## A General Theory of Sharing Graphs (1998)

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Venue: | Theoret. Comput. Sci |

Citations: | 5 - 4 self |

### BibTeX

@TECHREPORT{Guerrini98ageneral,

author = {Stefano Guerrini},

title = {A General Theory of Sharing Graphs},

institution = {Theoret. Comput. Sci},

year = {1998}

}

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### Abstract

this paper. 12 Conclusions and further work The box nesting property is the minimal (and natural) requirement under which the sharing reductions can be used to get a local and distributed implementation of fi-like rules. It is our aim to study how the class of the calculi having the box nesting property relates with the classes already studied by the term graph rewriting community. As shown by the relevant examples of

### Citations

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268 |
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(Show Context)
Citation Context ... categorical justification of Gonthier's technique; Asperti and Laneve gave a generalization of the methodology to the so-called Interaction Systems, the subclass of the Combinatory Reduction Systems =-=[Klo80]-=- for which it is possible to find a Curry-Howard analogy with a suitable intuition10 istic logic. Furthermore, Asperti used sharing graphs for the implementation of an optimal version of an ML-like fu... |

177 |
Interaction nets
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(Show Context)
Citation Context ...on of sharing graphs into GOI. For instance, the proof technique of Gonthier et al. rests on the fact that the optimal rules define an interaction system (for the definition of interaction system see =-=[Laf90]-=-) and that the optimal rules are sound w.r.t. the paths definable in GOI. In fact, the paths of GOI give a way to extract the normal form of a proof net without reducing it or, in the case of -calculu... |

134 | An algorithm for optimal lambda calculus reduction
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- 1990
(Show Context)
Citation Context ...erve mux labeling, we might argue that two muxes should be matching when they face with the same label and non-matching otherwise. Unfortunately, Lamp5 ing (who firstly introduced sharing graphs, see =-=[Lam90]-=-) has shown that this solution does not work: there are cases in which two muxes that have been inserted by the same redex are non-matching. The right solution requires a dynamical labeling of muxes, ... |

121 |
Semantics and Pragmatics of the Lambda-Calculus
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(Show Context)
Citation Context ...nodes accessing a shared subterm to the root of the subterm. That techniques have a main drawback: in order to avoid unwanted sideeffects, variable substitution requires a careful implementation (see =-=[Wad71]-=-). In fact, replacing a term T s for the occurrences of a variable x in a term T t by substituting a pointer to T s for any pointer to x would apply the variable substitution in every term T u that sh... |

105 | The geometry of optimal lambda reduction - Gonthier, Abadi, et al. - 1992 |

90 |
Optimal reductions in the lambda calculus
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- 1980
(Show Context)
Citation Context ...graph without muxes for which the box nesting property holds. 1.4 Optimality and other related works Sharing graphs were introduced by Lamping [Lam90] to implement Levy's optimal reductions of -terms =-=[Lev80]-=-. Several re nements of them where successively proposed by Gonthier et al. [GAL92a,GAL92b], and by Asperti and Laneve [AL94,Asp95]. The work of Gonthier et al. addressed how Lamping's formalism could... |

87 | The Optimal Implementation of Functional Programming Languages - Asperti, Guerrini - 1998 |

61 |
Lambda-Calcul et Reseaux
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(Show Context)
Citation Context ...es, see [Gir87] for their complete set. However, the only one relevant for our purposes is the exponential cut-elimination that will be depicted in Figure 22. 3.3 Pure proof nets Pure proof nets (see =-=[Reg92]-=-) are the nets corresponding to the interpretation of -calculus inside linear logic by means of the isomorphism !O ( O ' O; that, using O in the place of (, becomes ?I OO ' O, where I and O are two co... |

