C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In IEEE Symp. on Logic in Computer Science, pp. 186-195, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a rational number, the process that performs output action a and then functions as Q, is de ned analogously. In this case, PO = QO [ fa g. For both input and output pre xing, we have: d (a r :P; a u :P) c j r u j. Probabilistic Choice P = Q r Q is the probabilistic choice combinator [JP89] that chooses Q with probability r and Q with probability 1 r. PO = QO [Q O . P = Q]Q . Now a (q; X]X a (q; X) if q 2 Q, and a (q; X ] X ) a (q; X ) if q 2 Q . We de ne an initial distribution : fq 0 g) r; fq 0 g) 1 r, referring the reader to ....

C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual IEEE Symposium On Logic In Computer Science, pages 186-195, 1989.

....a computationally minded approach 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 ....

C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual IEEE Symposium On Logic In Computer Science, pages 186--195, 1989.

....which is a maximal Cartesian closed full subcategory of CONT if one restricts attention to pointed dcpo s (those with ) FS contains FC, but it is an open question whether FS and FC are different or coincide. The probabilistic power domain construction P , introduced by Jones and Plotkin [4,3], is an endofunctor of CONT. However, it is not known whether it is an endofunctor of any Cartesian closed full subcategory of CONT. Several candidate categories are ruled out by explicit counterexamples. The remaining ones are FC and FS, yet it is still unknown whether any of these two is closed ....

C. J. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Logic in Computer Science LICS '89, pages 186--195. IEEE Computer Society Press, 1989.

....xy . Thus, the given set is A xy f 1 i (B xy ) and thus is open, as f i is Lawson continuous. The authors accept responsibility for any errors and typos in this rendition. 27 5. 2 The domain Proc We fix a (countable) set L of labels and use the Jones Plotkin probabilistic powerdomain [JP89, Jon90] For notational convenience, we write L D for the product D indexed by the set of labels. Processes are given by the recursive domain equation: Pr (Proc) We will write for the partial order in the domain. Proposition 5.13 The domain equation Pr (Proc) can be solved in ....

C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual IEEE Symposium On Logic In Computer Science, pages 186--195, 1989.

....order to develop the corresponding applications. These developments include domain theoretic approaches to dataflow networks (e.g. Mat94] and [Mat95] logic programming (e.g. Sed96] domain theoretic approaches to integration (e.g. Eda94] models for probabilistic languages (e.g. Jon89] and [JP89]) models for real number computation ( EEP97] as well as models which incorporate complexity analysis (e.g. Sch96] and [RS96] Each of these applications involve real number measurements in some sense, and hence the adjective quantitative is used as opposed to the adjective qualitative ....

....spaces is resolved by the addition of a new topology. In [Smy91] the totally bounded spaces have been introduced, for which the notion of completion simplifies to the bicompletion and for which the induced topology is the Scott topology. Other approaches include the use of valuations (e.g. [JP89] and [Eda94] as well as the use of partial metrics (e.g. Mat94] In [O N97] the question is raised as to which domains are quantifiable in the sense that there exists a partial metric which induces the Scott topology. A similar question has been raised by Heckmann in [Hec96] It is shown in ....

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C. Jones, G. Plotkin, A probabilistic powerdomain of evaluations. In: LICS '89, IEEE Computer Society Press, 186 - 195.

.... of pairs x = fhs; x s ig s2State : This construction can be seen as a quantitative analogue of the usual powerset construction P(State) often used in the denotational semantics of nondeterministic languages [57,70] and is alternative to the probabilistic power domain construction introduced in [37,36]. Example 3.1 Assume State = N . Then V(State) V(N) is the in nite dimensional vector space R with typical element i2N i; where x i 2 R and i represents the base vector corresponding to the natural number i. Each set in the powerset of N can be represented by a vector in V(N) ....

....means of the probabilistic choice construct, while Kozen s approach considers random assignments in the language syntax and random variables with their associated probability distribution, in the semantical model. Another approach to the semantics of probabilistic languages has been pioneered in [62,36,37] and taken over in [39] This approach is based on probabilistic powerdomain, an extension of the powerdomain construction with measure theoretic concepts which allow for modelling probabilistic choice. This framework allows to accommodate uncountable domains, and seems therefore particularly ....

Jones, C. and G. Plotkin, A probabilistic powerdomain of evaluations, in: Symposium on Logic in Computer Science (LICS), IEEE Computer Society Press, 1989, pp. 186-195.

