N. Saheb-Djahromi, Cpo's of measures for nondeterminism, Theoretical Computer Science 12 (1980), 19-37.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....implies that the set satis es the Heine Borel property) Locally compact sober spaces are a broad class of T 0 spaces with rich structural properties [20] Continuous dcpo s are perhaps the most important examples of such spaces. In this context, the extension result was studied by Saheb Djahromi [51] for algebraic dcpo s and by Jones and Plotkin [24] 23] for continuous dcpo s (though these proofs contained some gaps) Norberg [43] established the results for nite valuations on second countable locally compact sober spaces; Lawson [30] for nite valuations on these same spaces and stably ....

N. Saheb-Djahromi, Cpo's of measures for nondeterminism, Theoretical Computer Science 12 (1980), 19-37.

.... source in order to determine how to optimally encode the information from that source A basic result in standard information theory says that L( H S holds when for any s 2 S: p s = 1=jEj) n with n 2 N [20] In our setting, where the randomized source is a program with probabilistic choice [18], this analysis corresponds precisely to statically analyze the probability of the objects produced by the program (see [16] for a general framework for probabilistic program analysis) When this probability lies in a rational power of type f1=ag N , then any encoding alphabet jEj = a can ....

N. Saheb-Djahromi. Cpo's of measures for nondeterminism. Theor. Comput. Sci., 12:19-37, 1980.

....F g : M 1 X # R with F g (#) # g d#, the latter being the Lebesgue integral of g with respect to #. See Section 6 for the definition of the Lebesgue integral. The weak topology on M 1 X is the coarsest topology which makes all these functionals continuous. A continuous valuation [16, 104, 86, 75, 67] is like a finite measure but is defined on open subsets. More precisely, a continuous valuation on a topological space Y is a mapping # : #Y # [0, 1] with (i) #(U ) #(V ) #(U # V ) #(U # V ) ii) #(#) 0. iii) U # V # #(U ) # #(V ) iv) For any directed subset A # #(Y ) ....

....PY is a dcpo in which lubs of directed subsets are computed pointwise. If Y is an # continuous dcpo with a countable basis B , then PY is an # continuous dcpo with a basis of simple valuations of the form # n i =1 r i # x with x i # B and rational r i 0 [76] Furthermore, Saheb Djahromi [104], Lawson [86] and Norberg [93] have independently shown that continuous valuations on di#erent classes of domains have unique extensions to Borel measures. It has recently been shown that any continuous valuation (and more generally any continuous 420 ABBAS EDALAT # finite valuation) on a ....

N. Saheb-Djahromi, Cpo's of measures for non-determinism, Theoretical Computer Science, vol. 12 (1980), no. 1, pp. 19--37.

....which perform probabilistic computation. Various approaches towards the semantics of probabilistic programs have been investigated up to now, each trying to capture the probabilistic feature in a suitable way. Early contributions in this area go back to the fundamental papers of Saheb Djahromi [34], and Kozen [29] More recent results are related to probabilistic power domains [25, 26, 27] probabilistic predicate transformers [31] and stochastic process calculi [2, 15] In [16, 17] the authors develop a probabilistic version of concurrent constraint programming [35] called Probabilistic ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19-37, 1980.

.... introduced in a previous work by the authors [10] which considers linear spaces structures (Banach or Hilbert spaces of measurable functions) as domain of denotations, in line with earlier important contributions in the area of probabilistic semantics like the fundamental papers of Saheb Djahromi [19], and Kozen [18] A rst result in this direction is the denotational model for average properties which refer to static quantities. These correspond to some real valued random variables whose de nition is xed when the process starts and doesn t change during all its execution [12] ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19-37, 1980.

....the two approaches [3, 13] we already mentioned in the introduction have been shown to be fully abstract only with respect to the results of nite computations. Additional investigations will compare our construction to other approaches towards the semantics of probabilistic programming languages [12, 9, 8], probabilistic predicate transformers [10] and logics and stochastic processes. ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19-37, 1980.

....1 Introduction Domain theory was introduced by Dana Scott in 1970 [Sco70] as a mathematical theory of computation in the semantics of programming languages. It has, since then, developed extensively in various areas of semantics, including probabilistic non determinism and probabilistic programs [Sah80, Koz81, Plo82, JP89]. In recent years, a new direction for applications of domain theory has emerged. It was shown in [Eda93] that, indeed, several branches of mathematics have natural domain theoretic structures. In particular, based on the probabilistic power domain, a constructive foundation for measure theory was ....

