Abbas Edalat. Domain theory and integration. Theoretical Computer Science, 151(1):163--193, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with the relative Scott topology is homeomorphic to X . Under quite mild conditions on D the set of normalized Borel measures on X , equipped with the weak topology, can be embedded into the set of maximal elements of the probabilistic powerdomain PD. This construction was utilized by Edalat [3, 4] to provide new results on the existence of attractors for iterated function systems, and to de ne a generalization of the Riemann integral to functions on metric spaces. In this paper we show that each measurement m : D [0; 1] has a natural extension to a measurement M : PD [0; 1] such that ....

A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163{ 193, 1995.

.... [30] Kamimura and Tang [15] and Abramsky [1] In the early 1990 s, Edalat demonstrated that a fair amount of classical mathematics takes place at the top of a domain, through a series of papers detailing satisfying connections between domain theory and fundamental themes in analysis (integration [7], measure theory [9] and dynamical systems [8] At the heart of such applications is that certain metric spaces X have domain theoretic models, i.e. there is a continuous dcpo MX such that X max(MX) where the maximal elements max(MX) are regarded a space in their relative Scott topology. One ....

A. Edalat. Domain theory and integration. Theoretical Computer Science 195 (1995) 163-193. 11

....to probabilistic processes. It has a way of meshing finite and continuous notions of computations which is not unlike domain theory. We expect far more interaction in the future between these theories than what is reported here. Work on probabilistic powerdomains [11] and integration on domains [9, 10] provides a beginning. Curiously enough the bulk of work in probabilistic process algebra rarely ever mentions averages or expectation values. We hope that the present paper stimulates the use of these methods by others. Outline. First we recall the definitions of our two basic objects of ....

Abbas Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

....measures on a Polish space X is itself a Polish space. Furthermore, the set of measures, whose supports are finite subsets of the countable set dense in X, is itself dense in the space of measures. Finally, if n , then fd n fd for all bounded continuous functions f . Edalat [Eda94, Eda95a, Eda95c, Eda95b] has exploited domain theoretic methods to define R integration and show that R integrability w.r.t. a bounded Borel measure extends Riemann integrability to compact metric spaces. The R integral of f with respect to measure is approximated by the sums of f w.r.t. simple ....

....paper, is not present in their paper. Thus they do not attempt to relate their domain to labelled Markov processes. 9 Conclusions Our work equips Markov processes with the structure required to apply extant theories of approximations of integrals. In a series of influential papers [Eda94, Eda95a, Eda95c, Eda95b] Edalat exploits domain theoretic methods to approximate the integrals of a large class of functions. The compact Polish space structure of LMP enables these theories to be applied to computing over LMP s. 36 . For probability measures over metric spaces, there is a large well ....

Abbas Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

....space currently known to possess a countably based model, and (2) The metric spaces with ideal models are exactly the completely metrizable spaces. 1 Introduction Over ten years before the realization that certain parts of mathematics could be executed in a purely domain theoretic manner ( 6][7][8] 9] there was interest in the space of maximal elements of a continuous dcpo, as indicated for instance by Scott [29] Kamimura and Tang [15] and Abramsky [1] But since this realization, one gets the distinct impression that interest in this topic may be growing ....

....as simple as an ideal domain can model so many spaces. It makes one wonder if ideal models can arise in any other way. For countably based domains, the answer is no. 15 5 Countably based ideal models The G Lemma 2. 16 establishes that the countably based domains considered by Edalat ( 8] [7]) Edalat and Heckmann [6] and Lawson [17] can all be used to construct ideal models (Theorem 4.1) Though it exploits regularity, separation is not required to prove that the maximal elements form a G set. Example 5.1 Let N 1 = fx n : n 0g [ f1 1 g and N 2 = fy n : n 0g [ f1 2 g ....

A. Edalat. Domain theory and integration. Theoretical Computer Science 195 (1995) 163-193.

.... that can be written as the sup of the directed family j=1 a ij x ij , then the integral R x2X f(x)d (as defined by Tix [11] for example, extending Jones [8] can be computed, not just characterized, as the sup of all finite sums a ij f(x ij ) This is one of the ingredients in [4], notably. For short, call quasi simple those valuations that are sups of directed families of simple valuations. The initial purpose of this paper is to refine our understanding of quasi simple valuations. This will be achieved by proving Theorem 1 below, and exploring some of its applications. ....

