Logical aspects of logical frameworks

This thesis provides a model–theoretic semantic analysis of aspects of the LF logical framework. The LF logical framework is the λΠ-calculus together with the judgements-as-types representation mechanism. A denotational semantics is provided for the λΠ-calculus in terms of Kripke λΠ-models. These are a generalization of the Kripke lambda models of Mitchell and Moggi to dependent types and are based on the contextual categories of Cartmell and their reformulation by Ritter. We analyse these models in terms of (Kripke) logical relations. We also present Kripke models of the internal logic of the λΠ-calculus, the {∀,⊃}-fragment of many-sorted minimal first-order logic, and show that the propositions-as-types correspondence induces an isomorphism between the two classes of Kripke models. We provide a proofand model-theoretic account of judged object-logics (logics suitable for representation in LF). We show that the judgements-as-types correspondence induces an epimorphism between these Kripke models and Kripke λΠ-models; which we use to provide model-theoretic proofs of faithfulness. We consider a variant of the LF logical framework which uses the worldsas-parameters representation mechanism. The generality of our account of the judgements-as-types correspondence allows us to treat the worlds-as-parameters representation mechanism as a special case. We interpret the syntactic ‘worlds’ introduced by worlds-as-parameters as worlds in our Kripke models and show that there exists a worlds-as-parameters epimorphism. We provide a semantic account of proof-search in the λΠ-calculus by identifying a class of Herbrand models and providing a least fixed-point construction corresponding, as usual, to resolution. Finally, we provide a characterization of abstract logic programming languages, as defined by Miller et al., in the LF logical framework.