## A Structural Approach to Reversible Computation (2001)

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Venue: | Theoretical Computer Science |

Citations: | 18 - 3 self |

### BibTeX

@ARTICLE{Abramsky01astructural,

author = {Samson Abramsky},

title = {A Structural Approach to Reversible Computation},

journal = {Theoretical Computer Science},

year = {2001},

volume = {347},

pages = {1--16}

}

### Years of Citing Articles

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### Abstract

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of low-level machine models. By contrast, we develop a more structural approach. We show how high-level functional programs can be mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1

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Citation Context ...le quite concrete, are based directly on ideas stemming from Linear Logic and Geometry of Interaction [18, 19, 20, 21], and developed in previous work by the present author and a number of colleagues =-=[2, 3, 5, 6, 7, 9, 10]-=-. Our work here can be seen as a concrete manifestation of these more abstract and foundational developments. However, no knowledge of Linear Logic or Geometry of Interaction is required to read the p... |

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Citation Context ...le quite concrete, are based directly on ideas stemming from Linear Logic and Geometry of Interaction [18, 19, 20, 21], and developed in previous work by the present author and a number of colleagues =-=[2, 3, 5, 6, 7, 9, 10]-=-. Our work here can be seen as a concrete manifestation of these more abstract and foundational developments. However, no knowledge of Linear Logic or Geometry of Interaction is required to read the p... |

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Citation Context ...ation 7: F !x !y = !(x y) Monoidal Functoriality 8: W x !y = x !y !y Contraction 6 The notion of LCA corresponds to a Hilbert style axiomatization of the f!; (g fragment of linear logic [6, 13,=-= 49-=-]. The principal types of the combinators correspond to the axiom schemes which they name. They can be computed by a Hindley-Milner style algorithm [32] from the above equations: 1: B : (s(s) ( ( (s) ... |

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Citation Context ...lso, combinator algebras form the basic ingredient for realizability constructions, which are a powerful tool for building models of very expressive type theories (for textbook presentations see e.g. =-=[12, 19]-=-). By our results in the previous section, a combinator program M can be compiled in a syntax-directed fashion into a biorthogonal automaton A. Moreover, note that the size of A is linear in that of M... |

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Citation Context ...h taken and the objectives are very dierent to ours, and there are no clear analogues to our results. In fact, our concrete view of Geometry of Interaction can be traced back via Ian Mackie's thesis [=-=29]-=-, written under the present author's supervision, to ideas originating in [2, 3]. 2 The Model We formulate our computational model as a kind of automaton with some simple term-rewriting capabilities. ... |

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Citation Context ...l inductive types [22]; and control structures including recursion, higher-order functions, and continuations [34]. A representation of database query languages in the pure -calculus is developed in [=-=23-=-]. The - calculus can be compiled into combinators, and in fact this has been extensively studied as an implementation technique [32]. Although combinatory weak reduction does not capture all ofs-redu... |

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Citation Context ... on reversible computing. Our approach also has conceptual interest in that our constructions, while quite concrete, are based directly on ideas stemming from Linear Logic and Geometry of Interaction =-=[24, 25, 26, 27, 43, 20]-=-, and developed in previous work by the present author and a number of colleagues [2, 3, 5, 6, 7, 9, 10]. Our work here can be seen as a concrete manifestation of these more abstract and foundational ... |

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Citation Context ...oth foundational and, in the medium term, for practical reasons, is by now well established. We quote from the excellent summary in the introduction to the recent paper by Buhrman, Tromp and Vitanyi [=-=15]: Rev-=-ersible Computation: R. Landauer [27] has demonstrated that it is only the \logically irreversible" operations in a physical computer that necessarily dissipate energy by generating a correspondi... |

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Citation Context ... on reversible computing. Our approach also has conceptual interest in that our constructions, while quite concrete, are based directly on ideas stemming from Linear Logic and Geometry of Interaction =-=[18, 19, 20, 21]-=-, and developed in previous work by the present author and a number of colleagues [2, 3, 5, 6, 7, 9, 10]. Our work here can be seen as a concrete manifestation of these more abstract and foundational ... |

11 |
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Citation Context ...attice is preserved by constructions such as function space, lifting, and inverse limit), while when constants are added, to be modelled by sums, inconsistency arises and the natural models are cpo’s =-=[1]-=-. This suggests that the pure λ-calculus itself provides the ultimate reversible simulation of the irreversible phenomena of computation. 8 Universality A minor variation of the ideas of the previous ... |

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Citation Context ...further attention, and we hope to study it in more detail in the future. 5.1 Pure vs. Applied -calculus Our discussion has been based on using the pure -calculus or CL, with no constants and -rules [2=-=4, 1-=-2]. Thus integers, booleans etc. are all to be represented as -terms. The fact that - calculus and Combinatory Logic can be used to represent data as well as control is an important facet of their uni... |

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Citation Context ... express itself in friction: computers will dissipate a lot of heat unless their mode of operation becomes reversible, possibly quantum mechanical. The previous approaches of which we are aware (e.g. =-=[28, 13, 14]-=-) proceed by showing that some standard, low-level, irreversible computational model such as Turing machines can be simulated by a reversible version of the same model. Our approach is more \structura... |

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Citation Context ...een as a concrete manifestation of these more abstract and foundational developments. However, no knowledge of Linear Logic or Geometry of Interaction is required to read the present paper. The paper =-=[1-=-6] contains some discussion of a reversible abstract machine for the -calculus. However, the approach taken and the objectives are very dierent to ours, and there are no clear analogues to our results... |

1 |
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Citation Context ...mputations are exactly the time-reversals of the computations of A. We then establish a connection to models of functional computation. We will show that our model gives rise to a combinatory algebras=-=[24-=-], and derive universality as an easy consequence. This method of establishing universality has potential signicance for the important issue of how to program reversible computations. To quote from [1... |

1 |
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Citation Context ... express itself in friction: computers will dissipate a lot of heat unless their mode of operation becomes reversible, possibly quantum mechanical. The previous approaches of which we are aware (e.g. =-=[27, 13, 14]-=-) proceed by showing that some standard, low-level, irreversible computational model such as Turing machines can be simulated by a reversible version of the same model. Our approach is more \structura... |