56 | Linear Logic Without Boxes - Gonthier, Abadi, et al. - 1992 |

43 |
Geometry of Interaction 1: Interpretation of System F
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(Show Context)
Citation Context ... [GAL92a,GAL92b], and by Asperti and Laneve [AL94,Asp95]. The work of Gonthier et al. addressed how Lamping's formalism can be interpreted inside the so-called Geometry of Interaction (GOI) of Girard =-=[Gir89]-=-; Asperti presented a more categorical justification of Gonthier's technique; Asperti and Laneve gave a generalization of the methodology to the so-called Interaction Systems, the subclass of the Comb... |

40 | Interaction Systems I: the theory of optimal reduction
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- 1994
(Show Context)
Citation Context ...re G is equal to R(G) (see Proposition 49).s11.3 Optimality The sharing implementations are tightly related to -calculus optimal reductions [L'ev80] and to their generalization to Interaction Systems =-=[AL94]-=-. Anyhow, because of the generality of the rewriting system oe, such a correspondence is restricted to the implementation of the fi-rule only (i.e., we cannot say anything on how to optimize the numbe... |

38 | Interaction combinators
- Lafont
- 1997
(Show Context)
Citation Context ...sed on an applicative principle, but to simulate the reduction of a -term inside Combinatory Logic may greatly increase the length of the reduction. In spite of this, Lafont's interaction combinators =-=[Laf97]-=- and the work of Fern'andez and Mackie [FM96b,FM96a] on how to encode term rewriting systems into interaction nets might give useful insights for the determination of the class of systems implementabl... |

19 | Optimality and inefficiency: What isn’t a cost model of the lambda calculus
- Lawall, Mairson
- 1996
(Show Context)
Citation Context ...ink that the interest in the definition of a suitable notion of cost and of complexity classes for sharing computations is even more appealing after the results of Asperti [Asp96], Lawall and Mairson =-=[LM96]-=-, and Asperti and Mairson [AM98] showing that L'evy families cannot be the cost model of -calculus reductions. Acknowledgments I wish to thank Simone Martini who supervised my thesis and always encour... |

17 | The Bologna optimal higher-order machine
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- 1996
(Show Context)
Citation Context ...ossible to find a Curry-Howard analogy with a suitable intuition10 istic logic. Furthermore, Asperti used sharing graphs for the implementation of an optimal version of an ML-like functional language =-=[AGN96]-=-. The main concern of all these studies have been the implementation of optimal reductions. Hence, the set of rules that they proposed was the maximal one preserving optimality. Here, we revert the po... |

16 | Parallel beta reduction is not elementary recursive
- Asperti, Mairson
(Show Context)
Citation Context ...inition of a suitable notion of cost and of complexity classes for sharing computations is even more appealing after the results of Asperti [Asp96], Lawall and Mairson [LM96], and Asperti and Mairson =-=[AM98]-=- showing that L'evy families cannot be the cost model of -calculus reductions. Acknowledgments I wish to thank Simone Martini who supervised my thesis and always encouraged me persisting, even when th... |

14 | On the fine structure of the exponential rule - Martini, Masini - 1994 |

12 | Coherence for sharing proof nets
- Guerrini, Martini, et al.
(Show Context)
Citation Context ... fits into a well-known approach in which dependencies between the formulas of a proof (net) are represented by means of indexes. For a detailed discussion of these connections we refer the reader to =-=[GMM97a]-=-. 1.5 Overview of the paper The body of the paper starts with a formal definition of the structures that we shall study (section 2). The main difference w.r.t. what we have done so far is that we shal... |

9 |
On the complexity of beta-reduction
- Asperti
- 1996
(Show Context)
Citation Context ... in complexity theory. We think that the interest in the definition of a suitable notion of cost and of complexity classes for sharing computations is even more appealing after the results of Asperti =-=[Asp96]-=-, Lawall and Mairson [LM96], and Asperti and Mairson [AM98] showing that L'evy families cannot be the cost model of -calculus reductions. Acknowledgments I wish to thank Simone Martini who supervised ... |