....dcpo with bottom . Every bounded continuous probability valuation on X , with its Scott topology, is the sup of a directed family of simple probability valuations i=1 a i x i on X . If X is algebraic, we may additionally require that x i is finite, 1 i n. Proof. Use Edalat s trick [4]: apply Corollaries 2 and 3 to the restriction of to the dcpo X = X nf g. Then is the sup of a directed family of simple sub probability valuations ( i ) on X , where i = a ij x ij , and a ij 1. Let i be a ij x ij ( a ij ) i.e. add the missing ....

A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

....of a stored pattern is a measure of how well represented it is in the synaptic couplings and should decrease with time in a forgetful neural network. However, we will derive computationally the embedding strength decay rate of the stored patterns using recent advances in domain theory by Edalat [13, 11, 10, 12]. This may seem like a strange marriage of ideas, since domain theory was introduced by Dana Scott [30] in 1970 as a mathematical theory of semantics of programming languages. The rise of domain theory over a quarter of a century was primarily motivated by the need to solve recursive procedures ....

....means fractured or broken, to describe these new objects. It is easy to see that the self similarity at different magnifications in fractals is a form of recursion and it is this feature that has made it amenable to a domain theoretic representation. In particular, Edalat devised an algorithm [11] to evaluate the expecta tion of continuous functions over fractal probability distributions. This is very pertinent to the problem at hand because it can be shown [12] that for random stored patterns, the probability distribution of the synaptic cou plings is a fractal and evaluating the ....

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A. Edalat. Domain theory and integration. Theoretical Computer Sci- ence, 151:163 193, 1995.

....6, 10] from the point of view of de nability. In this approach equality is usually used as a basic relation so a computable function can be discontinuous. It diverges from the situation in concrete computability over the reals, in particularly, in computable analysis. The second approach ( e.g. [17, 28 30, 16, 7, 8, 11, 35 37]) is closely related to computable analysis. In this approach computation is an in nite process which produces approximations closer and closer to the result. We work in the framework of the second approach. The main result of our paper is an application of de nability theory, which was originally ....

....ordered reals with equality, or to the strictly ordered reals without equality. In Section 3 we introduce notions of second order computability via domain theory. In this work, to construct computational models for operators and functionals we will use continuous domains. Continuous domains, e.g. [29, 30, 16, 7, 8, 11, 35 37], are generalisation of algebraic domains, e.g. 2, 27, 32, 33] The continuous domain (more precisely, the interval domain) for the reals was rst proposed by Dana Scott [29] and later was applied to mathematics, physics and real number computation in [7, 8, 36, 37, 27] and other publications. In ....

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A. Edalat, Domain Theory and integration, Theoretical Computer Science, 151, 1995, pages 163-193.

....domain representations for topological spaces. We are specially interested in the case where a topological space can be realised as a subset of the maximal elements of a continuous dcpo [67] 11] 33] 34] 39] The interest in this kind of representations arises from Edalat s work in integration [8] and domain theory [9] Edalat s Rintegral [8] later extended by Edalat and Negri [12] is constructed by embedding a second countable locally compact Hausdor space as the subset of maximum elements of an continuous dcpo; then extending locally nite Borel measures on the space to continuous ....

.... We are specially interested in the case where a topological space can be realised as a subset of the maximal elements of a continuous dcpo [67] 11] 33] 34] 39] The interest in this kind of representations arises from Edalat s work in integration [8] and domain theory [9] Edalat s Rintegral [8] (later extended by Edalat and Negri [12] is constructed by embedding a second countable locally compact Hausdor space as the subset of maximum elements of an continuous dcpo; then extending locally nite Borel measures on the space to continuous valuations on the dcpo. The main feature of ....

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A. Edalat, Domain theory and integration, Theoretical Computer Science 151 (1995), 163-193.

....and metric topology. We refer to [16,4] for a further discussion on the use of complete metric spaces and Banach s Theo3 rem in the area of programming language semantics. The advantage of using metric spaces is also reflected in a number of semantical investigations related to probability. See [7,17,5] for example. The mathematical background is sketched in the next section. For our study we have used a construction for obtaining a metric space of measures from a given metric space that is borrowed from [23,11] A similar construction is available for valuations, cf. 3] The paper contributes ....

