Needed Reduction and Spine Strategies for the Lambda Calculus
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector INFORMATION AND COMPUTATION 75, 191-231 (1987) Needed Reduction and Spine Strategies for the Lambda Calculus H. P. BARENDREGT* University of Ngmegen, Nijmegen, The Netherlands J. R. KENNAWAY~ University qf East Anglia, Norwich, United Kingdom J. W. KLOP~ Cenfre for Mathematics and Computer Science, Amsterdam, The Netherlands AND M. R. SLEEP+ University of East Anglia, Norwich, United Kingdom A redex R in a lambda-term M is called needed if in every reduction of M to nor- mal form (some residual of) R is contracted. Among others the following results are proved: 1. R is needed in M iff R is contracted in the leftmost reduction path of M. 2. Let W: MO+ M, + Mz --t .._ reduce redexes R,: M, + M,, ,, and have the property that Vi.3j> i.R, is needed in M,. Then W is normalising, i.e., if M, has a normal form, then Ye is finite and terminates at that normal form. 3. Neededness is an undecidable property, but has several efhciently decidable approximations, various versions of the so-called spine redexes. r 1987 Academic Press. Inc. 1. INTRODUCTION A number of practical programming languages are based on some sugared form of the lambda calculus. Early examples are LISP, McCarthy et al. (1961) and ISWIM, Landin (1966). More recent exemplars include ML (Gordon etal., 1981), Miranda (Turner, 1985), and HOPE (Burstall * Partially supported by the Dutch Parallel Reduction Machine project. t Partially supported by the British ALVEY project.
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