## On Köthe sequence spaces and linear logic (2001)

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Venue: | Mathematical Structures in Computer Science |

Citations: | 30 - 9 self |

### BibTeX

@INPROCEEDINGS{Ehrhard01onköthe,

author = {Thomas Ehrhard},

title = {On Köthe sequence spaces and linear logic},

booktitle = {Mathematical Structures in Computer Science},

year = {2001},

pages = {579--623}

}

### Years of Citing Articles

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### Abstract

We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed -calculus and dierential calculus can be combined; we give a few examples of computations. 1

### Citations

365 |
Theorie Des Distributions
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Citation Context ...pidly decreasing smooth functions would have been replaced by a space of entire mappings (many very good texts are available on the theory of distributions, let us just mention Schwartz's famous book =-=[Sch66-=-]). With this intuition in mind, we can adopt a more convenient notation for the application of an element S of A(E) 0 On K othe sequence spaces and linear logic 37 to an element h of A(E): following ... |

216 |
Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics
- Lane
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Citation Context ...vergence" (entire functions), although Kothe spaces are probably more expressive than that. We just want to illustrate the fact that rather simple topological vector spaces can quite easily be 5 =-=See [Mac7-=-1], chapter 7, section 8 and also the introduction of [Gir99]. On K othe sequence spaces and linear logic 5 used for modeling standard linear logic and typed -calculus 6 . The paper remains at an elem... |

198 | The Convenient Setting of Global Analysis - Kriegl, Michor - 1997 |

141 | Proofs and types, volume 7 of Cambridge Tracts - Girard, Taylor, et al. - 1989 |

99 | What is a categorical model of intuitionistic linear logic
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(Show Context)
Citation Context ...nd the commutation of the Mac Lane pentagonal diagram are easily checked too. So the category KK is symmetric monoidal closed. It is actually a ?-autonomous category in the sense of [Bar79] (see also =-=[B-=-ie95]). Let f : E Y ! EX and g : EZ ! E T be linear continuous maps between Kothe spaces. Then we can dene a linear continuous map f ( g : EX(Z ! E Y (T by its matrix (indexed by (jXj jZj) (jY j jT... |

87 | Autonomous categories and linear logic - Barr - 1991 |

73 |
Locally Convex Spaces
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(Show Context)
Citation Context ...e elements of E ? ): these spaces are Kothe sequence spaces 4 (which are classically dened as the orthogonals of certain subsets of K I called Kothe sets), a notion due to Kothe and Toeplitz; see [Jar=-=81-=-], which will be our basic reference for locally convex spaces. Another classical reference book on this topic is [Sch73], see chapter 4 for a few informations on Kothe sequence spaces. More detailed ... |

51 | Finiteness spaces
- Ehrhard
- 2005
(Show Context)
Citation Context ...s in staying in the discrete setting (still considering subsets of the webs) and saying that two subsets are orthogonal if their intersection issnite; this leads to the notion ofsniteness spaces, see =-=[Ehr00]. More pre-=-cisely, an at most countable set I being given (the web), we replace the notion of \subset of I" by the notion of \element of K I ". Then, given x; x 0 2 K I we replace the notion of \cardin... |

44 | The differential lambda-calculus
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(Show Context)
Citation Context ... systems and typed -calculus by dierential constructions. With L. Regnier, we have been able in that way to dene a dierential lambda-calculus where lambda-terms can be formally dierentiated, see [ER01=-=]-=-. 5 Towards an intrinsic theory All the spaces introduced so far were equipped with explicit bases (the webs) which, from the viewpoint of classical functional analysis, seems to be a major drawback o... |

30 | R.: Probabilistic Games Semantics
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- 2000
(Show Context)
Citation Context ... by Barr in [Bar79, Bar90]. Other semantics of typed lambda-calculi with real coecients have been considered, among which we would like to mention in particular Danos and Harmer's probabilistic games =-=[DH00]-=-, whose non-deterministic character seems to be very much in the spirit of the model presented here. We propose a quite concrete approach based on a standard notion in the theory of locally convex top... |

19 |
T.: On phase semantics and denotational semantics: the exponentials
- Bucciarelli, Ehrhard
- 2001
(Show Context)
Citation Context ...pt to denotational semantics, one must consider settings where the exponentials of linear logic are interpreted in a non-uniform way, as in the models induced by the symmetric product phase spaces of =-=[BE99b]-=-. The usual coherence space semantics indeed is uniform: this means that the elements of the web of the coherence space !E are thesnite cliques (or multi-cliques) of the coherence space E. For that re... |

16 | Coherent Banach spaces: a continuous denotational semantics
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(Show Context)
Citation Context ...e must associate innite dimensional spaces and for that reason, topological vector spaces must be considered. Several models based on topological vector spaces have already been presented, by Girard [=-=Gir99]-=- and Blute [Blu96], in a setting consideredsrst by Barr in [Bar79, Bar90]. Other semantics of typed lambda-calculi with real coecients have been considered, among which we would like to mention in par... |

