Theoretical Computer Science
An algorithm for testing conversion in type theory
Logical frameworks
A framework for defining logics
Journal of the ACM (JACM)
Logic programming in a fragment of intuitionistic linear logic
Papers presented at the IEEE symposium on Logic in computer science
LICS '96 Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
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The λΛ-calculus is a dependent type theory with both linear and intuitionistic dependent function spaces. It can be seen to arise in two ways. Firstly, in logical frameworks, where it is the language of the RLF logical framework and can uniformly represent linear and other relevant logics. Secondly, it is a presentation of the proof-objects of BI, the logic of bunched implications. BI is a logic which directly combines linear and intuitionistic implication and, in its predicate version, has both linear and intuitionistic quantifiers. The λΛ-calculus is the dependent type theory which generalizes both implications and quantifiers. In this paper, we describe the categorical semantics of the λΛ-calculus. This is given by Kripke resource models, which are monoid-indexed sets of functorial Kripke models, the monoid giving an account of resource consumption. We describe a class of concrete, set-theoretic models. The models are given by the category of families of sets, parametrized over a small monoidal category, in which the intuitionistic dependent function space is described in the established way, but the linear dependent function space is described using Day's tensor product.