## Recursion from Iteration (1994)

Venue: | Lisp and Symbolic Computation |

Citations: | 10 - 0 self |

### BibTeX

@INPROCEEDINGS{Filinski94recursionfrom,

author = {Andrzej Filinski},

title = {Recursion from Iteration},

booktitle = {Lisp and Symbolic Computation},

year = {1994},

pages = {11--38}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In a simply-typed, call-by-value (CBV) language with first-class continuations, the usual CBV fixpoint operator can be defined in terms of a simple, infinitelylooping iteration primitive. We first consider a natural but flawed definition, based on exceptions and "iterative deepening" of finite unfoldings, and point out some of its shortcomings. Then we present the proper construction using full first-class continuations, with both an informal derivation and a proof that the behavior of the defined operator faithfully mimics a "built-in" recursion primitive. In fact, given an additional uniformity assumption, the construction is a two-sided inverse of the usual definition of iteration from recursion. Continuing, we show that the CBV looping primitive is in fact the direct-style equivalent of a continuation-passing-style fixpoint, and that this correspondence extends all the way to traditional definitions of these operators in terms of reflexive types. 1. Introduction 1.1. Background ...

### Citations

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- Reynolds
- 1972
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- Gunter
- 1992
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- 1975
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- 1991
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- 1986
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1 | extended version to appear - Revised |