.... as a means of formulating the semantics of computation [22, 23] Ramsey and Pfeffer [29] present a stochastic lambda calculus whose denotational semantics is based upon the monad of probability measures [10] Jones [17] presents a probabilistic metalanguage based upon the monad of evaluations [18]. Both monads do not distinguish discrete and continuous probability distributions, and provide a unified representation scheme for all kinds of probability distributions. The above languages, however, do not fully implement their monads. The di#culty is that a probability measure or an ....

....on a measurable space# is a mapping from not# but its # algebra B# to [0.0, 1. 0] Since it is computationally infeasible to keep track of probabilities assigned to all events from an infinite discrete domain or a continuous domain, they provide only a binary choice construct (e1 orp e2 in [18] and choose p e1 e2 in [29] As a result, both languages essentially implement the probability monad in [6] which is capable of specifying only probability distributions over finite domains. In this paper, we propose a monadic probabilistic language capable of specifying probability ....

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C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings, Fourth Annual Symposium on Logic in Computer Science, pages 186-- 195. IEEE Computer Society Press, June 1989.

....that the algebras are the continuous frames and the homomorphisms are the frame homomorphisms. There is thus the possibility of doing locale theory in Infosys to treat what, classically, are the locally compact locales. Probabilistic power domains Jones [11] see also Jones and Plotkin [12]) has described how to construct, for an arbitrary dcpo P, a probabilistic power domain E(P) also a dcpo. Its points are evaluations , functions mapping tip to the unit closed real interval [0, 1] and satisfying certain conditions similar to those in measure theory. 0, 1] with its Scott ....

C. Jones and G.D. Plotkin, A probabilistic powerdomain of evaluations, LICS 89.

....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 w 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 which satisfies: i) a ) ii) v( O, and (iii) a C b v(a) v(b) A continuous valuation [Lawson, 1982; Jones and Plotkin, 1989; Jones, 1989] is a valuation such that whenever A C Q(Y) is a directed set (wrt C ) of open ....

....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 which satisfies: i) a ) ii) v( O, and (iii) a C b v(a) v(b) A continuous valuation [Lawson, 1982; Jones and Plotkin, 1989; Jones, 1989] is a valuation such that whenever A C Q(Y) is a directed set (wrt C ) of open sets of Y, then ( J o) supo(o) OEA For any b C Y, the point valuation based at b is the valuation z b: Q(Y) 0, oe) defined by 1 if boO z b(O) 0 otherwise. Any finite linear combination of ....

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C. Jones and G. Plotkin. A Probabilistic Powerdomain of Evaluations. In Logic in Computer Science, pages 186 195. IEEE Computer Society Press, 1989.

....where semantic objects are trees with three kinds of nodes: probabilistic, nondeterministic and action nodes. Another probabilistic extension of CSP is presented in [23] where a denotational semantics of CSP is de ned by applying the probabilistic powerdomain construction of Jones and Plotkin [16, 15] over a directed complete partial order. Probabilistic processes are considered to be probability distributions over processes of CSP. In [20] a generalisation of Larsen and Skou s [21] reactive probabilistic transition system is presented. This model considers three kinds of choices: ....

C. Jones and G.D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of 4th Annual Symposium on Logic in Computer Science, 1989.

....where semantic objects are trees with three kinds of nodes: probabilistic, nondeterministic and action nodes. Another probabilistic extension of CSP is presented in [14] where a denotational semantics of CSP is de ned by applying the probabilistic powerdomain construction of Jones and Plotkin [9] over a directed complete partial order. Probabilistic processes are considered to be probability distributions over processes of CSP. Based on ACP, we can cite [1] In that paper, a probabilistic version of ACP is presented leading to a language that combines probability and nondeterminism. An ....

C. Jones and G.D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of 4th Annual Symposium on Logic in Computer Science, 1989.

....of this semantic universe, in due contrast with traditional denotational semantics. Related work The most obvious precursor to this work is the study of probabilistic powerdomains and their application to denotational semantics as in the early work of SahebDjahromi [22, 23] and of Jones Plotkin [14] amongst others. More recent work [7, 8] has investigated bisimulation in a probabilistic context. It would be interesting to consider more closely the connection between that work and our own as this remains rather unclear for now. Another possible connection is with exact real arithmetic [9] ....

C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings, fourth Annual IEEE Symposium on Logic In Computer Science, 1989.

....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 compact spaces, which implies the extension for ....

....result together with the extension result imply that every locally nite continuous valuation on the space extends uniquely to a Borel measure and determines a continuous valuation on a continuous dcpo. Continuous valuations on continuous dcpos can be approximated via chains of simple valuations [24]. For our purposes a domain representation will be an embedding from a topological space into a continuous dcpo topologised with the Scott topology. Other authors have proposed di erent notions of domain representations [58] 52] 5] 4] The last part of the paper is concerned with characterising ....

