams. With this method, first a naive and then more sophisticated semantics for synchronous co-operation are developed, which include such features as interleaving and synchronization. Then we refine the semantics to deal with a bounded number of processors, subatomic actions, maximal parallelism and a real-time operator. Finally, it is indicated how to apply these ideas to branching-time models, where it becomes possible to analyze deadlock behaviour as well as a form of true concurrency. John-Jules Meyer received his Master's degree in Mathematics in 1979 from the University of Leiden, and his Ph.D. degree in 1985 from the Free University Amsterdam. He is currently a Professor of Theoretical Computer Science, both at the Free University Amsterdam and at the University of Nijmegen. His current research interests include semantics of programming languages and logics for computer science, in particular artifical intelligence. Erik de Vink received the M.S. degree in Mathematics from the University of Amsterdam. He is currently a Junior Researcher at the Department of Mathematics and Computer Science of the Free University Amsterdam. At the moment his main research concerns the semantics of concurrent and logic programming languages.     

Hierarchies of modal and temporal logics with reference pointers

We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: point of reference-reference pointer which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, Kamp's and Stavi's temporal operators, as well as nominals (names, clock variables), are definable in them. Universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and a strong completeness theorem is proved for them and extended to some classes of their extensions.     

Temporalizing Epistemic Default Logic

We present an epistemic default logic, based on the metaphore of a meta-level architecture. Upward reflection is formalized by a nonmonotonic entailment relation, based on the objective facts that are either known or unknown at the object level. Then, the meta (monotonic) reasoning process generates a number of default-beliefs of object-level formulas. We extend this framework by proposing a mechanism to reflect these defaults down. Such a reflection is seen as essentially having a temporal flavour: defaults derived at the meta-level are projected as facts in a next object level state. In this way, we obtain temporal models for default reasoning in meta-level formalisms which can be conceived as labeled branching trees. Thus, descending the tree corresponds to shifts in time that model downward reflection, whereas the branching of the tree corresponds to ways of combining possible defaults. All together, this yields an operational or procedural semantics of reasoning by default, which admits one to reason about it by means of branching-time temporal logic. Finally, we define sceptical and credulous entailment relations based on these temporal models and we characterize Reiter extensions in our semantics.     

Fodorian semantics,pathologies, and “Block's Problem”

In two recent books, Jerry Fodor has developed a set of sufficient conditions for an object X to non-naturally and non-derivatively mean X. In an earlier paper we presented three reasons for thinking Fodor's theory to be inadequate. One of these problems we have dubbed the Pathologies Problem. In response to queries concerning the relationship between the Pathologies Problem and what Fodor calls Block's Problem, we argue that, while Block's Problem does not threatenFodor's view, the Pathologies Problem does.We would like to thank Ray Elugardo, Pat Manfredi, and Donna Summerfield for helpful comments on an earlier paper on Fodorian Semantics, X means X: Semantics Fodor-Style. We would especially like to thank Ned Block for extended e-mail conversations about Block's Problem. Block agrees that his problem is not the same as our pathologies problem. Contrary to what we say here, he still maintains that his objection can ultimately be made to work to defeat Fodor's theory of meaning. His elaboration of Block's Problem is different than the one we present here. Versions of a related paper were presented at the 1991 Annual Meeting of the Southern Society for Philosophy and Psychology as well as the Canadian Society for History and Philosophy of Science.     

Timing Analysis of Combinational Circuits in Intuitionistic Propositional Logic

Classical logic has so far been the logic of choice in formal hardware verification. This paper proposes the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes. The model-theoretic properties are exploited to handle the second-order nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof-theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way.We present a natural Kripke-style semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic (Information and Computation, Vol. 137, No. 1, 1–33, 1997), in which validity is validity up to stabilisation, and implication comes out as boundedly gives rise to. We show that this semantics is equivalently characterised by a notion of realisability with stabilisation bounds as realisers. Following this second point of view an intensional semantics for proofs is presented which allows us effectively to compute quantitative stabilisation bounds.We discuss the application of the theory to the timing analysis of combinational circuits. To test our ideas we have implemented an experimental prototype tool and run several examples.     

A Simple Stereo Algorithm to Recover Precise Object Boundaries and Smooth Surfaces

For recovering precise object boundaries in area-based stereo matching, there are two problems. One is the so-called occlusion problem. This can be avoided if we can select only visible cameras among many cameras used. Another one is the problem called boundary overreach, i.e. the recovered object boundary turns out to be wrongly located away from the real one due to the window's coverage beyond a boundary. This is especially harmful to segmenting objects using depth information. A few approaches have been proposed to solve this problem. However, these techniques tend to degrade on smooth surfaces. That is, there seems to be a trade-off problem between recovering precise object edges and obtaining smooth surfaces.In this paper, we propose a new simple method to solve these problems. Using multiple stereo pairs and multiple windowing, our method detects the region where the boundary overreach is likely to occur (let us call it BO region) and adopts appropriate methods for the BO and non-BO regions. Although the proposed method is quite simple, the experimental results have shown that it is very effective at recovering both sharp object edges at their correct locations and smooth object surfaces. We also present a sound analysis of the boundary overreach which has not been clearly explained in the past.     

Modalities in linear logic weaker than the exponential “of course”: Algebraic and relational semantics

We present a semantic study of a family of modal intuitionistic linear systems, providing various logics with both an algebraic semantics and a relational semantics, to obtain completeness results. We call modality a unary operator on formulas which satisfies only one rale (regularity), and we consider any subsetW of a list of axioms which defines the exponential of course of linear logic. We define an algebraic semantics by interpreting the modality as a unary operation on an IL-algebra. Then we introduce a relational semantics based on pretopologies with an additional binary relationr between information states. The interpretation of is defined in a suitable way, which differs from the traditional one in classical modal logic. We prove that such models provide a complete semantics for our minimal modal system, as well as, by requiring the suitable conditions onr (in the spirit of correspondence theory), for any of its extensions axiomatized by any subsetW as above. We also prove an embedding theorem for modal IL-algebras into complete ones and, after introducing the notion of general frame, we apply it to obtain a duality between general frames and modal IL-algebras.     

Computations in fragments of intuitionistic propositional logic

This article is a report on research in progress into the structure of finite diagrams of intuitionistic propositional logic with the aid of automated reasoning systems for larger calculations. Afragment of a propositional logic is the set of formulae built up from a finite number of propositional variables by means of a number of connectives of the logic, among which possibly non-standard ones like ¬¬ or which are studied here. Thediagram of that fragment is the set of equivalence classes of its formulae partially ordered by the derivability relation. N.G. de Bruijn's concept of exact model has been used to construct subdiagrams of the [p, q, , , ¬]-fragment.     

A modal contrastive logic: The logic of ‘but’

In this paper we present a modal approach to contrastive logic, the logic of contrasts as these appear in natural language conjunctions such as but. We use a simple modal logic, which is an extension of the well-knownS5 logic, and base the contrastive operators proposed by Francez in [2] on the basic modalities that appear in this logic. We thus obtain a logic for contrastive operators that is more in accord with the tradition of intensional logic, and that, moreover — we argue — has some more natural properties. Particularly, attention is paid to nesting contrastive operators. We show that nestings of but give quite natural results, and indicate how nestings of other contrastive operators can be done adequately. Finally, we discuss the example of the Hangman's Paradox and some similarities (and differences) with default reasoning. But but us no buts, as they say.Also partially supported by Nijmegen University, Toernooiveld, 6525 ED Nijmegen, The Netherlands.