We show that a type system based on the intuitionistic modal logic S4 provides an expressive framework for specifying and analyzing computation stages in the context of functional languages. Our main technical result is a conservative embedding of Nielson & Nielson's two-level functional language in our language Mini-ML, which in

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 offers a synthetic overview of, and original contributions to, the use of logics and formal methods in the analysis of hybrid systems

The last years have seen two major advances in Knowledge Representation and Reasoning. First, many interesting problems (ranging from Semi-structured Data to Linguistics) were shown to be expressible in logics whose main deductive problems are EXPTIME-complete. Second, experiments in automated reasoning have substantially broadened the meaning of “practical tractability”. Instances of realistic size for PSPACE-complete problems are now within reach for implemented systems. Still, there is a gap between the reasoning services needed by the expressive logics mentioned above and those provided by the current systems. Indeed, the algorithms based on tree-automata, which are used to prove EXPTIME-completeness, require exponential time and space even in simple cases. On the other hand, current algorithms based on tableau methods can take advantage of such cases, but require double exponential time in the worst case. We propose a tableau calculus for the description logic ALC for checking the satisfiability of a concept with respect to a TBox with general axioms, and transform it into the first simple tableaubased decision procedure working in single exponential time. To guarantee the ease of implementation, we also discuss the effects that optimizations (propositional backjumping, simplification, semantic branching, etc.) might have on our complexity result, and introduce a few optimizations ourselves.

A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known ‘relational’ translation makes the modal logic’s possible worlds structure explicit by introducing a distinguished predicate symbol to represent the accessibility relation. In the second approach, the ‘functional ’ translation method, paths in the possible worlds structure are represented by compositions of functions which map worlds to accessible worlds. On the syntactic level this means that every flexible symbol is parametrized with particular terms denoting whole paths from the initial world to the actual world. The ‘target logic’ for the translation is a first-order many-sorted logic with built in equality. Therefore the ‘source logic’ may also be first-order many-sorted with built in equality. Furthermore flexible function symbols are allowed. The modal operators may be parametrized with arbitrary terms and particular properties of the accessibility relation may be specified within the

Normal modal logics can be defined axiomatically as Hilbert systems, or semantically in terms of Kripke's possible worlds and accessibility relations. Unfortunately there are Hilbert axioms which do not have corresponding first-order properties for the accessibility relation. For these logics the standard semantics-based theorem proving techniques, in particular, the relational translation into first-order predicate logic, do not work. There is an alternative translation, the so-called functional translation, in which the accessibility relations are replaced by certain terms which intuitively can be seen as functions mapping worlds to accessible worlds. In this paper we show that from a certain point of view this functional language is more expressive than the relational language, and that certain second-order frame properties can be mapped to first-order formulae expressed in the functional language. Moreover, we show how these formulae can be computed automatically from the ...

this paper, we present a logic based framework for the representation of actions. Actions can change the meaning of properties of existing constants and their relationships. For instance "take" changes the properties of "on" and "hold": if somebody takes an object lying on a table then he/she will be holding this object after performing the action and the object will no longer be on the table. If somebody builds a wall, then the properties of bricks and cement will change. In logic oriented approaches, actions are frequently described by their preconditions and their results. The preconditions are represented by a formula which has to be true before the execution of the action and the results are represented by a formula which will be true after the action has been performed. In addition, the general laws of the world are described by formulas which have always to be true. An action occurs in a state of the world, which is also described by a formula, and yields a new state of the world which is obtained from the old state. Logic based action theories encounter typically the following problems:

We examine the role played by proof rules in general axiomatisations for hybrid logic. We prove three main results. First, all known axiomatisations for the basic hybrid language

In the two parts of this article a transformational approach to the design of distributed real-time systems is presented. The starting point are global requirements formulated in a subset of Duration Calculus called implementables and the target are programs in an occam dialect PL. In the first part we show how the level of program specifications represented by a language SL can be reached. SL combines regular expressions with ideas from action systems and with time conditions, and can express the distributed architecture of the implementation. While Duration Calculus is state-based, SL is event-based, and the switch between these two worlds is a prominent step in the transformation from implementables to SL. Both parts of the transformational calculus rely on the mixed term techniques by which syntax pieces of two languages are mixed in a semantically coherent manner. In the first part of the article mixed terms between implementables and SL and in the second part of the article mixed terms between SL and PL are used. The approach is illustrated by the example of a computer controlled gas burner.

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. Introduction In the study of non-classical propositional logics (and especially of modal logics) there are two main ways of defining interpretations or models. One possibility is to use algebras -- usually lattices with operators -- as models. Propositional variables are interpreted over elements of these algebraic models, an...