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C. Jones and G. Plotkin, A probabilistic powerdomain of evaluations, Logic in Computer Science (LICS), IEEE Computer Society Press, Silver Spring, MD, 1989, pp. 186-195. 36

....of probabilistic and nondeterministic choice is studied in the context of two process languages, a nonconcurrent and a concurrent one. Various operational, denotational and axiomatic models are proposed in the literature to describe the probabilistic operator of PCCS (cf. 8] See, e.g. [15,18,19,9,22,17]. Often the nondeterminacy is removed or restricted when treating probability, and the parallel operator is interpreted as a synchronous product. In a number of these references, though, both probability on the one hand, and nondeterminism and or concurrency on the other are treated on equal ....

C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proc. LICS'89, pages 186--195. Asilomar, 1989.

....dcpo if it has a basis and an continuous dcpo if it has a countable basis. In a continuous dcpo D for all C D the set C is open. A continuous dcpo with the Scott topology is always sober and locally quasicompact but not necessarily coherent. Given a topological space (X; a valuation [4, 16, 14] is a map : 0; 1] which satis es: 0 (strictness) Q 1 Q 2 ) Q 1 ) Q 2 ) monotonicity) Q 1 [ Q 2 ) Q 1 Q 2 ) Q 1 ) Q 2 ) modularity) A valuation is said to be continuous [16, 14] if for any D , which is 3 directed with respect to , we have ....

....coherent. Given a topological space (X; a valuation [4, 16, 14] is a map : 0; 1] which satis es: 0 (strictness) Q 1 Q 2 ) Q 1 ) Q 2 ) monotonicity) Q 1 [ Q 2 ) Q 1 Q 2 ) Q 1 ) Q 2 ) modularity) A valuation is said to be continuous [16, 14] if for any D , which is 3 directed with respect to , we have ( S D) sup Q2D (Q) For any a 2 X we de ne the point valuation based at a as the function a : 0; 1) such that a (Q) 1 if a 2 Q 0 otherwise. A simple valuation is any nite linear combination P n i=1 r i ....

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C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Logic in Computer Science (LICS), pages 186-195. IEEE Computer Society Press, Silver Spring, MD, 1989.

....number, the process that performs output action a and then functions as Q, is de ned analogously. In this case, PO = QO [ fa g. For both input and output pre xing, we have: d c (a r :P; a u :P) c j r u j. 19 Probabilistic Choice P = Q r Q 0 is the probabilistic choice combinator [JP89] that chooses Q with probability r and Q 0 with probability 1 r. PO = QO [Q 0 O . P = Q]Q 0 . Now P a (q; X]X 0 ) a (q; X) if q 2 Q, and P a (q; X ] X 0 ) 0 a (q; X 0 ) if q 2 Q 0 . We de ne an initial distribution : fq 0 g) r; fq 0 0 g) 1 r, referring ....

C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual IEEE Symposium On Logic In Computer Science, pages 186-195, 1989.

.... bounded complete domains the result followed from the work of Lawson [12] who showed that a continuous valuation defined on a distributive continuous lattice L has a unique extension to a regular Borel measure on L (on the Borel oe algebra of its Lawson topology) Finally Jones and Plotkin [10,11], following the approach in [17] claimed the result for continuous domains, without presenting a correct proof. The gaps 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 ....

....if for all x 2 D the set B x = B # #x is directed and x = F B x . D is a continuous domain if it has a basis and a continuous domain if it has a countable basis. In a continuous domain D for all C D the set C is Scott open. Given a topological space (X; Omega X) a valuation (see [3,12,11]) is a map : Omega X [0; 1] which satisfies: ffl ( 0 (strictness) ffl O 1 O 2 ) O 1 ) O 2 ) monotonicity) ffl (O 1 [ O 2 ) O 1 O 2 ) O 1 ) O 2 ) modularity) 2 Alvarez Manilla, Edalat and Saheb Djahromi A valuation is said to be (Scott) continuous (see [12,11] ....

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C. Jones and G. Plotkin, A probabilistic powerdomain of evaluations, Logic in Computer Science (LICS) (IEEE Computer Society Press, Silver Spring, MD, 1989) 186--195.

....Thus, the given set is S xy A xy f 1 i (B xy ) and thus is open, as f i is Lawson continuous. 12 The authors accept responsibility for any errors and typos in this rendition. 25 5. 2 The domain Proc We x a (countable) set L of labels and use the Jones Plotkin probabilistic powerdomain [JP89, Jon90] For notational convenience, we write L D for the product Y L D indexed by the set of labels. Processes are given by the recursive domain equation: Proc = L P Pr (Proc) We will write v for the partial order in the domain. Proposition 5.13 The domain equation Proc = L P Pr ....

