V.Yu. Sazonov. Expressibility of functions in D. Scott's LCF language. Algebra i Logika, 15:308-- 330, 1976. Russian.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....u] where [A c B] is the cpo of continuous functions from cpo A to cpo B ordered pointwise. Full continuous models are important because they can be used to obtain models of programming languages that are fully abstract, i.e. models in which equality coincides with observational congruence (cf. [4, 11, 14]) Adding algebraic terms: An algebraic signature is a signature whose constants have first order type, viz. types of the form ( The function constants have curried types so, for example, if we want to include the addition function as an algebraic constant, we give it type ....

....underlying algebra (N; 0; 1; fails the regularity condition because of the equation ( x 0) 0. 4.4 Observational congruence In order to apply the corollaries to PCF, we need to build a model of PCF. The full continuous model over N is a good candidate, but the model is not fully abstract [11, 14] that is, there are congruences that hold in PCF that are not equivalences in the model. To get around this problem, we need to define a model that precisely captures PCF observational congruences. The PCF term model is a model constructed out of observational congruence classes of closed PCF ....

V.Yu. Sazonov. Expressibility of functions in D. Scott's LCF language. Algebra i Logika, 15:308-- 330, 1976. Russian.

....part by NSF Grant No. 8511190 DCR, ONR grant No. N00014 83 K 0125, and NSF Graduate Fellowships. semantics is then called fully abstract. The language PCF, when observing numerals, has a well matched denotational semantics. Plotkin and Sazonov show that the Scott style, cpo model A V is adequate [11, 12]. Moreover, although A V is not fully abstract, the addition of a parallel conditional operator pcond to PCF makes the model fully abstract under this notion of observation [11, 12] There may be other plausible choices for observations, e.g. in a language with stores, one could observe the ....

....has a well matched denotational semantics. Plotkin and Sazonov show that the Scott style, cpo model A V is adequate [11, 12] Moreover, although A V is not fully abstract, the addition of a parallel conditional operator pcond to PCF makes the model fully abstract under this notion of observation [11, 12]. There may be other plausible choices for observations, e.g. in a language with stores, one could observe the contents of memory cells. Other notions of observation can open a morass of problems. In PCF, for example, one might wish to observe terms at higher type, e.g. printing a message when ....

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V. Sazonov. Expressibility of functions in D. Scott's LCF language. Algebra i Logika, 15:308-- 330, 1976. (Russian).

....where [A c B] is the cpo of continuous functions from cpo A to cpo B ordered pointwise. Full continuous models are important because they can be used to obtain models of programming languages that are fully abstract, i.e. models in which equality coincides with observational congruence (cf. [5, 13, 17]) 2.2 Algebraic terms and equations Recall that a (single sorted) algebraic signature is a set of constants Sigma alg = ff 0 ; f 1 ; g with an arity for each constant. For simplicity, we will only consider single sorted algebraic signatures, although we expect the generalization to ....

V. Sazonov. Expressibility of functions in D. Scott's LCF language. Algebra i Logika, 15:308--330, 1976. Russian.

....also does not solve the problem because the presence of error values in the ground domain violates another antecedent of the Riecke Subrahmanyam meta theorem [Riecke: private communication, 15 October 1992] 3 The idea for this equation is due to John Gateley. 4 Also compare Sazonov s paper [25] on the expressibility of PCF. He proved that PCF can only express the sequential functions of its continuous function model, and conjectured that adding parallel or and parallelexists would make PCF a computationally complete language. His discovery of parallel or and parallel exists was ....

Sazonov, V.Y. Expressibility of functions in D.Scott's LCF language. Algebra i Logika 15(3), 1976, 308--330.

....HSF . We give another interpretation for the sequential strategies as partial continuous functionals, denoted by [ p] The functionals [ p] are called sequential and the set of these functionals SF . The sequential functionals seem to be essentially the same as the serial functionals in [Saz76]. We give two definitions for elements of HSF (SF ) to be computable. We have the recursive elements of HSF (SF ) being extensions of recursive sequential strategies. They correspond to the effectively serial functionals in [Saz76] Secondly we consider the class of Kleene recursive functionals, ....

....seem to be essentially the same as the serial functionals in [Saz76] We give two definitions for elements of HSF (SF ) to be computable. We have the recursive elements of HSF (SF ) being extensions of recursive sequential strategies. They correspond to the effectively serial functionals in [Saz76]. Secondly we consider the class of Kleene recursive functionals, i.e. definable by Kleene s schemata (S1) S8) S11) Kle59] over HSF (SF ) which is strongly related to definability in PCF [Pla66] We show that both notions coincide. So we have found a computation model for higher types ....

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V. Yu. Sazonov. Expressibility of functions in D. Scott's LCF language. Algebra and Logic (English translation), 15:192--206, 1976.

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