We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

The oe-calculus adds explicit substitutions to the -calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises the oe-calculus to provide a linear calculus of explicit substitutions, called xDILL, which analogously describes the implementation of linear functional programming languages. Our main observation is that there are non-trivial interactions between linearity and explicit substitutions and that xDILL is therefore best understood as a synthesis of its underlying logical structure and the technology of explicit substitutions. This is in contrast to the oe-calculus where the explicit substitutions are independent of the underlying logical structure. Keywords: -calculus, explicit substitutions, linear logic 1 Introduction This paper combines the technologies of explicit substitutions and linearity in a mathematically consistent way. We start by describing these technologies and the...

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

We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency design, etc. Both systems have so far been studied mainly from a type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

This paper is about relating traditional Kripke-style semantics for intuitionistic modal logics to their corresponding categorical semantics. Both forms of semantics have important applications within computer science. One of our aims is to persuade traditional modal logicians that categorical semantics is easy, fun and useful; just like Kripke semantics.

The-calculus [1] adds explicit substitutions to the-calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises the-calculus to provide a linear calculus of explicit substitutions which analogously describes the implementation of linear functional programming languages. 1