Programming in higher-order typed lambda-calculi

Typed A-calculi have been objects of theoretical study for many years. Recently, it has been shown that all the inductively defined types (including numbers, booleans, lists, and trees, as well as more complex structures like typed terms and proofs) can be represented in higher-order A-calculi with no built-in types or type constructors. This raises the possibility of designing practical programming languages based on pure typed A-calculi. This tutorial presents a hierarchy of increasingly powerful languages, beginning with Church's simply typed A-calculus (F x ) and the second-order polymorphic A-calculus of Girard and Reynolds, and culminating in a fragment of Girard's o>-order polymorphic A-calculus (F^) . Our focus throughout is on the unusual style of programming that arises in these languages, where all functions are total and the primary control construct is iteration rather than general recursion. This work was supported in part by the Office of Naval Research and the Defense Advanced Research Projects Agency (DOD) under contract number N00014-84-K-0415 The contained in this document are those of the authors and . J ^ U S ^ ^ ^ the official policies, either expressed or implied, of DARPA or the U.S. Government. r e p r e S e n t m g

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