A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

....In the case of the upper space UX , as it is called, one nds that X may be recovered from UX via X maxUX = ffxg : x 2 Xg UX where the topology on maxUX UX is the relative Scott topology inherited from UX . Because of this homeomorphism, we say that UX is a model of the space X. In [4], Abbas Edalat used only this link to the classical world to de ne a generalization of the Riemann integral for bounded functions f : X R on a compact metric space X. Once it was observed that X could be represented as the maximal elements of the continuous dcpo UX , all other details were ....

A. Edalat. Domain Theory and Integration. Theoretical Computer Science 151 (1995) p.163-193.

....on a Polish space X is itself a Polish space. Furthermore, the set of measures, whose supports are nite subsets of the countable set dense in X, is itself dense in the space of measures. Finally, if n , then R fd n R fd for all bounded continuous functions f . Edalat [Eda94, Eda95a, Eda95c, Eda95b] has exploited domain theoretic methods to de ne R integration and show that R integrability w.r.t. a bounded Borel measure extends Riemann integrability to compact metric spaces. The R integral of f with respect to measure is approximated by the sums of f w.r.t. simple ....

....our paper, is not present in their paper. Thus they do not attempt to relate their domain to labelled Markov processes. 9 Conclusions Our work equips Markov processes with the structure required to apply extant theories of approximations of integrals. In a series of in uential papers [Eda94, Eda95a, Eda95c, Eda95b] Edalat exploits domain theoretic methods to approximate the integrals of a large class of functions. The compact Polish space structure of LMP enables these theories to be applied to computing over LMP s. 34 For probability measures over metric spaces, there is a large well ....

Abbas Edalat. Domain theory and integration. Theoretical Computer Science, 151:163-193, 1995.

....theory. We confine ourselves to integration on a compact metric space X . Assume f : X # R is bounded and # # M 1 X is a probability measure. Let P 1 UX be the subdcpo of the normalised valuations on UX . This is again an # continuous dcpo with a basis of normalised simple valuations [32]. For any dcpo Y which has bottom, in particular for UX when X is compact, the information ordering on simple valuations in P 1 Y has an interesting physical interpretation. For two simple valuations # 1 = # b#B r b # b # 2 = # c#C s c # c in P 1 Y , where B, C are finite subsets of Y , ....

....in particular for UX when X is compact, the information ordering on simple valuations in P 1 Y has an interesting physical interpretation. For two simple valuations # 1 = # b#B r b # b # 2 = # c#C s c # c in P 1 Y , where B, C are finite subsets of Y , we have by the splitting lemma [76, 32]: # 1 # # 2 i#, for all b # B and all c # C , there exists a non negative number t b,c such that #b # B # # c#C t b,c = r b # #c # C # # b#B t b,c = s c # and t b,c #= 0 implies b # c. We can consider any b # B as a source with mass r b , any c # C as a sink with mass s ....

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, Domain theory and integration, Theoretical Computer Science, vol. 151 (1995), pp. 163--193.

....functor M 1 is defined on ultrametric spaces, and the Borel # algebras and associated measures are taken with respect to the metric topology. Our reasons for considering metric spaces rather than the, in semantical contexts, more standard use of ordered structures, such as [Jon89, JP89] and [Eda95a, Eda95b], are twofold. Firstly, one can resort to the rich literature on standard measure theory for metric spaces (see, e.g. KV84] Secondly, we can use the recently developed coalgebraic theory on metric spaces [AR89, RT94] which seems to be better suited to describe (both ordinary and ....

A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

....all measures on a Polish space X is itself a Polish space. Furthermore, the set of measures, whose supports are finite subsets of the countable set dense in X , is itself dense in the space of measures. Finally, if n , then R fd n R fd for all bounded continuous functions f . Edalat [16, 17, 19, 18] has exploited domain theoretic methods to define R integration and show that R integrability w.r.t. a bounded Borel measure extends Riemann integrability to compact metric spaces. The R integral of f with respect to measure is approximated by the sums of f wrt simple valuations less than . ....

Abbas Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

....and metric topology. We refer to [15, 3] for a further discussion on the use of complete metric spaces and Banach s Theorem in the area of programming language semantics. The advantage of using metric spaces is also reflected in a number of semantical investigations related to probability. See [6, 16, 4] for example. The mathematical background is sketched in the next section. For our study we have used a construction for obtaining a metric space of measures from a given metric space that is borrowed from [22, 10] A similar construction is available for valuations, cf. 2] The paper contributes ....

