We present a general method to transform a compositional specification of a specializer for a functional programming language into a set of combinators that can be used to perform the same specialization more efficiently. The main transformation steps are the transition to higher-order abstract syntax and untagging. All transformation steps are proved correct. The resulting combinators can be implemented in any functional language, typed or untyped, pure or impure. They may also be considered as forming a domain-specific language for meta-programming. We demonstrate the generality of the method by applying it to several specializers of increasing strength. We demonstrate its efficiency by comparing it with a traditional specialization system based on self-application.

In this paper we demonstrate that a broad class of higher-order functional programs can be transformed into semantically equivalent multidimensional intensional programs that contain only nullary variable definitions. The proposed algorithm systematically eliminates user-defined functions from the source program, by appropriately introducing context manipulation (i.e. intensional) operators. The transformation takes place in M steps, where M is the order of the initial functional program. During each step the order of the program is reduced by one, and the final outcome of the algorithm is an M-dimensional intensional program of order zero. As the resulting intensional code can be executed in a purely tagged-dataflow way, the proposed approach offers a promising new technique for the implementation of higher-order functional languages.

This theoretical pearl is about the closed term model of pure untyped lambda-terms modulo β-convertibility. A consequence of one of the results is that for arbitrary distinct combinators (closed lambda terms) M, M′, N, N′ there is a combinator H such that

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The general result, which comes from Statman (1998), is that uniformly r.e. partitions of the combinators, such that each ‘block’ is closed under β-conversion, are of the form {H−1{M}}M∈ΛΦ. This is proved by making use of the idea behind the so-called Plotkin-terms, originally devised to exhibit some global but non-uniform applicative behaviour. For expository reasons we present the proof below. The following consequences are derived: a characterization of morphisms and a counter-example to the perpendicular lines lemma for β-conversion.