N. Saheb-Djahromi. Cpo's of measures for non-determinism. Theoretical Computer Science, 12(1):19--37, 1980.

....of UX. It sends any open subset of X to a G ffi subset (i.e. a countable intersection of open subsets) of UX and any Borel subset of X into a Borel subset of UX (Edalat 1995a) We will consider the probabilistic power domain PUX of the upper space of X. Recall the basic definitions. A valuation (Saheb Djahromi 1980; Lawson 1982; Jones 1989) on a topological space Y is a map : Omega Y [0; 1) where Omega Y is the lattice of open subsets, which satisfies: i) a) b) a [ b) a b) modularity) ii) 0, and (iii) a b ) a) b) A continuous valuation (Lawson 1982; Jones and Plotkin ....

Saheb-Djahromi, N. (1980) Cpo's of measures for non-determinism. Theoretical Computer Science, 12(1):19--37.

....Hausdorff space X , can be embedded into its upper space UX which can be given an effective structure. We would like to have a similar embedding for the set of bounded Borel measures on X . For this, we use the probabilistic power domain of UX . 2. 2 The Probabilistic Power Domain Recall from [7, 24, 21, 16] that a valuation on a topological space Y is a map : Omega Y [0; 1) which satisfies: i) a) b) a [ b) a b) ii) 0 (iii) a b ) a) b) A continuous valuation [21, 17, 16] is a valuation such that whenever A Omega Gamma Y ) is a directed set (wrt ) of open sets of Y ....

....im(d) then D = d Gamma1 (b) satisfies the following properties: i) x v y v z x; z 2 D ) y 2 D. ii) For any directed set hx i i i2I with F i x i 2 D, we have x i 2 D for some i 2 I . iii) For any directed set hx i i i2I with x i 2 D for all i 2 I , we have F i x i 2 D. It follows [24] that D is a crescent, i.e. D = v Gamma w for some open sets v; w 2 Omega Y . Now consider the map Pd : PY PY , induced by the probabilistic power domain functor P on the deflation d v 1 Y . We have Pd( ffi d Gamma1 v , since d Gamma1 (O) O for all O 2 Omega Y . Consider ....

N. Saheb-Djahromi. Cpo's of measures for non-determinism. Theoretical Computer Science, 12(1):19--37, 1980.

....can be defined on a universe of semantic domains which is closed under the usual constructions. What we find, in particular, is that the probabilistic powerdomain construction is in conflict with function spaces. The probabilistic powerdomain was first defined by Saheb Djahromi in 1980, [25]. It has since been studied extensively by Plotkin, Graham, Jones, Kirch, Heckmann and the second author, 24, 8, 13, 12, 20, 9, 10, 27] Originally, the probabilistic powerdomain was introduced as a tool in denotational semantics but, more recently, Edalat demonstrated its usefulness in more ....

....ordering between valuations 0 if (O) 0 (O) for all O 2 : Valuations have a long history in measure and lattice theory, see [21] and the references given there. As a construction in denotational semantics, the probabilistic powerdomain was first defined by Saheb Djahromi in [25], with the additional restriction (X) 1. The definition we have chosen is the one of [12] It was later shown by Kirch, 20] that one can extend the range of valuations to R 0; or even R 0; f1g, retaining the core properties. This extension has the advantage that we can freely add ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

....equational characterizations (which do exist for some of them) are rather bizarre and do not give us much insight. The hyperspace approach is developed in logical form in Section 7.3. We should also mention the various attempts to define a probabilistic version of the powerdomain construction, see [Saheb Djahromi, 1980, Main, 1985, Graham, 1988, Jones and Plotkin, 1989, Jones, 1990] As an aside, these cannot be restricted to algebraic domains; the wider concept of continuous domain is forced upon us through the necessary use of the unit interval [0; 1] They do have an equational description in some sense ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

....0 9 d x A; c p 9 d 0 x B; c t 9xd 0 R5 p(y) c 1 9 ff (ffi yff 9x(ffi ffx A) c p(x) GammaA 2 P Table 2. The transition system for PCCP. by a distribution (function) on the constraint system, that is a function from the set of constraints C into the real interval [0; 1] see [25, 8]) Our approach generalises the approach in [4] as in order to deal with recursion we define a similar operator Phi which transforms interpretations into interpretations. By constructing the appropriate limit of a sequence of interpretations, i.e. a fixed point of Phi, we are then able to ....

