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

In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.

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this paper. This calculus satises cut-elimination, as for instance shown (in a more complicated form) in [Wij90]. This calculus is dierent from what is usually taken as the basic constructive system K, as we do not assume the distribution of possibility (3) over disjunctions neither in its binary form 3(A _ B) ! (3A _ 3B) nor in its nullary form 3? ! ? The sequent calculus above corresponds to an axiomatic formulation given by axioms for intuitionistic logic, plus axioms: 2(A ! B) ! (2A ! 2B) 2(A ! B) ! (3A ! 3B) 2A3B ! 3(A B) together with rules for Modus Ponens and Necessitation: ` A ! B ` A ` B MP ` A ` 2A Nec Wijesekera proved a Craig interpolation theorem, one of the usual consequences of syntactic cut-elimination and produced Kripke, algebraic and topological semantics for a calculus very similar to the one above. The only dierence is that he does assume 3? ! ?. From our \wish list" for logical systems only a natural deduction formulation and a categorical semantics are missing. These we proceed to discuss

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It is shown that the modal logic S4, simple -calculus and modal -calculus admit a realization in a very simple propositional logical system LP , which has an exact provability semantics. In LP both modality and -terms become objects of the same nature, namely, proof polynomials. The provability interpretation of modal -terms presented here may be regarded as a system-independent generalization of the Curry-Howard isomorphism of proofs and -terms. 1 Introduction The Logic of Proofs (LP , see Section 2) is a system in the propositional language with an extra basic proposition t : F for "t is a proof of F ". LP is supplied with a formal provability semantics, completeness theorems and decidability algorithms ([3], [4], [5]). In this paper it is shown that LP naturally encompasses -calculi corresponding to intuitionistic and modal logics, and combinatory logic. In addition, LP is strictly more expressive because it admits arbitrary combinations of ":" and propositional connectives. The id...

We consider a labelled tableau presentation of constant domain first-order S5 and prove a strong cut-elimination theorem. Keywords: strong cut-elimination, modal logic S5, tableau calculi 1 Introduction The present note is devoted to a proof of strong cut-elimination in a labelled tableau calculus for the constant domain modal predicate logic S5. Modal tableau calculi which build in the accessibility relation of possible worlds models were first introduced by Kripke (1963) and were later `linearized' by various authors, notably Fitting (1972, 1983, 1993) and Mints (1992). As in Gabbay's (1994) theory of labelled deductive systems, the basic declarative unit of these tableau calculi is not just a formula A, but rather a formula plus label (oe; A). In case of the modal logic S5, the label oe may just be a single positive integer, whereas in general it is a non-empty finite sequence of positive integers. Moreover, for S5 the accessibility relation between labels may be universal and h...

In this chapter we present the DenK project, a long-term effort where the aim is to build a generic cooperative human-computer interface combining multiple input and output modalities. We discuss the view on human-computer interaction that underlies the project and the emerging DenK system. The project integrates results from fundamental research in knowledge representation, communication, natural language semantics and pragmatics, and object-oriented animation. Central stage in the project is occupied by the design of a cooperative and knowledgeable electronic assistant that communicates in natural language and that has internal access to an application domain which is presented visually to the user. This electronic `Cooperative Assistant' has an information state that is represented in a rich form of type theory, a formalism that enables us to model the inherent cognitive dynamics of a dialogue participant.