A. Edalat and M.H. Escardo. Integration in Real PCF (extended abstract). In Proceedings of the 11th Annual IEEE Symposium on Logic In Computer Science, pages 382-393, New Brunswick, New Jersey, USA, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....seems that certain natural operations (e.g. Riemann integration for discontinuous functions) are not computable in PCF but become computable in the intensionally sequential setting. Here we expect our ideas to have an impact on current research in real number computation by Edalat, Escard o et al. [EE96]. However, the connections between their setting and ours have yet to be explored in detail. II. Other concrete datatypes As mentioned above, many interesting datatypes such as lazy lists and trees arise as retracts of finite types. This means that notions of computability for the finite ....

A. Edalat and M.H. Escard'o. Integration in Real PCF (extended abstract). In Proc. 11th Annual IEEE Symposium on Logic in Computer Science, 1996.

.... possible to approximate points of the space (total information, e.g. a probability measure) with domain elements (partial information, e.g. a linear combination of point measures) In that project, the introduction of partial elements has led to significant contributions to numerical integration [13], the design of new image compression algorithms based on work in dynamical systems and fractals [12] and the derivation of novel semantics and implementations of exact real arithmetic [14] Although we propose a non standard version of an established methodology, we hasten to point out that this ....

A. Edalat and M. H. Escardo. Integration in Real PCF. In IEEE Symposium on Logic in Computer Science. IEEE Computer Society, IEEE Computer Society Press, 1996.

....in the technical development that follows. We also briefly discuss applications, introduce preliminary background and give a summary of the main results of this paper. 1. 1 Embedding spaces into domains In applications of domain theory [2] to denotational semantics [16, 13] and integration [8, 11], one starts by implicitly or explicitly embedding given spaces X , Y , Z, into appropriate domains C, D, E, endowed with the Scott topology. One of the simplest examples is given by the embedding of the discrete space of natural numbers into the so called flat domain N of natural ....

....j : X C and k : Y D are subspace embeddings into injective spaces, then the continuous functions C D capture the continuous maps X Y in the sense of the (co)extension property. When one considers higher order maps [X Y ] Z, such as the integration and supremum operators discussed in [11], one is led to consider the case in which the function space [C D] captures the the function space [X Y ] in the stronger sense of having it embedded as a subspace, via some continuous (co)extension map [X Y ] C D] By the above remarks, it suffices to consider continuous extension ....

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A. Edalat and M.H. Escard'o. Integration in Real PCF (extended abstract). In Proceedings of the Eleventh Annual IEEE Symposium on Logic In Computer Science, pages 382--393, New Brunswick, New Jersey, USA, July 1996.

....L ; tail R : 0; 1] 0; 1] by cons L (x) x=2; tail L (x) min(2x; 1) cons R (x) x 1) 2; tail R (x) max(0; 2x Gamma 1) and define head : I T by head(x) x 1=2) where the inequality map was defined in Section 1. For motivation and a detailed discussion about these functions see [8, 7, 5], where effective computation rules for them are given. The main property of the functions cons L ; cons R ; tail L ; tail R : I I and head : I T is given by the following lemma: Lemma 1.2 The identity of I is the unique continuous function f : I I such that f(x) pif head(x) then cons L ....

A. Edalat and M.H. Escard'o. Integration in Real PCF. In Proceedings of the Eleventh Annual IEEE Symposium on Logic In Computer Science, New Brunswick, New Jersey, USA, July 1996.

....map x 7 fxg is a topological embedding of the Euclidean real line into the partial real line endowed with its Scott topology. The partial real line has been used to model exact real number computation in the framework of the programming language Real PCF [9,10] including computation of integrals [4]. To appear in Theoretical Computer Science 28 November We introduce induction principles and recursion schemes for the partial unit interval (the domain of closed subintervals of the unit interval with end points 0 and 1) which allow us to verify that Real PCF programs meet their ....

....to be an algebra homomorphism from constr to pif ffi (id B Theta aL Theta aR ) and the result again follows from Proposition 24. 2 More examples of definitions of elementary functions by dyadic recursion can be found in [9,11] and a recursive definition of Riemann integration can be found in [4]. 5.3 Bifurcated binary expansions The canonical solution of the domain equation D = TD is the domain BTree of infinite binary trees with nodes labeled by truth values, ordered nodewise, together with the bifree algebra mktree : T(BTree) BTree that maps a list hp; s; ti to the tree with ....

A. Edalat and M.H. Escard'o. Integration in Real PCF (extended abstract). In Proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science, pages 382--393, New Brunswick, New Jersey, USA, July 1996.

.... on the partial real line coincides with the Hausdorff topology considered in interval analysis [15] as it is shown in [9] The partial real line has been used to model exact real number computation in the framework of the programming language Real PCF [6, 8] including computation of integrals [3]. We introduce induction principles and recursion schemes for the partial unit interval (the domain of closed subintervals of the unit interval with end points 0 and 1) which allow us to verify that Real PCF programs meet their specification. The induction principles and recursion schemes ....