....namely by using a random choice construct, while the approach of [10] is based on some kind of fuzzy data, i.e. on the use of random variables. Related work on the denotational semantics of probabilistic imperative programming languages based on measure theoretic notions has been done by [25, 15, 13]. As for further work: Additional investigations should compare PCCP to other approaches to the semantics of probabilistic computation, e.g. probabilistic predicate transformers [20, 19] probabilistic logics and model checking [2, 12] or probabilistic process algebras [3] A possible extension ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

....choice construct, while the approach of [8] is based on some kind of fuzzy data, i.e. on the use of random variables. Further work will lead to the formulation of a denotational semantics for PCCP. It is natural to base this semantics on measure theoretic notions, e.g. on spaces of measures, cf. [21, 13, 11]. We are already able to define a Banach space based semantics for a restricted version of PCCP in which all constraints in the probabilistic choice construct are true. We called this language Probabilistic Constraint Logic Programming (PLCP) As a next step we plan to define a denotational ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

....equational characterizations (which do exist for some of them) are rather bizarre and do not give us much insight. The hyperspace approach is developed in logical form in Section 7.3. We should also mention the various attempts to define a probabilistic version of the powerdomain construction, see [Saheb Djahromi, 1980, Main, 1985, Graham, 1988, Jones and Plotkin, 1989, Jones, 1990] As an aside, these cannot be restricted to algebraic domains; the wider concept of continuous domain is forced upon us through the necessary use of the unit interval [0; 1] They do have an equational description in some sense ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

....the distribution of synaptic couplings in these networks using the action of an iterated function system on a probabilistic power domain. We then obtain algorithms to compute the decay of the embedding strength of the stored patterns. 1 Introduction The probabilistic power domain was introduced in [21] and developed in [20,14] for studying probabilistic computation, in order to provide semantics for probabilistic programming languages. In [3,5] a general framework was established for application of the probabilistic power domain in computation beyond semantics. It was shown that the ....

N. Saheb-Djahromi. Cpo's of measures for non-determinism. Theoretical Computer Science, 12(1):19--37, 1980.

....case the transition probability could be made dependent on the quality of the entailment of a certain constraint. Further work will also lead to the formulation of a denotational semantics for PCCP. It is natural to base this semantics on measure theoretic notions, e.g. on spaces of measures, cf. [16, 9, 8]. Additional investigations will relate PCCP to other semantic approaches towards probabilistic computation, e.g. probabilistic predicate transformers [13, 11] probabilistic logics and model checking [2, 7] and probabilistic process algebras [3] ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

....the two approaches [3, 13] we already mentioned in the introduction have been shown to be fully abstract only with respect to the results of finite computations. Additional investigations will compare our construction to other approaches towards the semantics of probabilistic programming languages [12, 9, 8], probabilistic predicate transformers [10] and logics and stochastic processes. ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

....a suitable setting for well behaved measures. But we have already seen that a locally compact second countable Hausdorff space also has the interesting property that its upper space is an continuous dcpo. This gives us a link with the theory of valuations. Definition 1. 4 [ Birkhoff, 1967; Saheb Djahromi, 1980; Lawson, 1982; Jones, 1989 ] A valuation on a topological space Y is a map : Omega Y [0; 1) which satisfies: i) a) b) a [ b) a b) ii) 0, and (iii) a b ) a) b) A continuous valuation [ Lawson, 1982; Jones and Plotkin, 1989; Jones, 1989 ] is a valuation such ....

N. Saheb-Djahromi. Cpo's of measures for nondeterminism. Theoretical Computer Science, 12(1):19--37, 1980.

....F g : M 1 X R with F g ( R g d , the latter being the Lebesgue integral of g with respect to . See Section 6 for the definition of the Lebesgue integral. The weak topology on M 1 X is the coarsest topology which makes all these functionals continuous. A continuous valuation [15, 104, 86, 75, 67] is like a finite measure but is defined on open subsets. More precisely, a continuous valuation on a topological space Y is a mapping : Omega Y [0; 1] with (i) U) V ) U [ V ) U V ) ii) 0 (iii) U V ) U) V ) iv) For any directed subset A Omega Gamma Y ) with ....

....PY is a dcpo in which lubs of directed subsets are computed pointwise. If Y is an continuous dcpo with a countable basis B, then PY is an continuous dcpo with a basis of simple valuations of the form P n i=1 r i ffi x i with x i 2 B and rational r i 0 [76] Furthermore, Saheb Djahromi [104], Lawson [86] and Norberg [93] have independently shown that continuous valuations on different classes of domains have unique extensions to Borel measures. It has recently been shown that any continuous valuation (and more generally any continuous oe finite valuation) on a continuous domain has ....

