## Interaction systems and linear logic -- A different . . . (2009)

### BibTeX

@MISC{Hyvernat09interactionsystems,

author = {Pierre Hyvernat},

title = {Interaction systems and linear logic -- A different . . . },

year = {2009}

}

### OpenURL

### Abstract

### Citations

261 |
Programming in Martin-Löf’s Type Theory, An introduction
- Nordström, Peterson, et al.
- 1990
(Show Context)
Citation Context ... 336.2 Containers In [3], the authors study the notion of container, a structure bearing several similarities with the notion of interaction system. They work in a variant of Martin-Löf type theory (=-=[21,22]-=-), a dependent predicative type theory. Mild knowledge about this type theory is assumed in this section. Simply, a container is given by the following: • a set A of shapes; • and for any a ∈ A, a set... |

193 | P.: Full abstraction for PCF
- Abramsky, Jagadeesan, et al.
- 2000
(Show Context)
Citation Context ...evier 25 May 2009Introduction Transition systems and simulation relations are well known tools in computer science. More recent is the use of games to give models for different programming languages =-=[4,5]-=-, or as an interesting tool for the study of other programming notions [6]. We devise a denotational model of linear logic based on those two ideas. Basically, a formula will be interpreted by an “alt... |

142 |
1984] Intuitionistic Type Theory Bibliopolis
- Martin-Löf
(Show Context)
Citation Context ... 336.2 Containers In [3], the authors study the notion of container, a structure bearing several similarities with the notion of interaction system. They work in a variant of Martin-Löf type theory (=-=[21,22]-=-), a dependent predicative type theory. Mild knowledge about this type theory is assumed in this section. Simply, a container is given by the following: • a set A of shapes; • and for any a ∈ A, a set... |

108 | Categories for the Working Mathematician. 2nd edition - Lane - 1998 |

87 |
Autonomous categories and linear logic
- Barr
- 1991
(Show Context)
Citation Context ...More precisely, for any state s, the function a ↦→ D(s, a) is constant.) PROOF. Just notice that w ⊥⊥ satisfies this property. ✷ 30Finally, we have Corollary 35 The category Int is ⋆-autonomous (see =-=[19]-=-), we can thus interpret classical linear logic. PROOF. Once we know that ⊥ is dualizing, the remaining condition are fairly easy to check: the following diagram should be commutative w1 ⊸ w2 ⊥ ✲ w ⊥ ... |

63 |
Containers: Constructing strictly positive types
- Abbott, Altenkirch, et al.
- 2005
(Show Context)
Citation Context ...constructive principle, that this category is in fact a model for full classical linear logic ; and we finally have a brief look at the related notions of predicate transformers ([2]) and containers (=-=[3]-=-). Key words: Denotational Semantics, Linear Logic, Differential λ-calculus, Interaction Systems, Simulations, Games Semantics, Containers Email address: pierre.hyvernat@univ-savoie.fr (Pierre Hyverna... |

56 |
Locally cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ...ut the idea is similar. There is currently some work being done on generalizing containers to a notion of dependent containers, i.e. interaction systems. The idea, following the original intuition of =-=[23]-=- and [24] is to define a dependent container in a locally cartesian closed category C as: • an object S in C; • an object A in C/S; • an object D in C/(ΣSA); 13 • a morphism n in C(ΣΣSAD , S). The app... |

51 | Finiteness spaces
- Ehrhard
- 2005
(Show Context)
Citation Context ...lculus. This is particularly interesting because the original model for differential λ-calculus (finiteness spaces) did not have a reflexive object: they could not interpret fixpoint combinators (see =-=[15]-=-). 5 Classical Linear Logic If one has in mind the definition of negation (see page 15), the next result can look quite surprising: interaction systems can interpret classical linear logic. In other w... |

39 | On the interpretation of type theory in locally cartesian closed categories
- Hofmann
- 1995
(Show Context)
Citation Context ...ea is similar. There is currently some work being done on generalizing containers to a notion of dependent containers, i.e. interaction systems. The idea, following the original intuition of [23] and =-=[24]-=- is to define a dependent container in a locally cartesian closed category C as: • an object S in C; • an object A in C/S; • an object D in C/(ΣSA); 13 • a morphism n in C(ΣΣSAD , S). The appropriate ... |

34 | Interactive programs in dependent type theory
- Hancock, Setzer
- 2000
(Show Context)
Citation Context ...raction Systems The definition of interaction system we are using was developed primarily by Peter Hancock and Anton Setzer. Their aim was to describe programming interfaces in dependent type theory (=-=[7,8]-=-). The ability to use dependent types makes it possible to add logical specifications to usual specification. The result is a notion of formal interface describing: hal-00387605, version 1 - 25 May 20... |

32 | Applying game semantics to compositional software modelling and verification, in: Tools and Algorithms for the Construction and Analysis
- Abramsky, Ghica, et al.
(Show Context)
Citation Context ...re well known tools in computer science. More recent is the use of games to give models for different programming languages [4,5], or as an interesting tool for the study of other programming notions =-=[6]-=-. We devise a denotational model of linear logic based on those two ideas. Basically, a formula will be interpreted by an “alternating transition system” (called an interaction system) and a proof wil... |

26 | On full abstraction for
- Hyland, Ong
- 2000
(Show Context)
Citation Context ...evier 25 May 2009Introduction Transition systems and simulation relations are well known tools in computer science. More recent is the use of games to give models for different programming languages =-=[4,5]-=-, or as an interesting tool for the study of other programming notions [6]. We devise a denotational model of linear logic based on those two ideas. Basically, a formula will be interpreted by an “alt... |