....I[1; 2] I[1; 2] by aL (x) p x and aR (x) p 2x. Then exp is an algebra homomorphism from cons to pif ffi (id Theta aL Theta aR ) and the result again follows from Proposition 5.4. More examples can be found in [6, 7] and a recursive definition of Riemann integration can be found in [3]. 6. Applications to Real PCF Real PCF [6] is an extension of PCF [21, 16] with a ground type for the partial real line and some primitive functions, which include the real number constructors and destructors discussed in Section 3. The remaining primitives are necessary to obtain the ....

A. Edalat and M.H. Escard'o. Integration in Real PCF (extended abstract). In Proceedings of the Eleventh Annual IEEE Symposium on Logic In Computer Science, New Brunswick, New Jersey, USA, July 1996.

....strings, and list structures [31] In recent years, however, Domain Theory [2] seems to have reached a level of maturity, that allows more challenging data types. One prominent such development has been the extensive work on the semantics of (exact) real number computation [8] and integration [6]. The theory of integration, of course, is based upon the notion of measures (see for example [11] Traditionally, a measure is defined on a oe algebra of sets. The set theoretic operations of such algebras, set complement, for example, cannot be easily reconciled with topological notions. The ....

A. Edalat and M. H. Escardo, Integration in Real PCF, in: IEEE Symposium on Logic in Computer Science, IEEE Computer Society (IEEE Computer Society Press, 1996).

.... or even the unit interval [vE86] One consequence of our choice of the meaning function is that the process of evaluating a formula in a given model must involve numerical calculations and approximation, a fact interesting in itself in the light of recent work on computation and approximation [Eda95b] Probabilistic logics of programs, mainly temporal, have been considered by a number of authors, see e.g. HSP83, Var85, Koz85, CY88, PZ93, ACD91, LS91, HJ94, Han94, BK96] The prevailing method is based on taking a non probabilistic logic and extending its syntax by labelling the existing ....

....would be the probability that the formula holds at a given state. This can be seen as providing a framework for exploiting numerical computation analysis to establish the confidence level of program properties holding, and is of particular interest because of the recent developments due to Edalat [Eda95a, Eda95b] concerning computational models of metric spaces and integration. Probabilistic model checking as outlined above should be seen as an alternative to the classical model checking, which in certain circumstances can be advantageous, but comes at a price known only too well to those ....

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A. Edalat and M. Escardo. Integration in Real PCF. In Proceedings of Logic in Computer Science (LICS), IEEE Computer Society Press, 1996.

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A. Edalat and M.H. Escardo. Integration in Real PCF (extended abstract). In Proceedings of the 11th Annual IEEE Symposium on Logic In Computer Science, pages 382-393, New Brunswick, New Jersey, USA, 1996.

....In Section 8 we define multiple integrals and show how to define them from interval Riemann integration. In Section 9 we show that Real PCF extended with maximization is universal. Several proofs have been omitted due to lack of space. For a full version of this paper containing all proofs see [13]. 2. Real PCF In this section we summarize the results of [16, 14] needed in this paper. We assume familiarity with PCF [23, 19] We are deliberately informal concerning syntax. For simplicity and without essential loss of generality, we restrict ourselves to the unit interval [0; 1] 2.1. ....

A. Edalat and M. Escardo. Integration in Real PCF. Available by ftp (theory.doc.ic.ac.uk:papers/Escardo/papers) and WWW (http://theory.doc.ic.ac.uk/smhe/), December 1995.

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A. Edalat and M.H. Escard'o. Integration in Real PCF (extended abstract). In Proceedings of the 11th Annual IEEE Symposium on Logic In Computer Science, New Brunswick, New Jersey, USA, July 1996.

....Borel measure if and only if its set of discontinuities has measure zero and that if the function is R integrable then it is also Lebesgue integrable and the two integrals coincide. This theory has had applications in exact computation of integrals [8] in the semantics of programming languages [12], in the 1 dimensional random fields Ising model in statistical physics [9] in forgetful neural networks [10] in stochastic processes [5] and in chaos theory [8] Apart from the domain theoretic integral, there are two other notions of generalized Riemann integrals in the literature, namely, the ....

A. Edalat and M. Escard'o, Integration in Real PCF, in the proceedings of Logic in Computer Science, IEEE Computer Society, 1996.

....implies that the Euclidean topology coincides with the relative Scott topology on the subspace of maximal elements. Any continuous function f : R R extends canonically to a Scott continuous function If : IR IR, defined on any compact interval a by (If) a) f(a) This is the maximal extension [39] of f on IR, in other words if g : IR IR satisfies g(fxg) ff(x)g for all x 2 R then g v If . In practice, for convenience, we usually denote the maximal extension If simply by f . The continuous domain IR can be equipped with a canonical effective structure by using the standard enumeration ....

....and their left inverses as well as the predicate for comparison of intervals with zero and the parallel conditional. This language again enjoys the equivalence of its operational and denotational semantics and, equipped with the existential quantifier, is also universal [51] Moreover, in [39], it is shown that Riemann integration can be introduced in the language. The above two approaches address the issue of formal computability rather than efficient computation. A fundamental question is whether a feasible setting for exact computation can be developed so that basic numerical ....

A. Edalat and M. Escard'o. Integration in real PCF. In Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS),. IEEE, 1996.

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