A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

....to construct computational models for real valued functions and functionals we will use continuous This research was supported in part by the RFBR (grants N 99 01 00485, N 00 0100810) and by the Siberian Division of RAS (a grant for young researchers, 2000) domains. Continuous domains, e.g. [31, 32, 14, 6, 7, 10, 38 40], are generalisation of algebraic domains, e.g. 2, 29, 34, 35] The continuous domain (more precisely, the interval domain) for the reals was rst proposed by Dana Scott [31] and later was applied to mathematics, physics and real number computation in [6, 7, 39, 40, 29] and others. In this ....

.... domains, e.g. 31, 32, 14, 6, 7, 10, 38 40] are generalisation of algebraic domains, e.g. 2, 29, 34, 35] The continuous domain (more precisely, the interval domain) for the reals was rst proposed by Dana Scott [31] and later was applied to mathematics, physics and real number computation in [6, 7, 39, 40, 29] and others. In this article we propose continuous domains, named as function domains to construct a computational model of operators and real valued functionals de ned on the set of continuous real valued functions. In Section 2, we recall basic de nitions and tools from [7] and introduce some ....

A. Edalat, Domain Theory and integration, Theoretical Computer Science, 151, 1995, pages 163-193.

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A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163-193, 1995.

....in [17] and [10] were pointed out in particular by O. Kirch and R. Tix. Continuous valuations have in recent years played a crucial role in the domain theoretic approach to classical measure theory which has led to a new generalization of the Riemann integral with applications in fractal geometry [5,4,6]. c fl1998 Published by Elsevier Science B. V. Alvarez Manilla, Edalat and Saheb Djahromi In this extended abstract we give a short and direct proof for a general result that includes the domain theoretic cases mentioned above. We show that if a oe finite valuation on a directed complete ....

A. Edalat, Domain theory and integration, Theoretical Computer Science 151 (1995), 163--193.

....the category of continuous information systems and approximable relations. 2.3. Eectively given domains. An # continuous domain can be e#ectively presented with respect to an enumeration of a basis by requiring that the way below relation restricted to the basis elements is recursively enumerable [123, 47]. This can be stated in terms of information systems. A continuous information system (A, #, #) with an enumeration of its elements A = a 0 , a 1 , a 2 , where a 0 = #, is e#ectively given with respect to this enumeration if the entailment relation am # a n is r.e. in m and n, i.e. ....

....This is also exactly how a computable number in the interval approach to computability on the real line is characterized as for example by Rogers [102, p. 371] We can define a continuous function f : R # R to be computable if it has a computable extension g : IR # IR. It is shown directly in [47] that our definition of computable real number and computable real function coincide with the well established notion by Pour El and Richards [101] which is equivalent to that of Weihrauch [122] and is based on the classical work of Grzegorczyk [62, 63] In fact, it was known from the work of ....

, A domain theoretic approach to computability on the real line, Theoretical Computer Science (1997), to appear.

....Of course, the operational semantics cannot evaluate a program denoting a real number in finitely many steps. However, it can compute an arbitrarily small rational interval containing the real number in a sufficiently large number of steps. Based on previous work on domain theory and integration [11, 8], we show how to handle integration in Real PCF. In domain theoretic integration, one obtains increasingly better approximations to the value of the integral of a real valued function. This has led to exact computations of integrals in various fields such as statistical physics [7] neural nets ....

....It follows that a program has some partial evaluation iff it does not denote bottom; it is important here that a cannot be bottom in a primitive operation cons a . 3. Interval Riemann integrals A generalization of the Riemann theory of integration based on domain theory was introduced in [8]. Essentially, a domain theoretic framework for the integration of realvalued functions w.r.t. any finite measure on a compact metric space was constructed using the probabilistic power domain of the upper space of the metric space. In this paper we will only be concerned with integration w.r.t. ....

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A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

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Abbas Edalat. Domain theory and integration. Theoretical Computer Science, 151(1):163--193, 1995.

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Abbas Edalat. Domain theory and integration. Theoretical Computer Science, 151(1):163--193, 13 November 1995.

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A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163-193, 1995.

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A. Edalat. Domain theory and integration. Theoretical Computer Science, 151:163--193, 1995.

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