8 | Proof nets, garbage, and computations - Guerrini, Martini, et al. - 2001 |

7 | From term rewriting to generalised interaction nets - Fernández, Mackie - 1996 |

6 | logic, comonads and optimal reductions - Linear - 1995 |

3 |
Local and asyncrhonous betareduction
- Danos, Regnier
- 1993
(Show Context)
Citation Context ...e of the integer functional ffi m;q . For this purpose, we need an axiomatization of lifting operators and of their inverse functionals (in the style of Danos and Regnier's dynamic algebra, e.g., see =-=[DR93]-=-) and a detailed study of their properties. 7.3 Left inverses of lifting operators The endomorphisms L[m; q; a] are injective (by equation (LO1)). Hence, each L[m; q; a] has a left inverse L[m; q; a].... |

3 | Interaction nets and term rewriting systems (extended abstract - Fernández, Mackie - 1996 |

3 |
sharing-morphisms and (optimal) - graph reductions
- Sharing-graphs
(Show Context)
Citation Context ...nt is not explicit in the paper, but it is implicit in the definition of least-shared-instance that we shall give. For the case of -calculus, a more direct presentation based on paths can be found in =-=[Gue97]-=-.) The proper sharing graphs defined in this way contain the ones obtainable as a result of a sharing reduction and, maybe, it could be proved that the two classes coincide. The result of this approac... |

2 |
Comparing -calculus translations in sharing graphs
- Asperti, Laneve
- 1995
(Show Context)
Citation Context ...ss if we proceed to fi-reduction in a global way; in the sharing implementation of -calculus, the boxing of -terms turns to be necessary, although the one presented here is not the only solution; see =-=[AL95]-=-.) Boxes and rewriting rules of the calculus must be compatible, i.e., the correctness of the boxing must be preserved by the fi-rule. Therefore, the boxing rules for the -calculus cannot be arbitrary... |

2 |
Optimality and ine ciency: What isn't a cost model of the lambda calculus
- Lawall, Mairson
- 1996
(Show Context)
Citation Context ...k that the interest in the de nition of a suitable notion of cost and of complexity classes for sharing computations is even more appealing after the results of Asperti [Asp96] and Lawall and Mairson =-=[LM96]-=- showing that Levy families are unlike to be the cost model of -calculus reductions. Acknowledgments I thank Simone Martini who supervised my thesis and always encouraged me persisting, even when thin... |

2 |
sharing-morphisms, and (optimal) λgraph reductions
- Sharing-graphs
- 1995
(Show Context)
Citation Context ...s not 11 explicit in the paper, but it is implicit in the definition of least-shared-instance that we shall give. For the case of λ-calculus, a more direct presentation based on paths can be found in =-=[Gue97]-=-.) The proper sharing graphs defined in this way contain the ones obtainable as a result of a sharing reduction and, maybe, it could be proved that the two classes coincide. The result of this approac... |

1 |
Interaction combinators. FTP
- Lafont
- 1995
(Show Context)
Citation Context ...sed on an applicative principle, but to simulate the reduction of a -term inside Combinatory Logic may greatly increase the length of the reduction. In spite of this, Lafont's interaction combinators =-=[Laf95]-=- and the work of Fern'andez and Mackie [FM96b,FM96a] on how to encode term rewriting systems into interaction nets might give useful insights for the determination of the class of systems implementabl... |

1 | logic, comonads and optimal reductions. Fundamentae Informaticae, 22:3{22 - Linear - 1995 |

1 |
Comparing λ-calculus translations in sharing graphs
- Asperti, Laneve
- 1995
(Show Context)
Citation Context ...s if we proceed to β-reduction in a global way; in the sharing implementation of λ-calculus, the boxing of λ-terms turns to be necessary, although the one presented here is not the only solution; see =-=[AL95]-=-.) Boxes and rewriting rules of the calculus must be compatible, i.e., the correctness of the boxing must be preserved by the β-rule. Therefore, the boxing rules for the λ-calculus cannot be arbitrary... |