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...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...

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...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...

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...(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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...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 ? ...

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...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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...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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...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...

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...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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...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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... 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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...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'...

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...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...

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...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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...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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...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...

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...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'...

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... 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...

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...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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... 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...