N. Saheb-Djahromi. Cpo's of measures for non-determinism. Theoretical Computer Science, 12(1):19--37, 1980.

.... dcpo with a bottom element , then P 1 Y is also an continuous dcpo with a bottom element ffi and has a basis consisting of simple valuations [26, 25, 15] Therefore, any 2 P 1 Y is the lub of an chain of normalised simple valuations and, hence by a lemma of Saheb Djahromi [33] can be uniquely extended to a Borel measure on Y which we denote for convenience by as well [33, page24] For 0 c 1, let P c Y denote the dcpo of valuations with total mass c, i.e. P c Y = f 2 PY j (Y ) cg. Since P c Y is obtained from P 1 Y by a simple rescaling, it shares the ....

N. Saheb-Djahromi. Cpo's of measures for non-determinism. Theoretical Computer Science, 12(1):19--37, 1980.

....choice construct, while the approach of [10] is based on some kind of fuzzy data, i.e. on the use of random variables. Further work will lead to the formulation of a denotational semantics for PCCP. It is natural to base this semantics on measure theoretic notions, e.g. on spaces of measures, cf. [23, 15, 13]. We are already able to define a Banach space based semantics for a restricted version of PCCP in which all constraints in the probabilistic choice construct are true [6] We called this language Probabilistic Constraint Logic Programming (PLCP) As a next step we plan to define a denotational ....

N. Saheb-Djahromi. CPO's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

....that is modular, monotone and such that ( 0 (see below) a continuous valuation is one which preserves directed suprema. For a directed complete partial orders (dcpo) the problem of extending a continuous valuation to a Borel measure reappeared in the context of probabilistic nondeterminism [23]. In [23] a proof was given for algebraic dcpo s but it contained a gap. Norberg established the result for nite valuations on continuous dcpo s and gave applications in random set theory and a proof of the Daniell Kolmogorov theorem for continuous lattices. Lawson [16] showed that a ....

....modular, monotone and such that ( 0 (see below) a continuous valuation is one which preserves directed suprema. For a directed complete partial orders (dcpo) the problem of extending a continuous valuation to a Borel measure reappeared in the context of probabilistic nondeterminism [23] In [23] a proof was given for algebraic dcpo s but it contained a gap. Norberg established the result for nite valuations on continuous dcpo s and gave applications in random set theory and a proof of the Daniell Kolmogorov theorem for continuous lattices. Lawson [16] showed that a continuous ....

[Article contains additional citation context not shown here]

N. Saheb-Djahromi. Cpo's of measures for nondeterminism. Theoretical Computer Science, 12:19-37, 1980.

.... lattice of open sets of a given topological space that is modular, monotone and such that ( 0 (see below) For domains, the problem of extending a (Scott) continuous valuation, i.e. one preserving directed suprema, to a Borel measure appeared in the context of probabilistic nondeterminism (see [17]) In [17] a proof was given for algebraic domains but it contained a gap. Norberg established the result for oe finite valuations on continuous domains and gave applications in random set theory and a proof of the Daniell Kolmogorov theorem for continuous lattices. For continuous bounded ....

.... open sets of a given topological space that is modular, monotone and such that ( 0 (see below) For domains, the problem of extending a (Scott) continuous valuation, i.e. one preserving directed suprema, to a Borel measure appeared in the context of probabilistic nondeterminism (see [17] In [17] a proof was given for algebraic domains but it contained a gap. Norberg established the result for oe finite valuations on continuous domains and gave applications in random set theory and a proof of the Daniell Kolmogorov theorem for continuous lattices. For continuous bounded complete ....

[Article contains additional citation context not shown here]

N. Saheb-Djahromi, Cpo's of measures for nondeterminism, Theoretical Computer Science 12 (1980), 19--37. 10

....that is modular, monotone and such that ( 0 (see below) a continuous valuation is one which preserves directed suprema. For a directed complete partial orders (dcpo) the problem of extending a continuous valuation to a Borel measure reappeared in the context of probabilistic nondeterminism [23]. In [23] a proof was given for algebraic dcpo s but it contained a gap. Norberg established the result for 1991 Mathematics Subject Classification 60B05, 06B35, 54F05 oe finite valuations on continuous dcpo s and gave applications in random set theory and a proof of the Daniell Kolmogorov ....

....modular, monotone and such that ( 0 (see below) a continuous valuation is one which preserves directed suprema. For a directed complete partial orders (dcpo) the problem of extending a continuous valuation to a Borel measure reappeared in the context of probabilistic nondeterminism [23] In [23] a proof was given for algebraic dcpo s but it contained a gap. Norberg established the result for 1991 Mathematics Subject Classification 60B05, 06B35, 54F05 oe finite valuations on continuous dcpo s and gave applications in random set theory and a proof of the Daniell Kolmogorov theorem ....

[Article contains additional citation context not shown here]

N. Saheb-Djahromi. Cpo's of measures for nondeterminism. Theoretical Computer Science, 12:19--37, 1980.

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