Hybrid Languages and Temporal Logic

Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the Sofia school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev) the method remains little known. In our view this has deprived temporal logic of a valuable tool. The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the first technical, the second conceptual. First, we show that hybridization gives rise to well-behaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full first-order expressive strength, is demonstrated for a weaker local language capable of defining the Until operator, we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths

Named Models in Coalgebraic Hybrid Logic

Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding

Named Models in Coalgebraic Hybrid Logic

Hybrid logic extends modal logic with support for reasoning about individual states, designated by so-called nominals. We study hybrid logic in the broad context of coalgebraic semantics, where Kripke frames are replaced with coalgebras for a given functor, thus covering a wide range of reasoning principles including, e.g., probabilistic, graded, default, or coalitional operators. Specifically, we establish generic criteria for a given coalgebraic hybrid logic to admit named canonical models, with ensuing completeness proofs for pure extensions on the one hand, and for an extended hybrid language with local binding on the other. We instantiate our framework with a number of examples. Notably, we prove completeness of graded hybrid logic with local binding

How generic language extensions enable ''open-world'' design in Java

By \emph{open--world design} we mean that collaborating classes are so loosely coupled that changes in one class do not propagate to the other classes, and single classes can be isolated and integrated in other contexts. Of course, this is what maintainability and reusability is all about. In the paper, we will demonstrate that in Java even an open--world design of mere attribute access can only be achieved if static safety is sacrificed, and that this conflict is unresolvable \emph{even if the attribute type is fixed}. With generic language extensions such as GJ, which is a generic extension of Java, it is possible to combine static type safety and open--world design. As a consequence, genericity should be viewed as a first--class design feature, because generic language features are preferably applied in many situations in which object--orientedness seems appropriate. We chose Java as the base of the discussion because Java is commonly known and several advanced features of Java aim at a loose coupling of classes. In particular, the paper is intended to make a strong point in favor of generic extensions of Java

Strong Completeness for Non-Compact Hybrid Logics

We provide a strongly complete infinitary proof system for hybrid logic. This proof system can be extended with countably many sequents. Thus completeness proofs are provided for infinitary hybrid versions of non-compact logics like ancestral logic and Segerberg's modal logic with the bounded chain condition. This extends the completeness result for hybrid logics by Blackburn and Tzakova