## Finiteness spaces (1987)

### Cached

### Download Links

- [www.pps.jussieu.fr]
- [www.pps.univ-paris-diderot.fr]
- [iml.univ-mrs.fr]
- DBLP

### Other Repositories/Bibliography

Venue: | Mathematical Structures in Computer Science |

Citations: | 50 - 12 self |

### BibTeX

@INPROCEEDINGS{Ehrhard87finitenessspaces,

author = {Thomas Ehrhard},

title = {Finiteness spaces},

booktitle = {Mathematical Structures in Computer Science},

year = {1987}

}

### Years of Citing Articles

### OpenURL

### Abstract

We investigate a new denotational model of linear logic based on the purely relational model. In this semantics, webs are equipped with a notion of “finitary ” subsets satisfying a closure condition and proofs are interpreted as finitary sets. In spite of a formal similarity, this model is quite different from the usual models of linear logic (coherence semantics, hypercoherence semantics, the various existing game semantics...). In particular, the standard fix-point operators used for defining the general recursive functions are not finitary, although the primitive recursion operators are. This model can be considered as a discrete version of the Köthe space semantics introduced in a previous paper: we show how, given a field, each finiteness space gives rise to a vector space endowed with a linear topology, a notion introduced by Lefschetz in 1942, and we study the corresponding model where morphisms are linear continuous maps (a version of Girard’s quantitative semantics with coefficients in the field). We obtain in that way a new model of the recently introduced differential lambda-calculus. Notations. If S is a set, we denote by M(S) = N S the set of all multi-sets over S. If µ ∈ M(S), |µ| denotes the support of µ which is the set of all a ∈ S such that µ(a) ̸ = 0. A multi-set is finite if it has a finite support. If a1,..., an are elements of some given set S, we denote by [a1,..., an] the corresponding multi-set over S. The usual operations on natural numbers are extended to multi-sets pointwise. If (Si)i∈I are sets, we denote by πi the i-th projection πi: ∏ j∈I Sj → Si.

### Citations

236 |
Categories for the working mathematician., volume 5 of Graduate text in Mathematics
- Lane
- 1971
(Show Context)
Citation Context ...i∈I Xi, a property which is completely standard in this kind of categories where morphisms can be added and where composition commutes to these sums (a category enriched over commutative monoids, see =-=[Mac71]-=-). 15Exponentials. Let us first introduce some additional notations concerning finite multi-sets. If µ is an element of Mfin(I), we define its size (or cardinality) as #µ = ∑ i∈I µ(i) ∈ N. We also de... |

150 |
The system F of variable types, fifteen years later, Theoret
- Girard
- 1986
(Show Context)
Citation Context ...are the finiteness spaces, and we also extend this semantics to the second order, using new ideas presented in [Bac00] for extending to non-uniform settings the constructions that Girard performed in =-=[Gir86]-=- for qualitative domains and coherence spaces. 1 Or successively; the multiset can be interpreted (e.g. in the forthcoming example) as giving the different values taken by the argument during the comp... |

149 |
Proofs and Types, volume 7 of Cambridge Tracts
- Girard, Lafont, et al.
- 1989
(Show Context)
Citation Context ...the theory of coherence space models (for instance), e.g. in the coherence space interpretation of system F (it is the essence of the so-called Moggi's observations that morphisms compose nicely, see =-=[GLT89] page 141)-=-, or in the "normal form theorems" which give a functional account of morphisms as in [BE99a]. It is also because of this property that coherence spaces and hypercoherences admit an interpre... |

98 | What is a categorical model of intuitionistic linear logic?’, Typed lambda calculi and applications
- Bierman
- 1995
(Show Context)
Citation Context ...ubset of the web of a finiteness space is finitary, but the converse is of course false in general. In that way, we build a new model of first order propositional linear logic, a linear category (see =-=[Bie95]-=-) whose objects are the finiteness spaces, and we also extend this semantics to the second order, using new ideas presented in [Bac00] for extending to non-uniform settings the constructions that Gira... |

59 | Hypercoherence: A strongly stable model of linear logic
- Ehrhard
- 1995
(Show Context)
Citation Context ...nce, finiteness spaces are very different from coherence spaces. It is not clear whether these various cases can be handeled within a common framework. Interestingly enough, the hypercoherence model (=-=[Ehr93]-=-) does not seem to admit such a synthetic description. 8This set can be assumed to be countable, a property preserved by all the constructions we consider. 4Orthogonal. The space X ⊥ is defined by |X... |

31 | Probabilistic game semantics
- Danos, Harmer
(Show Context)
Citation Context ... Or successively; the multiset can be interpreted (e.g. in the forthcoming example) as giving the different values taken by the argument during the computation. In the non-deterministic game model of =-=[DH00]-=- the same information is made available, but moreover, the temporal scheduling (the order in which these values are taken) is specified. 2 This property is not only mathematically appealing, but it al... |

