## A Fully Complete PER Model for ML Polymorphic Types (2000)

Venue: | Proceedings of CSL 2000, Springer LNCS Volume 1862 |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Abramsky00afully,

author = {Samson Abramsky and Marina Lenisa},

title = {A Fully Complete PER Model for ML Polymorphic Types},

booktitle = {Proceedings of CSL 2000, Springer LNCS Volume 1862},

year = {2000},

pages = {140--155},

publisher = {Springer}

}

### OpenURL

### Abstract

. We present a linear realizability technique for building Partial Equivalence Relations (PER) categories over Linear Combinatory Algebras. These PER categories turn out to be linear categories and to form an adjoint model with their co-Kleisli categories. We show that a special linear combinatory algebra of partial involutions, arising from Geometry of Interaction constructions, gives rise to a fully and faithfully complete model for ML polymorphic types of system F. Keywords: ML-polymorphic types, linear logic, PER models, Geometry of Interaction, full completeness. Introduction Recently, Game Semantics has been used to define fully-complete models for various fragments of Linear Logic ([AJ94a,AM99]), and to give fully-abstract models for many programming languages, including PCF [AJM96,HO96,Nic94], richer functional languages [McC96], and languages with non-functional features such as reference types and non-local control constructs [AM97,Lai97]. All these results are cru...

### Citations

214 | On full abstraction for PCF - Hyland, Ong - 2000 |

201 | Full abstraction for PCF
- Abramsky, Jagadeesan, et al.
- 2000
(Show Context)
Citation Context ...ibly fully-complete/fullyabstract) models. Examples of this latter use of realizability appear in this paper, and in [AL99], where a fully-abstract PER model for PCF, alternative to the game model of =-=[AJM96]-=-, is provided using the algebra of well-bracketed strategies. A categorical model of a type theory (or logic) is said to be fully-complete ([AJ94a]) if, for all types (formulae) A; B, all morphisms f ... |

168 |
Traced monoidal categories
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Citation Context ...e. for defining !ff. We now explain how application is computed geometrically, using the language of "boxes and wires" which arises in the general setting of traced symmetric monoidal catego=-=ries (see [JSV96]-=- for an abstract treatment). Let us represent an one-input/one-output function ff 2 [N * N] by the following one-input-port/one-output-port box (see Fig. 1(i) below). In order to define the applicatio... |

103 |
Towards a geometry of interaction
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- 1989
(Show Context)
Citation Context ... union of graph relations, and (fi; ff 11 ) ? denotes S n0 (fi; ff 11 ) n . The above formula for computing the application is essentially the Execution Formula from Girard's Geometry of Interaction (=-=[Gir89]-=-). The definition of the !-operation on our applicative structure is quite simple. The operation ! is intended to produce, from a single copy of ff, infinitely many copies of ff. These are obtained by... |

82 |
Categories for Types
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- 1994
(Show Context)
Citation Context ...t this model satisfies the axioms in the axiomatization of fully-complete models for ML-types given in [AL99a]. This axiomatization is given on the models of system F which are called hyperdoctrines (=-=[Cro93]-=-). In particular, it works in the context of adjoint models. It consists of two main steps. The first is an axiomatization of the fact that every morphism f : 1 ! [[T ]], where T is an ML-type generat... |

73 | New foundations for the geometry of interaction - Abramsky, Jagadeesan - 1992 |

68 | Full abstraction for functional languages with control - Laird |

46 |
DI-domains as a model of polymorphism
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- 1986
(Show Context)
Citation Context ...complete denotational models only for a small subclass of ML-types, i.e. the algebraic types (see [HRR90]). In the literature, there are two fully-complete models for the whole system F: i.e. that of =-=[BC88]-=-, and that of [Hug99]. The first model is based on a quotient of a term model, the latter is a game model. But both these models still have a somewhat syntactical flavor, and their constructions are e... |

36 | Full abstraction for idealized algol with passive expressions - Abramsky, McCusker - 1999 |

16 | Axioms for full abstraction and full completeness - Abramsky - 1997 |

13 |
and full abstraction for FPC
- Games
- 1996
(Show Context)
Citation Context ...y-complete models for various fragments of Linear Logic ([AJ94a,AM99]), and to give fully-abstract models for many programming languages, including PCF [AJM96,HO96,Nic94], richer functional languages =-=[McC96]-=-, and languages with non-functional features such as reference types and non-local control constructs [AM97,Lai97]. All these results are crucially based on the linear analysis of the intuitionistic a... |

