## Relational Properties of Domains (1996)

Venue: | Information and Computation |

Citations: | 99 - 5 self |

### BibTeX

@ARTICLE{Pitts96relationalproperties,

author = {Andrew M. Pitts},

title = {Relational Properties of Domains},

journal = {Information and Computation},

year = {1996},

volume = {127},

pages = {66--90}

}

### Years of Citing Articles

### OpenURL

### Abstract

New tools are presented for reasoning about properties of recursively defined domains. We work within a general, category-theoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property of mixed initiality/finality is transferred to a corresponding property of invariant relations. The existence of invariant relations is proved under completeness assumptions about the notion of relation. We show how this leads to simpler proofs of the computational adequacy of denotational semantics for functional programming languages with user-declared datatypes. We show how the initiality/finality property of invariant relations can be specialized to yield an induction principle for admissible subsets of recursively defined domains, generalizing the principle of structural induction for inductively defined sets. We also show how the initiality /finality property gives rise to the co-induct...

### Citations

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Citation Context ...10). The rule for recursive function declarations is: \Gamma; f:oe 0 !oe; x 0 :oe 0 ` e:oe \Gamma ` e 0 :oe 0 : \Gamma ` letrec fx 0 = e:oe in fe 0 : oe We omit the other type assignment rules. (See (=-=Gunter, 1992-=-, Tables 2.1, 3.1 and 4.1) for example.) Let Expr oe denote the collection of closed expressions of type oe, i.e. the expressionsse for which the typing assertion ; ` e:oe holds. Let Can oe denote the... |

234 | The formal semantics of programming languages: an introduction - Winskel - 1993 |

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Citation Context ...(private communication), that the Curry fixed point combinator coincides with the least fixed point operator in the canonical domain model of the lazy lambda calculus studied in (Abramsky, 1990) and (=-=Abramsky and Ong, 1993-=-). 3 Separation of positive and negative occurrences of variables. This is a key aspect of Freyd's recent work on recursive types via his notion of `algebraic compactness' (Freyd, 1991, 1992). Lets\Ph... |

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Citation Context ...embedding-projection pair; and as loc. cit. shows, this is sufficient to establish the co-induction result, which includes Abramsky's `internal full abstraction' theorem for his domain model of SCCS (=-=Abramsky, 1991, Pro-=-position 3.11) as an instance. However, Rutten (1993, Sect. 5) obtains Abramsky's result (modulo the use of chain-complete simulations) just using the local continuity of the functor P " : Cpo ? ... |

102 | Parametricity and local variables
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Citation Context ...tion principles Many kinds of relational structure on Cpo ? possess distinguished `identity' relations. The following definition axiomatizes the properties we require of such identity relations (cf. (=-=O'Hearn and Tennent, 1993-=-, Sect. 2)). Definition 6.1 A unitary relational structure, R, on a category C is a relational structure in the sense of Definition 4.1 together with a family of Rrelations I D 2 R(D) (for each C-obje... |

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Citation Context ...and (Abramsky and Ong, 1993). 3 Separation of positive and negative occurrences of variables. This is a key aspect of Freyd's recent work on recursive types via his notion of `algebraic compactness' (=-=Freyd, 1991-=-, 1992). Lets\Phi(ff \Gamma ; ff + ) denote the result of replacing all positive occurrences of ff in \Phi by a new variable ff + , and replacing all negative occurrences by a different variable ff \G... |

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Citation Context ...ader to (Pitts, 1994) for further discussion and applications of this co-inductive characterization of the partial order on recursively defined cppos. (See also (Fiore, 1993) and (Rutten, 1993); and (=-=Paulson, 1993-=-) for related applications.) The existence of a simulation (i.e. a relation satisfying (37)) containing two elements of a recursively defined cppo establishes that the order relation holds between the... |

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Citation Context ...nship, but do not preserve the operation of taking inverse images of relations. Similarly, the notion of composition of (binary) relations is fundamental to much work on categories of relations (see (=-=Freyd and Scedrov, 1990-=-) for example), but plays little role here because it is not preserved by the functors we need to consider. 15 Here are some examples of relational structures on the category Cpo ? of cppos and strict... |

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Citation Context ... by (4). Secondly, such minimal solutions are needed to ensure denotational semantics of programming language expressions are computationally adequate (in the technical sense described in the survey (=-=Meyer and Cosmodakis, 1988-=-)) for their operational behaviour. We give a concrete example of this in Sect. 5. In order to express a minimality condition on solutions of (4), one needs some notion of comparison between cppos tha... |

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Citation Context ...particular notion of `relation ' being used and by the particular way domain constructors act on these relations. A preliminary version of the results in this paper appeared in the extended abstract (=-=Pitts, 1993-=-). I would like to thank one of the referees, whose comments helped to improve the exposition of this paper. 2 Overview The purpose of this section is not just to provide the busy reader with some und... |

12 |
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Citation Context ...s of (abstract) relations on the recursively defined domains that arise in the denotational semantics of such datatypes. The key idea which is explored in this paper, and which goes back at least to (=-=Milne, 1973-=-; Plotkin, 1973; Reynolds 1974), is to take account of the fact that various domain constructors can be extended to corresponding constructors for relations on domains. Traditionally one considers onl... |

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Citation Context ...ast fixed point operator in a certain domain model of the untyped lambda calculus. The argument is due to Plotkin (private communication) and should be compared with the proofs of similar results in (=-=Barendregt, 1984-=-, Exercise 18.4.20) and (Winskel, 1993, Sect. 13.10.2). Example 3.6 Let D be rec ff:(ff ! ff) ? , the canonical domain model of the lazy lambda calculus studied in (Abramsky, 1990) and (Abramsky and O... |

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Citation Context ...roof, due to Plotkin (private communication), that the Curry fixed point combinator coincides with the least fixed point operator in the canonical domain model of the lazy lambda calculus studied in (=-=Abramsky, 1990-=-) and (Abramsky and Ong, 1993). 3 Separation of positive and negative occurrences of variables. This is a key aspect of Freyd's recent work on recursive types via his notion of `algebraic compactness'... |

2 |
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Citation Context ... lazy lambda calculus. We refer the reader to (Pitts, 1994) for further discussion and applications of this co-inductive characterization of the partial order on recursively defined cppos. (See also (=-=Fiore, 1993-=-) and (Rutten, 1993); and (Paulson, 1993) for related applications.) The existence of a simulation (i.e. a relation satisfying (37)) containing two elements of a recursively defined cppo establishes t... |

2 |
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Citation Context ...vantage of yielding solutions up to equality rather than just up to isomorphism; and there is the related approach using least fixed points of continuous operators on universal domains, described in (=-=Gunter and Scott, 1990-=-). These various constructions serve not only to prove the existence of some solution to a domain equation, but in fact produce a solution that is minimal, in 9 a sense which we review in this section... |

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