29 | On köthe sequence spaces and linear logic
- Ehrhard
(Show Context)
Citation Context ...e functions, showing that we have obtained in that way a model of the recently introduce differential lambdacalculus [ER01], categorically completely similar to the model of Köthe spaces presented in =-=[Ehr02]-=-. All these constructions can be seen as rephrasing Girard’s quantitative semantics of the lambda-calculus presented in [Gir88] (see also [Has02]) where lambda-terms are interpreted as normal functors... |

20 |
On phase semantics and denotational semantics: the exponentials
- Bucciarelli, Ehrhard
(Show Context)
Citation Context ...ls are interpreted as the operation which maps a set E to the set of all finite multi-sets with domain included in E (taking here finite sets instead of multi-sets would not give rise to a model). In =-=[BE99b]-=-, we have shown how to equip this model with various generalized and phase-valued notions of coherence; we have obtained in that way a non-uniform version of coherence spaces, but a lot of hitherto un... |

16 |
Two applications of analytic functors
- Hasegawa
(Show Context)
Citation Context ...tely similar to the model of Köthe spaces presented in [Ehr02]. All these constructions can be seen as rephrasing Girard’s quantitative semantics of the lambda-calculus presented in [Gir88] (see also =-=[Has02]-=-) where lambda-terms are interpreted as normal functors which are power series whose coefficients are (possibly infinite) sets; the role of our additional finiteness space structure is to keep these c... |

15 |
Normal functors, power series and λ-calculus
- Girard
- 1988
(Show Context)
Citation Context ...tegorically completely similar to the model of Köthe spaces presented in [Ehr02]. All these constructions can be seen as rephrasing Girard’s quantitative semantics of the lambda-calculus presented in =-=[Gir88]-=- (see also [Has02]) where lambda-terms are interpreted as normal functors which are power series whose coefficients are (possibly infinite) sets; the role of our additional finiteness space structure ... |

11 | Hopf algebras and linear logic - Blute - 1996 |

5 |
Algebraic topology. Number 27 in American mathematical societey colloquium publications
- Lefschetz
- 1942
(Show Context)
Citation Context ...t we tend to consider this as a rather interesting feature: after all we do not have so many simple denotational models introducing natural divides between computational primitives. 5 In the sense of =-=[Lef42]-=-. This is a notion of topology for vector spaces or modules where basic neighborhoods are linear subspaces and which is therefore quite different from the usual notions considered in functional analys... |

5 |
logic, totality and full completeness
- Linear
- 1994
(Show Context)
Citation Context ...d model of coherence spaces; • u ∩ u ′ is not empty, which gives rise to a quite simple model of non uniform totality; • u ∩ u ′ has exactly one element, which gives rise to Loader’s totality spaces (=-=[Loa94]-=-); • another natural choice, suggested by one of the referees of this paper, might be to require u ∩ u ′ to be cofinite. We have no idea about the resulting model, if any. Due, maybe, to the logical c... |

4 | Quantitative semantics revisited (extended abstract - Barreiro, Ehrhard - 1999 |

1 |
On phase semantics and denotational semantics: the second order. Submitted for publication. Available on http://iml.univ-mrs.fr/~bac
- Bac
- 2000
(Show Context)
Citation Context .... This model can be considered as a discrete version of the Kothe space model presented in [Ehr00]. We also extend this semantics to the second order, following methods developed by A. Bruasse Bac in =-=[Bac00]-=-. Introduction The simplest denotational model of linear logic is doubtlessly the category of sets and relations. There, the additive operations are interpreted as the disjoint sum of sets, the multip... |

1 |
On Kothe sequence spaces and linear logic. IML technical report, submitted for publication
- Ehrhard
- 2000
(Show Context)
Citation Context ...e.g. in PCF) the general recursive functions are not finitary, although the primitive recursion operators are. This model can be considered as a discrete version of the Kothe space model presented in =-=[Ehr00]-=-. We also extend this semantics to the second order, following methods developed by A. Bruasse Bac in [Bac00]. Introduction The simplest denotational model of linear logic is doubtlessly the category ... |

1 | Duality of vector spaces. Cahiers de Topologie et Géométrie Différentielles - Barr - 1976 |

1 |
The differential lambda-calculus. Technical report, Institut de Mathématiques de Luminy
- Ehrhard, Regnier
- 2001
(Show Context)
Citation Context ..., pure lambda-terms should admit a finitary interpretation. Second, the finiteness space model presented here is perfectly adapted for interpreting the simply typed differential lambda-calculus 15 of =-=[ER03]-=-, but this calculus admits a natural untyped version whose denotational semantics would require something like a finitary model of the pure lambda-calculus. The solution might be a non standard interp... |

1 |
26 Ryu Hasegawa. The generating functions of lambda terms (extended abstract
- Solum
- 2001
(Show Context)
Citation Context ...rogram. But here, since the typical ever-looping program which is (Y )λx x cannot be interpreted, one should maybe rather consider the empty set as a kind of “daemon”, in the sense of Girard’s ludics =-=[Gir01]-=-, that is, a pure termination, without resulting information. One peculiarity of this model is that the finiteness space associated with a formula A of linear logic has as web the set interpreting A i... |