Archive for the 'Delphin/Elphin' Category

The nabla-calculus. Functional programming with higher-order encodings.

By Carsten Schurmann, Adam Poswolsky and Jeffrey Sarnat, from TLCA 2005, available from the Elphin website:

Higher-order encodings use functions provided by one language to represent variable binders of another. They lead to concise and elegant representations, which historically have been difficult to analyze and manipulate.
In this paper we present the nabla-calculus, a calculus for defining general recursive functions over higher-order encodings. To avoid problems commonly associated with using the same function space for representations and computations, we separate one from the other. The simply-typed λ-calculus plays the role of the representation level. The computation level contains not only the usual computational primitives but also an embedding of the representation-level. It distinguishes itself from similar systems by allowing recursion under representation-level λ-binders while permitting a natural style of programming which we believe scales to other logical frameworks. Sample programs include bracket abstraction, parallel reduction, and an evaluator for a simple language with first-class continuations.

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Practical programming with higher-order encodings and dependent types

By Adam Poswolsky and Carsten Schurmann, from ESOP 2008, available from the Delphin website:

Higher-order abstract syntax (HOAS) refers to the technique of representing variables of an object-language using variables of a meta-language. The standard first-order alternatives force the programmer to deal with superficial concerns such as substitutions, whose implementation is often routine, tedious, and error-prone. In this paper, we describe the underlying calculus of Delphin. Delphin is a fully implemented functional-programming language supporting reasoning over higher-order encodings and dependent types, while maintaining the benefits of HOAS.

More specifically, just as representations utilizing HOAS free the programmer from concerns of handling explicit contexts and substitutions, our system permits programming over such encodings without making these constructs explicit, leading to concise and elegant programs. To this end our system distinguishes bindings of variables intended for instantiation from those that will remain uninstantiated, utilizing a variation of Miller and Tiu’s ∇-quantifier [1].