## A finiteness structure on resource terms (2010)

Venue: | IN LICS |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Ehrhard10afiniteness,

author = {Thomas Ehrhard},

title = {A finiteness structure on resource terms},

booktitle = {IN LICS},

year = {2010},

pages = {402--410},

publisher = {IEEE Computer Society}

}

### OpenURL

### Abstract

We study the Taylor expansion of lambda-terms in a non-deterministic or algebraic setting, where terms can be added. The target language is a resource lambda calculus based on a differential lambda-calculus we introduced recently. This operation is not possible in the general untyped case where reduction can produce unbounded coefficients. We endow resource terms with a finiteness structure (in the sense of our earlier work on finiteness spaces) and show that the Taylor expansions of terms typeable in Girard’s system F are finitary by a reducibility method.

### Citations

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(Show Context)
Citation Context ...xive coherence relation, observing that each set T (M) is a clique for this coherence relation and proving that NF can be seen as a stable and linear function on this coherence space (in the sense of =-=[Gir86]-=-). These properties are lost in the present setting and superpositions can occur and even lead to infinite sums, as in the Taylor expansion (that we do not compute here) of the termM = (Θ)λx(x+z) wher... |

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(Show Context)
Citation Context ...gebraic lambda-calculus3 ), then we cannot expect Taylor expansions to be cliques for that coherence relation. Instead, we equip the set of resource terms with a finiteness structure (in the sense of =-=[Ehr05]-=-) which is defined in such a way that for any “finitary” linear combination ∑ sαss of resource lambda-terms, the sum ∑ sαsas always makes sense, whatever be the choices of as such that s beta-reduces ... |

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Citation Context ... multiplicative, structural and costructural rules: the resource calculus that we introduced in [ER08]. Similar calculi already existed in the literature, such as Boudol’s calculi with multiplicities =-=[Bou93]-=- or with resources [BCL99], and also Kfoury’s calculi [Kfo00], introduced with different motivations and with different semantic backgrounds. The intuition behind our calculus with resources is as fol... |

16 |
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(Show Context)
Citation Context ... termsand the size S(S) of a simple poly-term by induction as follows: 3 There are other algebraizations of the lambda-calculus, we think in particular of the calculus considered by Arrighi and Dowek =-=[AD08]-=- which is quite different from ours because application is right-linear in their setting 3• S(x) = 1 • S(λxs) = 1+S(s) • S(〈s〉S) = 1+S(s)+S(S) • S(s1···s2) = S(s1)+···+S(sn). 1.1.1 Extended syntax. G... |

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Citation Context ...l and costructural rules: the resource calculus that we introduced in [ER08]. Similar calculi already existed in the literature, such as Boudol’s calculi with multiplicities [Bou93] or with resources =-=[BCL99]-=-, and also Kfoury’s calculi [Kfo00], introduced with different motivations and with different semantic backgrounds. The intuition behind our calculus with resources is as follows. The first thing to s... |

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10 |
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(Show Context)
Citation Context ...rce calculus that we introduced in [ER08]. Similar calculi already existed in the literature, such as Boudol’s calculi with multiplicities [Bou93] or with resources [BCL99], and also Kfoury’s calculi =-=[Kfo00]-=-, introduced with different motivations and with different semantic backgrounds. The intuition behind our calculus with resources is as follows. The first thing to say is that types should be thought ... |

8 |
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(Show Context)
Citation Context ...icative, structural and promotion rules. But we can also consider a lambda-calculus corresponding to the multiplicative, structural and costructural rules: the resource calculus that we introduced in =-=[ER08]-=-. Similar calculi already existed in the literature, such as Boudol’s calculi with multiplicities [Bou93] or with resources [BCL99], and also Kfoury’s calculi [Kfo00], introduced with different motiva... |

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Citation Context ...fferential LL by means of the Taylor expansion of promotion rules. hal-00448431, version 1 - 19 Jan 2010 Resource lambda-calculus. This operation is more easily understood in the lambda-calculus (see =-=[Tra08]-=- for the connection between lambda-terms and nets in differential LL). Roughly speaking, the ordinary lambda-calculus correspond to the fragment of LL which contains the multiplicative, structural and... |

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1 |
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Citation Context ... for which one checks easily that addition and scalar multiplication are continuous (k being equipped with the discrete topology). Actually k〈X〉 is a linearly topologized vector space in the sense of =-=[Lef42]-=-: the topology is generated by neighborhoods of 0 which are linear subspaces (for instance, the V0(e ′) we introduced above). This topology is Hausdorff: for anya ∈ k〈X〉, if a = 0 one cant find a (li... |