10 | Retracing some paths - Abramsky - 1996 |

10 |
Hypergame semantics: full completeness for system F
- Hughes
- 2000
(Show Context)
Citation Context ...l models only for a small subclass of ML-types, i.e. the algebraic types (see [HRR90]). In the literature, there are two fully-complete models for the whole system F: i.e. that of [BC88], and that of =-=[Hug99]-=-. The first model is based on a quotient of a term model, the latter is a game model. But both these models still have a somewhat syntactical flavor, and their constructions are extremely complex. The... |

9 |
Realizability models based on history-free strategies. Unpublished draft
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(Show Context)
Citation Context ...ctions (e.g. in [AM99]), and as a technique for building directly interesting (possibly fully-complete/fullyabstract) models. Examples of this latter use of realizability appear in this paper, and in =-=[AL99]-=-, where a fully-abstract PER model for PCF, alternative to the game model of [AJM96], is provided using the algebra of well-bracketed strategies. A categorical model of a type theory (or logic) is sai... |

8 |
Categories, Types, and Structures. Foundations of Computing Series
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- 1991
(Show Context)
Citation Context ...l form ? does not appear. This result is used in the proof of full completeness of the model of PERs over the LCA of partial involutions. We assume that the reader is familiar with System F (see e.g. =-=[AL91]-=-). The class of ML-polymorphic types of system F corresponds to the limited kind of polymorphism allowed in the language ML. Definition 1 (ML-types). The class ML-Type of ML-types is defined by: ML-Ty... |

7 | Games semantics for untyped - calculus - Gianantonio, Franco, et al. - 1998 |

7 |
definable functionals and fij-conversion
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- 1983
(Show Context)
Citation Context ...calculus. In fact, any theory on ML-terms induces a theory satisfying Typical Ambiguity. The following is a result about simply typed -calculus with infinitely many type variabless1 , first proved in =-=[Sta88]-=-. Theorem 1 (Statman's Typical Ambiguity). Let T be a type ofs1 s.t. FV (T ) ` fX 1 ; : : : ; Xn g. If 6` M = fij N : T , then, there exist types S 1 ; : : : ; Sn , and Y 2 TV ar, and a term L s.t. ` ... |

6 | Axiomatizing Fully Complete Models for ML Polymorphic Types
- Abramsky, Lenisa
- 2000
(Show Context)
Citation Context ...nformation between input and output ports. The proof of full completeness consists in showing that this model satisfies the axioms in the axiomatization of fully-complete models for ML-types given in =-=[AL99a]-=-. This axiomatization is given on the models of system F which are called hyperdoctrines ([Cro93]). In particular, it works in the context of adjoint models. It consists of two main steps. The first i... |

5 |
and Full Completeness for Multiplicative Linear Logic
- Games
- 1994
(Show Context)
Citation Context ... PER model for PCF, alternative to the game model of [AJM96], is provided using the algebra of well-bracketed strategies. A categorical model of a type theory (or logic) is said to be fully-complete (=-=[AJ94a]-=-) if, for all types (formulae) A; B, all morphisms f : [[A]] ! [[B]], from the interpretation of A into the interpretation of B, are denotations of a proof-term of the entailment A ` B, i.e. if the in... |

5 |
Interprétation functionelle et élimunation des coupures de l’arithmètique d’ordre supérieur, Thèse d’Etat, Université Paris VII
- Girard
- 1972
(Show Context)
Citation Context ...enting operations which are often needed in defining game categories and models. Moreover, PER categories offer simple natural models for second order (polymorphic) -calculus, i.e. Girard's System F (=-=[Gir72]-=-). Recently, there has been much interest in realizability techniques, and in particular in linear realizability, especially in connection with full completeness and full abstraction problems. Realiza... |

4 | More Universal Game Models of Untyped - Calculus: The Bohm Tree Strikes Back - Ker, Ong - 1999 |

3 |
Geometry of Interaction and Models of Combinatory Logic
- Abramsky, Panangaden
- 1998
(Show Context)
Citation Context ...by the "combinatory version" of Girard's translation of Intuitionistic Logic into Linear Logic. We define: ff ffl s fi = ffffl!fi (standard combinators can be defined in terms of the linear =-=ones, see [AHPS98]-=- for details). We recall that a BCI-algebra is an applicative structure (A; ffl) with B; C; I combinators. In the next definition, we define a PER category over a BCIalgebra, which turns out to be sym... |

3 |
Algebraic types
- Hyland, Rosolini
- 1990
(Show Context)
Citation Context ...r is interesting, since, until now, the research on full completeness for System F has produced fully-complete denotational models only for a small subclass of ML-types, i.e. the algebraic types (see =-=[HRR90]-=-). In the literature, there are two fully-complete models for the whole system F: i.e. that of [BC88], and that of [Hug99]. The first model is based on a quotient of a term model, the latter is a game... |

2 | sequential functionals - Hereditarily - 1994 |