## A Characterization Of Lambda Definability In Categorical Models Of Implicit Polymorphism (1995)

Venue: | Theoretical Computer Science |

Citations: | 11 - 0 self |

### BibTeX

@ARTICLE{Alimohamed95acharacterization,

author = {Moez Alimohamed},

title = {A Characterization Of Lambda Definability In Categorical Models Of Implicit Polymorphism},

journal = {Theoretical Computer Science},

year = {1995},

volume = {146},

pages = {5--23}

}

### OpenURL

### Abstract

. Lambda definability is characterized in categorical models of simply typed lambda calculus with type variables. A category-theoretic framework known as glueing or sconing is used to extend the Jung-Tiuryn characterization of lambda definability [JuT93], first to ccc models, and then to categorical models of the calculus with type variables. Logical relations are now a well-established tool for studying the semantics of various typed lambda calculi. The main lines of research are focused in two areas, the first of which strives for an understanding of Strachey's notion of parametric polymorphism. The main idea is that a parametricly polymorphic function acts independently from the types to which its type variables are instantiated, and that this uniformity may be captured by imposing a relational structure on the types [OHT93, MSd93, MaR91, Wad89, Rey83, Str67]. The other line of research concerns lambda definability and the full abstraction problem for various models of languag...

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Citation Context ...ow \Omega t n [X ; \Phi 1 ]; : : : ; t n [X ; \Phi k ] ff in B (n; k). Each ccc representation t n : Pn \Gamma! Fn is defined by induction on the structure of terms using the ccc structure of Fn (see =-=[Cro93]-=- for details). 3. Sconing with iml-categories The construction of sconing with iml-categories is facilitated by introducing certain functors \Delta n k that lie conceptually close to the notion of imp... |

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Citation Context ... permutation [Lau70] was the most obvious of these since lambda definable functions cannot speak about particular elements, but this was not enough for a complete characterization. A later attempt in =-=[Plo80]-=- introduced the idea of invariance under a logical relation and then, in the same paper, the notion of invariance under I-relation by which Plotkin succeeded in characterizing lambda definability in c... |

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Citation Context ... R on T . The second part of the work presents an analog of \Gamma-relation and full completeness for iml-categories. The approach is similar to the D-relators of [MSd93] or the construction given in =-=[MaR91]-=-. There, the sconing construction is carried out on the zeroth fiber of an iml-category hB ; Fi via a glueing functor \Gamma : F 0 \Gamma! D. However, this characterizes lambda definability only for c... |

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Citation Context ...n b R; \Sigma = b T (or equivalently, R; \Sigma = T ). This suggests the following definition which generalizes the notion of logical relation to include I-relations [Plo80], Kripke logical relations =-=[MiM91]-=- and also Kripke logical relations with varying arity [JuT93]. Definition 1. Let F(C ) be the free ccc on a small category C , let T : F(C ) \Gamma! A be any ccc representation and let \Gamma : A \Gam... |

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Citation Context ...definable if f = T [x:oe . M: ] for some term x:oe . M: . Central to characterizing the lambda definable terms is the following proposition often known as the Fundamental Theorem of Logical Relations =-=[Sta85]-=-. Proposition 1. Let R be a logical relation on A. Then each f = T [x:oe . M: ] satisfies R. Proof. By definition, T is the unique ccc representation that satisfies J ; T = b T . The logical relation ... |

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Citation Context ...ince then Riecke and O'Hearn have modified this result to an extension of simply typed lambda calculus having basic arithmetic constructs and general recursion to provide fully abstract models of PCF =-=[OHR94]-=-. In this work, a notion of logical relation is presented that characterizes lambda definability in categorical models of simply typed lambda calculus with type variables. This language, called implic... |

26 | Tiuryn, A New Characterization of Lambda Definability
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(Show Context)
Citation Context ...orical models of simply typed lambda calculus with type variables. A category-theoretic framework known as glueing or sconing is used to extend the Jung-Tiuryn characterization of lambda definability =-=[JuT93]-=-, first to ccc models, and then to categorical models of the calculus with type variables. Logical relations are now a well-established tool for studying the semantics of various typed lambda calculi.... |

25 | Notes on sconing and relators
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(Show Context)
Citation Context ...on and type application are not defined explicitly in the syntax of terms, this form of polymorphism is called implicit polymorphism and the corresponding categorical models are called iml-categories =-=[MSd93]-=-. These categorical models are a simpler form of the hyperdoctrine models [Pit87, See87] of the Girard-Reynolds polymorphic lambda calculus. To keep this paper self-contained, the definitions of iml-c... |

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Citation Context ...80, Lau70]. Early attempts to characterize lambda definability in the full type hiearchy focused on invariance properties of functions that are definable by lambda terms. Invariance under permutation =-=[Lau70]-=- was the most obvious of these since lambda definable functions cannot speak about particular elements, but this was not enough for a complete characterization. A later attempt in [Plo80] introduced t... |

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2 |
Categories & Machines. Th`ese de Doctorat, Universit'e Paris VII
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(Show Context)
Citation Context .... A Kripke logical relation with varying arity on T is nothing more than a \Gammarelation on T , where \Gamma : A \Gamma! S W ffi is a glueing functor similar to the one that appears in Appendix C of =-=[Laf88]-=-. This \Gamma differs from the one used for I-relations in that \Gamma(A) is no longer a constant functor. The particular Kripke presheaf category that is used by Jung and Tiuryn has a poset W that co... |

2 |
Typed lambda calculus and cartesian closed categories
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(Show Context)
Citation Context ...ar, when the family consists of a single logical relation then the two notions coincide. As a cautionary note, completeness and full completeness should not be confused with the term as it appears in =-=[MSt89]-=-, where completeness refers to equational completeness for fij-conversion. That notion corresponds to a family of ccc representations forming a collective embedding [FrS90]. In [Plo80], Plotkin proves... |

1 |
An intersection problem for finite automata
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(Show Context)
Citation Context ...rtain results have been established: for those models T where A ' is infinite, Plotkin in [Plo80] finds a family of \Gamma-relations on T that is complete on a certain subset of Ob(F(f'g). Statman in =-=[SaS88]-=- shows that the restriction on T cannot be removed. Even when A ' is infinite, the question remains open whether there exists a glueing functor \Gamma : A \Gamma! S and a family of \Gamma-relations on... |