13 |
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Citation Context ...onstruction presented in that book (and simplied later with the help of Chu spaces) yields a ?-autonomous category (a model of multiplicative-additive linear logic) which, using the constructions of [=-=Bar90-=-], should lead to a model of full propositional linear logic based on topological vector spaces. These 1 On K othe sequence spaces and linear logic 2 constructions however are fairly abstract, and we ... |

12 |
autonomous categories. Number 752
- Barr
- 1979
(Show Context)
Citation Context ...te dierent from Blute's model as the topology we shall consider on the underlyingseld (theseld of real numbers, or theseld of complex numbers) will be the standard topology, not the discrete one. In [=-=Bar79-=-], chapter 4, example 1, topological vector spaces over theseld of real or complex numbers with the usual topology are also considered, and the construction presented in that book (and simplied later ... |

11 |
Hopf algebras and linear logic
- Blute
- 1996
(Show Context)
Citation Context ...nnite dimensional spaces and for that reason, topological vector spaces must be considered. Several models based on topological vector spaces have already been presented, by Girard [Gir99] and Blute [=-=Blu96]-=-, in a setting consideredsrst by Barr in [Bar79, Bar90]. Other semantics of typed lambda-calculi with real coecients have been considered, among which we would like to mention in particular Danos and ... |

6 |
Partially Additive Categories and Fully Complete Models of Linear Logic
- Haghverdi
- 2001
(Show Context)
Citation Context ...s intuitively to the superimposition of various possible results of a (non-deterministic) computation: this use of the sum operation has already been considered by Arbib and Manes in [AM86], see also =-=[Hag01] wher-=-e Haghverdi uses sums (at the \geometry of interaction" level) for building fully complete models of multiplicative linear logic. Possible connections between our model and Haghverdi's constructi... |

5 |
Normal functors� power series� and ��calculus
- Girard
- 1988
(Show Context)
Citation Context ...ample, the coecient 2 corresponds to a component of f which would not exist in a uniform setting. It must be pointed out that this eect was already present in Girard's work on quantitative semantics [=-=Gir8-=-8], since this model had no uniformity constraints, and Ryu Hasegawa showed in [Has97] how to compute eectively the entire coecients associated to the interpretations of -terms using generating functi... |

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

4 |
Topological Vector Spaces, volume 3 of Graduate Texts in Mathematics
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(Show Context)
Citation Context ...in subsets of K I called Kothe sets), a notion due to Kothe and Toeplitz; see [Jar81], which will be our basic reference for locally convex spaces. Another classical reference book on this topic is [S=-=ch7-=-3], see chapter 4 for a few informations on Kothe sequence spaces. More detailed informations on this topic can also be found in [Pie72] and of course in [Kot66] (in German). The topological vector sp... |

3 |
Linear spaces and di erentiation theory
- Frohlicher, Kriegl
- 1988
(Show Context)
Citation Context ...eem weird, it is completely standard in the theory of locally convex spaces. For instance, the extension of dierentiability to innite dimensional vector spaces developed by Frolicher and Kriegl in [FK=-=88] is a-=-pparently based on the idea that smoothness is not a topological notion, but a \bornological" notion. This viewpoint is exploited in the book by Kriegl and Michor [KM97]. At the end of the paper,... |

2 |
Nuclear locally convex spaces, volume 66 of Ergebnisse der Mathematik und ihrer Grenzgebiete
- Pietsch
- 1972
(Show Context)
Citation Context ...convex spaces. Another classical reference book on this topic is [Sch73], see chapter 4 for a few informations on Kothe sequence spaces. More detailed informations on this topic can also be found in [=-=Pie7-=-2] and of course in [Kot66] (in German). The topological vector spaces so dened are not always (more precisely, they are almost never) Banach spaces, they can even not be metrizable, but they behave r... |

1 |
Arbib and Ernest Manes. Algebraic approaches to program semantics
- Michael
- 1986
(Show Context)
Citation Context ...spaces corresponds intuitively to the superimposition of various possible results of a (non-deterministic) computation: this use of the sum operation has already been considered by Arbib and Manes in =-=[AM86], see-=- also [Hag01] where Haghverdi uses sums (at the \geometry of interaction" level) for building fully complete models of multiplicative linear logic. Possible connections between our model and Hagh... |

1 |
The generating functions of lambda terms (extended abstract
- Hasegawa
- 1997
(Show Context)
Citation Context ... (which is then necessarily unique). This interpretation of intuitionistic proofs as entire maps is of course completely in the spirit of Girard's quantitative semantics (see [Gir88, BE99a], and also =-=[Has97] whe-=-re a method is developed for computing the coecients of the \power series" associated to -terms in Girard's quantitative semantics). The space !X has a co-algebraic structure, as it is standard i... |

1 | Topologische Lineare Raume I. Number 107 in Die Grundlehren der mathematischen Wissenschaften - Kothe - 1966 |