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Polymorphic dynamic typing — aspects of proof theory and inference
, 1995
"... We study dynamic typing in continuation of Henglein’s dynamically typed λcalculus, with particular regard to proof theoretic aspects and aspects of polymorphic completion inference. Dynamically typed λcalculus provides a formal framework within which we can reason in a precise manner about proper ..."
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We study dynamic typing in continuation of Henglein’s dynamically typed λcalculus, with particular regard to proof theoretic aspects and aspects of polymorphic completion inference. Dynamically typed λcalculus provides a formal framework within which we can reason in a precise manner about properties of the process of completion for higher order programming languages. Completions arise from raw programs by insertion of type coercions which model runtime type operations of tagging and checking/untagging. Central among the problems studied in dynamic typing are the notions of minimization of runtime type coercions in completions and safety of completions. From the monomorphic framework of Henglein’s system, we work towards a polymorphic generalization which eventually comprises HindleyMilner style polymorphism, discriminative, tagged sum types, regular recursive types and socalled coercive types with a notion of coercion parameterization. The resulting system can be seen as a form of polymorphic qualified type system which aims at a common generalization of dynamic typing and certain systems of soft typing.
Category Theory and the SimplyTyped lambdaCalculus
, 1996
"... This report deals with the question on how to provide a categorical model for the simplytyped calculus. We first introduce cartesian closed categories and work in detail a number of results concerning this construction. Next, we present the basic concepts related with the typed calculus, i.e., co ..."
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This report deals with the question on how to provide a categorical model for the simplytyped calculus. We first introduce cartesian closed categories and work in detail a number of results concerning this construction. Next, we present the basic concepts related with the typed calculus, i.e., concrete syntax for terms, occurrence of variables, context substitution and equivalence of terms. Then we present the typing rules and an equational proof system together with reduction rules that model the execution of expressions (programs). The chapter ends with the presentation of a categorical semantics for the calculus and a soundness proof for the equational proof system. The main technical result of this proof is the substitution lemma, which says, basically, that the (operational) concept of substitution can be understood (algebraically) as a composition of two suitable morphisms in a (cartesian closed) category. Contents 1 Cartesian closed categories 2 1.1 Exponentials . . . . ....