D.Scott: Data types as lattices, SIAM Journal of Computing, 4, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....still possible to identify a first phase of static (although possibly not terminating) typechecking, followed by a second execution phase which does not require any typechecking. All the key ideas presented here are fairly well known. Semantic domains where Type:Type holds were defined by Scott [Scott 76] The basic language semantics problems were solved by McCracken [McCracken 79] Dependent types come from intuitionistic type theory [Martin Lf 80] and their denotational semantics is studied by Barendregt and Rezus [Barendregt 83] Rezus 85] Relevant languages have been proposed such as ....

.... B: AType. c: S(A) B) B(lft(A) B) c) The right projection of a pair is very useful for defining the Any type and the parametric modules operators in [MacQueen 86] Thus we have existential types as a primitive construct. 7. The meaning of Type A type is, in first approximation, a retraction [Scott 76] A retraction is similar to a coercion, as we can see from the following example of a boolean retraction (in the untyped l calculus) Bool = lx. if x then true else false Page 17 This retraction coerces an arbitrary object x to true or false (or diverges if x diverges) Note that booleans are ....

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D.Scott: Data types as lattices, SIAM Journal of Computing, 4, 1976.

....the formal semantics of types. The most fundamental one is concerned with devising mathematical models for types, normally by mapping every Page 12 type expression into a set of values (the values having that type) the basic difficulty here is in finding a mathematical meaning for the operator [Scott 76] Milner 78] MacQueen 84] The other, complementary, approach is to define a formal system of axioms and inference rules, in which it is possible to prove that an expression has some type. The relationship between models and formal systems is very strong. A semantic model is often a guide in ....

D.S.Scott: Data types as lattices, SIAM Journal of Computing, 4, 1976.

....say much more about inverse limits and the solution of such equations below. Another important model was introduced by Gordon Plotkin [1972] Scott then modified the form of Plotkin s construction slightly to get what he called the graph model which he used to derive a very detailed analysis [1976] of the use of continuous lattices (which are the retracts of the graph model) for domain theory. Actually, the ideas were already known in recursive function theory under the name of enumeration operators, but the exact connection CHAPTER 1. BACKGROUND 11 with calculus had not been realized ....

....model for the polymorphic calculus, which also has importance for programminglanguage semantics. This quick review hardly touches on the extent of the literature, and the reader must consult the sources mentioned as well as such books as [Barendregt 1984] for a more complete exposition. Plotkin [1976] introduced a quite different category of domains which he called SFP. This is a large category and was needed because the more established categories did not have the desired closure properties. Thereafter, it was soon discovered that the property of having least upper bounds is not preserved ....

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Data types as lattices. SIAM Journal of Computing, vol. 5 (1976), pp. 522--587.

....but much of our discussion could be carried over, and sometimes even improved, if we chose to refer to other models. The idea of types as parameters is fully developed in the second order l calculus [Bruce 84] The (only known) denotational models the second order l calculus are retract models [Scott 76] Here, types are not sets of objects but special functions (called retracts) these can be interpreted as identifying sets of objects, but are objects themselves. Because of the property that types are objects, retract models can more naturally explain explicit type parameters, while ideal ....

D.Scott: Data types as lattices, SIAM Journal of Computing, Vol 5, No 3, September 1976, pp. 523-587.

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