Domain-theoretic Foundations of Functional Programming by Thomas Streicher PDF

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By Thomas Streicher

ISBN-10: 9812701427

ISBN-13: 9789812701428

This textbook presents a foundation for a PhD path on domain-theoretic semantics of useful programming languages and their meta-mathematical houses. It introduces uncomplicated area concept and the means of logical family as constructed via Scott and Plotkin. the answer of recursive area equations is defined intimately. a whole dialogue of the well-known complete abstraction challenge for PCF (a useful Kernel language because of Scott and Plotkin) is given together with a building of the absolutely summary Milner version utilizing Kripke logical kinfolk. a last bankruptcy introduces computability in Scott domain names and exhibits that this version is absolutely summary and common for acceptable extensions of PCF by way of parallel language constructs.

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Additional info for Domain-theoretic Foundations of Functional Programming

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M Qa N) A (N Qa M) . t. the Scott model introduced in the previous section. Notice, however, that we always assume free variables to be typed by a context which we prefer to leave implicit 3 for sake of readability. M2 O Vx:<7. M(a;) =<,_•T M provided x is not free in M Y ff (M) =ff M(YCT(M)) Vx:a. M(x) Qa x => Ya{M) CCT x provided x is not free in M P(na) A (Vi: P(M(x))) =» P(Y (—) where A is an arbitrary formula without free occurrences of the variable x.

4 There are no PCF definable functions of type satisfying the constraint (f). Dt^L^t Proof. Suppose / is PCF definable and satisfies (f). Then /0_L = 0 = /-L0 and, therefore, we have /-Li. = /(o n ±)(_L n o) = /OJ. 3. However, by (t) we have / l l = 1 and, thus, by monotonicity of / it follows that 0 C 1 which is impossible. 3 was taken as a starting point by G. Berry, who in his These d' Etat [Berry 1979] introduced and investigated a category of so-called "stable domains" where all morphisms are required not only to be Scott continuous but also stable in the sense that x]y => f(xny) = f{x)nf(y) for all arguments x and y.

The claim of the previous theorem may be formulated more abstractly as follows: for every continuous f:CxA—>B there is a unique continuous g : C —> [A—>JB] such that the following diagram commutes [A-^B]xA gx\AA 6V . B s CxA where (<7xid,i)(c,a) = (g(c),a). This requirement makes sense in every category with (binary) cartesian products and characterises the exponential [A—>B] uniquely up to isomorphism. g. [Scott 1980]). One often writes A(/) for the unique map g with / = evo (^xid^). We will see later that projections, ev and A provide enough structure for interpreting the simply typed A-calculus in the category of predomains and continuous maps (and, actually, in an arbitrary cartesian closed category).

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