## Relational Properties of Domains (1996)

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Venue: | Information and Computation |

Citations: | 102 - 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

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### 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

1619 | The definition of Standard ML - Milner, Tofte, et al. - 1997 |

969 | Categories for the Working Mathematician - Lane - 1998 |

405 | LCF considered as a programming language - Plotkin - 1977 |

273 |
Semantics of Programming Languages: Structures and Techniques
- Gunter
- 1992
(Show Context)
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... |

243 | The Formal Semantics of Programming Languages. An Introduction - Winskel - 1993 |

223 | A powerdomain construction - Plotkin - 1976 |

203 |
The Lambda Calculus: its Syntax and Semantics (revised ed
- Barendregt
- 1984
(Show Context)
Citation Context ...leastsxed 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 LetD be rec:(!) ? , the canonical domain model of the lazy lambda calculus studied in (Abramsky, 1990) and (Abramsky and Ong, 199... |

176 | The category-theoretic solution of recursive domain equations - Smyth, Plotkin - 1982 |

137 | Full abstraction in the lazy lambda calculus
- Abramsky, Ong
- 1993
(Show Context)
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... |

125 | A logic for parametric polymorphism - Plotkin, Abadi - 1993 |

123 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
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 ? ... |

104 | Parametricity and local variables
- O’Hearn, Tennent
- 1995
(Show Context)
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... |

100 | The relation between computational and denotational properties for Scott’s D∞-models of the lambda-calculus - Wadsworth |

95 |
Algebraically complete categories
- Freyd
- 1990
(Show Context)
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... |

78 | Report on the Programming Language Haskell (version 1.2 - Hudak, Jones, et al. - 1992 |

56 |
Equality in hyperdoctrines and the comprehension schema as an adjoint functor
- Lawvere
- 1970
(Show Context)
Citation Context ...ements to related elements. Making this the fundamental notion contrasts with the usual axiomatic approach to predicates in categorical logic where, beginning with Lawvere's work on `hyperdoctrines' (=-=Lawvere, 1970-=-), the notion of the inverse image of a predicate along a map is fundamental. There are close connections between the two notions (see Remark 4.15). The principle reason for starting with relational s... |

48 | On the relation between direct and continuation semantics - REYNOLDS - 1974 |

44 | Mechanizing coinduction and corecursion in higher-order logic
- Paulson
- 1997
(Show Context)
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... |

43 | New Foundations for Fixpoint Computations: FIX-Hyperdoctrines and the FIX-Logic - Crole, Pitts - 1990 |

42 |
A.: Categories, Allegories
- Freyd, Scedrov
- 1990
(Show Context)
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... |

42 | A co-induction principle for recursively defined domains - Pitts - 1994 |

29 | Lectures on predomains and partial functions - Plotkin - 1985 |

22 |
Semantical paradigms: notes for an invited lecture
- Meyer, Cosmodakis
- 1988
(Show Context)
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... |

22 | Computational adequacy via “mixed” inductive definitions - Pitts - 1994 |

22 | Lambda-definability and logical relations - Plotkin - 1973 |

20 | Edinburgh LCF - Gordon, Milner, et al. - 1979 |

15 | Relational properties of recursively defined domains
- Pitts
- 1993
(Show Context)
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... |

15 |
The lazy lambda calculus, in \Research Topics in Functional Programming
- Abramsky
- 1990
(Show Context)
Citation Context ...le proof, due to Plotkin (private communication), that the Currysxed point combinator coincides with the leastsxed 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'... |

12 |
The formal semantics of computer languages and their implementations
- Milne
- 1973
(Show Context)
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... |

6 |
The Lambda Calculus: its syntax and semantics," Number 103
- Barendregt
- 1994
(Show Context)
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... |

5 | New foundations for computations: FIX-hyperdoctrines and the FIX-logic - Crole, Pitts - 1992 |

4 | Remarks on algebraically compact categories, in \Applications of Categories in Computer Science - Freyd - 1992 |

4 |
Semantic domains, in \Handbook of Theoretical Computer Science
- Gunter, Scott
- 1990
(Show Context)
Citation Context ...advantage of yielding solutions up to equality rather than just up to isomorphism; and there is the related approach using leastsxed 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... |

3 | Algebraic specification of datatypes: a synthetic approach - Lehmann, Smyth - 1981 |

2 |
The lazy lambda calculus, in "Research Topics in Functional Programming
- Abramsky
- 1990
(Show Context)
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 |
A coinduction principle for recursive datatypes based on bisimulation
- Fiore
- 1993
(Show Context)
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 |
Semantic Domains, in "Handbook of Theoretical Computer Science
- Gunter, Scott
- 1990
(Show Context)
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... |

2 | Inductive inference in reflexive domains - Jensen - 1981 |

1 | Remarks on algebraically compact categories, in "Applications of Categories in Computer Science - Freyd - 1992 |

1 | Types, abstraction and parametric polymorphism, in "Information Processing 83 - Reynolds - 1983 |

1 | A structural co-induction theorem, Report CSR9346 - Rutten - 1993 |

1 | Domains for denotational semantics, in "Proc. 9th Internat. Coll. on Automata, Languages and Programming - Scott - 1982 |

1 | A logic for Miranda, Formal - Thompson - 1989 |

1 | Inductive inference in re domains - Jensen - 1981 |

1 | Algebraic speci of datatypes: a synthetic approach - Lehmann, Smyth - 1981 |

1 |
Relational properties of recursively de domains
- Pitts
- 1993
(Show Context)
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... |