23 | Categorical models of linear logic revisited. Available at http://www.pps.jussieu.fr/mellies/papers/catmodels.ps.gz
- Mellies
- 2002
(Show Context)
Citation Context ...l model for intuitionistic linear logic. The details of categorical models for linear logic are quite intricate, and there are several notions, not all of which are equivalent. We refer to the survey =-=[13]-=- and the references given there. In the case of Int, the situation is however quite simple: proposition 21 makes Int into a “Lafont category” (see [14]). Corollary 22 With the construction defined in ... |

16 |
Calculi for synchrony and asynchrony, Theoret
- Milner
- 1983
(Show Context)
Citation Context ...ce of inverses is irrelevant. 91.3.3 Synchronous Product. There is an obvious tensor construction reminiscent of the synchronous product found in SCCS (synchronous calculus of communicating systems, =-=[12]-=-): Definition 12 Suppose w1 and w2 are interaction systems on S1 and S2. Define the interaction system w1 ⊗ w2 on S1 × S2 as follows: Aw1⊗w2((s1, s2)) = A1(s1) × A2(s2) Dw1⊗w2((s1, s2), (a1, a2)) = D1... |

10 | Differential categories
- Blute, Cockett, et al.
- 2005
(Show Context)
Citation Context ...2009 The general notion of categorical model for the differential λ-calculus (or differential proof nets, or “differential linear logic”) is only beginning to emerge. The main paper on the subject is =-=[17]-=-, where the categorical notion of “differentiation combinator” is studied in details. No real soundness theorem is however proved there because the authors work in a more general setting: the base cat... |

8 |
The differential lambda-calculus. Theoretical Computer Science, 309(1-3):1–41, 2003. 14 Thomas Ehrhard and Laurent Regnier. Böhm trees, Krivine machine and the Taylor expansion of ordinary lambda-terms
- Ehrhard, Regnier
(Show Context)
Citation Context ...tional translations, serve as a model for the simply typed λ-calculus. We use some of the additional structure of the category to extend this to a model of the simply typed differential λ-calculus of =-=[1]-=-. Once this is done, we go a little further by constructing a reflexive object in this category, thus getting a concrete non-trivial model for the untyped differential λ-calculus. We then show, using ... |

7 | Predicate transformers and linear logic: yet another denotational model
- Hyvernat
- 2004
(Show Context)
Citation Context ...ux’s “differential λµ-calculus” (see [20]), either in Vaux’s setting (typed) or in an untyped setting. 316 Related Notions 6.1 Predicate Transformers The category Int has a very concrete feeling. In =-=[2]-=-, we have developed a model for full linear logic with a different intuition: the category of predicate transformers with forward data-refinements: Definition 36 If S is a set, a predicate transformer... |

2 |
Programming interfaces and basic topology, Annals of Pure and Applied Logic 137
- Hancock, Hyvernat
- 2006
(Show Context)
Citation Context ...ulation from this lower level interaction system to stacks: for each of the stacks commands, we need to provide a witness command in the low level world in such a way as to guarantee simulation. (See =-=[9]-=- and [10] for a more detailed description of programming in terms of interaction systems.) Recall that the composition of two relations is given by: (s1, s3) ∈ r2 · r1 ⇔ (∃s2) (s1, s2) ∈ r1 and (s2, s... |

2 |
catégories et machines, Thèse de doctorat, Université Paris 7
- Lafont, Logiques
- 1988
(Show Context)
Citation Context ... of which are equivalent. We refer to the survey [13] and the references given there. In the case of Int, the situation is however quite simple: proposition 21 makes Int into a “Lafont category” (see =-=[14]-=-). Corollary 22 With the construction defined in the previous sections, Int is a Lafont category. In particular, “! ” is a comonad; and we have for any w1 and w2, we have the following natural isomorp... |

1 |
A logical investigation of interaction systems, Thèse de doctorat, Institut mathématique de Luminy, Université Aix-Marseille II
- Hyvernat
- 2005
(Show Context)
Citation Context ...from this lower level interaction system to stacks: for each of the stacks commands, we need to provide a witness command in the low level world in such a way as to guarantee simulation. (See [9] and =-=[10]-=- for a more detailed description of programming in terms of interaction systems.) Recall that the composition of two relations is given by: (s1, s3) ∈ r2 · r1 ⇔ (∃s2) (s1, s2) ∈ r1 and (s2, s3) ∈ r2 I... |

1 |
λ-calculus in an algebraic setting, unpublished note
- Vaux
- 2006
(Show Context)
Citation Context ... of λ-terms. One consequence is that we need a notion of sum of terms, interpreted as a non-deterministic choice. It is also possible to only add sums (and coefficients) to the usual λ-calculus as in =-=[16]-=-. hal-00387605, version 1 - 25 May 2009 It is not the right place to go into the details of the differential λ-calculus and we refer to [1] for motivations and a complete description. A complete defin... |

1 |
Differential λµ-calculus, technical report
- Vaux
- 2005
(Show Context)
Citation Context ...uctive and almost counter-intuitive. Putting the model of the differential λ-calculus with the dualizing object ⊥, it is expected that we get a model for Lionel Vaux’s “differential λµ-calculus” (see =-=[20]-=-), either in Vaux’s setting (typed) or in an untyped setting. 316 Related Notions 6.1 Predicate Transformers The category Int has a very concrete feeling. In [2], we have developed a model for full l... |