Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary meta-theory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are self-justifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic first-order logic. It is also established that, in many cases, the natural deduction systems induce well-known intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their

this paper are algebraic dcpos, and many of the points discussed here will be needed later in the special case. 2 They provide a simple example to illustrate the "Display categories" in Section 3.2

The "observational content" of geometric logic is discussed and it is proposed that geometric logic is an appropriate basis for a Z-like specification language in which schemas are used as geometric theory presentations.

We divide attempts to give the structural proof theory of modal logics into two kinds, those pure formulations whose inference rules characterise modality completely by means of manipulations of boxes and diamonds, and those labelled formulations that leverage the use of labels in giving inference rules. The widespread adoption of labelled formulations is driven by their ability to model features of the model theory of modal logic in its proof theory. We describe

. By adopting theories as primitive components of a logic and recognizing that formulae are just presentation details we arrive at the concept of topological institution. In a topological institution, we have, for each signature, a frame of theories, a set of interpretation structures and a satisfaction relation. More precisely, we have, for each signature, a topological system. We show how to extract a topological institution from a given institution and establish an adjunction. Illustrations are given within the context of equational logic. We study the compositionality of theories. Formulae are recovered when we establish a general technique for presenting topological institutions. Topological institutions with finitely observable theories are shown to be useful in temporal monitoring applications where we would like to be able to characterize the properties of the system that can be monitored. Namely, an invariant property (G') cannot be monitored because it cannot be positively es...

Solid theoretical foundations of object oriented databases (OODBs) are still missing. The work reported in this paper contains results on a formally founded object oriented datamodel (OODM) and is intended to contribute to the development of a uniform mathematical theory of OODBs. A clear distinction between objects and values turns out to be essential in the OODM. Types and classes are used to structure values and objects repectively. This can be founded on top of any underlying type system. We outline different approaches to type systems and their semantics and claim that OODB theory on top of arbitrary type systems leads to type theory with topos-theoretically defined semantics. On this basis the known solutions to the problems of unique object identification and genericity can be generalized. It turns out that extents of classes must be completely representable by values. Such classes are called value-representable. As a consequence object identifiers degenerate to a pure...

Conceptual modelling requires a solid mathematical theory of concepts concerning the collection of concepts used in a specific, but broad enough field. The field considered in this paper is database modelling. Here object orientation in the widest sense has been identified as a unifying conceptual umbrella that encompasses all relevant datamodels. The theory of object oriented databases has brought to light the fundamental distinction between the concepts of objects and values and correspondingly types and classes. This can be founded on top of any underlying type system. Thus, expressiveness of a datamodel basically depends on the type concept, from which the other concepts can be derived. In order to achieve a uniform mathematical theory we outline different type systems and their semantics and claim that OODB theory on top of arbitrary type systems leads to type theory with topos-theoretically defined semantics. On this basis the known solutions to the problems of unique ...

By adopting theories as primitive components of a logic and recognizing that formulae are just presentation details we arrive at the concept of topological institution. In a topological institution, we have, for each signature, a frame of theories, a set of interpretation structures and a satisfaction relation. More precisely, we have, for each signature, a topological system. We show how to extract a topological institution from a given institution. Illustrations are given within the context of both equational and propositional logics. Already within the general setting of topological institutions we study the compositionality of theories and provide a characterization of both soundness and completeness. Formulae are recovered when we establish a general technique for presenting topological institutions. Equational and propositional topological institutions with finitely observable theories appear as simple applications of this technique. We consider and compare two notio...

. The importance of compositionality in program construction is being accepted quite well. With this respect, the relational programming has clear advantages over the functional programming. Unfortunately, there is no general technique of relational programming efficient enough to compete with the existing functional programming techniques. Here we discuss structural synthesis of programs - a method of synthesis of functional programs explainable in terms of higher-order functional constraint nets, simple types or intuitionistic logic. This method has been used in implementation of declarative languages that allow us to specify concepts as relations and use them in specifications more flexibly than functions. 1. About the SSP Structural synthesis of programs (SSP) has been known for quite a number of years and has been used at least in two commercial systems: XpertPriz and PRIZ (4). During the years, SSP has been applied and extended in several ways. Here we summarize the recent exte...