C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual IEEE Symposium On Logic In Computer Science, pages 186-195, 1989.

....any topological space Y , the poset 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 ....

....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 ....

C. Jones and G. Plotkin, A probabilistic powerdomain of evaluations, Logic in computer science, IEEE Computer Society Press, 1989, pp. 186--195.

....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 Concurrent Constraint Programming (PCCP) The denotational semantics of PCCP is given in terms ....

C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Symposium on Logic in Computer Science (LICS), pages 186-195. IEEE Computer Society Press, 1989.

....p.4 A Per Model of Secure Information Flow in Sequential Programs 5 Section 5 considers the problem of preventing unwanted probabilistic information flows in programs. We show how this can be solved in the same framework by utilising a probabilistic semantics based on the probabilistic powerdomain [21]. Section 6 shows how the probabilistic security specification satisfies compositionality properties which facilitates straightforward proofs of correctness for compositional analyses. We illustrate the usefulness of the compositional nature of the security specifications by presenting a ....

....to be secure, and the security condition implicit in their correctness argument is not directly comparable due to the fact that they consider parallel deterministic threads and a noncompositional semantics. To counter the problem indicated by the example we consider probabilistic powerdomains [21] which allow the probabilistic nature of choice to be reflected in the semantics of programs, and hence enable us to capture the fact that varying the value of h causes a change in the probability distribution of values of l. hosc.tex; 2 10 2000; 19:28; p.23 24 Andrei Sabelfeld and David Sands ....

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Jones, C. and G. D. Plotkin: 1989, `A Probabilistic Powerdomain of Evaluations '. In: Proceedings, Fourth Annual Symposium on Logic in Computer Science. Asilomar Conference Center, Pacific Grove, California, pp. 186--195.

....M 1 is defined on ultrametric spaces, and the Borel oe 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, as studied, e.g. by Jones and Plotkin [JP89] and by Edalat [Eda94] are twofold. Firstly, one can resort to the rich literature for standard measure theory on metric spaces. Secondly, we can apply the recently developed theory on coalgebraic bisimulation and final coalgebras in the metric setting [AM89,RT94] Notably, we shall see that M 1 ....

C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proc. LICS'89, pages 186--195. Asilomar, 1989.

....As mentioned above, the functor M 1 is defined on ultrametric spaces, and the Borel oe 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 [Eda94, Eda95] are twofold. Firstly, one can resort to the rich literature for standard measure theory on metric spaces (see, e.g. KV84] Secondly, we can apply the recently developed theory on coalgebraic bisimulation and final coalgebras in the metric setting [AM89, RT94] Notably, we ....

C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proc. LICS'89, pages 186--195. Asilomar, 1989.

....mentioned above, the 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 ....

C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proc. LICS'89, pages 186--195. Asilomar, 1989.

....we devise. There is a long history of modeling probabilistic choice in this area, dating back to the seminal work of Saheb Djarhomi [24] in which a now standard construction of a cpo supporting probabilistic choice was given, beginning with an underlying cpo. This work led to the results in [11, 12] that clari ed and expanded the nature of Saheb Djarhomi s construction, and also showed that this construction, when applied to a continuous domain, yields a continuous domain. This is the construction used in [20] for probabilistic CSP, which is simply the probabilistic power domain PPr (FD ) of ....

....maps. The probabilistic power domain: We now describe the construction that allows probabilistic choice operators to be added to a domain. This construction was rst investigated by Saheb Djarhomi [24] who showed that the family he de ned yields a cpo. The construction later was re ned by Jones [11, 12] where it also was shown that the probabilistic power domain of a continuous domain is again continuous. The de nition of the more general construction goes as follows. De nition 1. If P is a dcpo, then a continuous valuation on P is a mapping : D [0; 1] de ned on the Scott open subsets of P ....

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C. Jones and G. Plotkin, A probabilistic powerdomain of evaluations, Proceedings of 1989 Symposium on Logic in Computer Science, IEEE Computer Society Press, 1989, pp. 186-195.

....this semantic universe, in due contrast with traditional denotational semantics. Related work The most obvious precursor to this work is the study of probabilistic powerdomains and their application to denotational semantics as in the early work of Saheb Djahromi [22, 23] and of Jones Plotkin [14] amongst others. More recently, Edalat et al. 8, 9] have investigated bisimulation in a probabilistic context. It would be interesting to consider more closely the connection between that work and our own as this remains rather unclear for now. Another possible connection is with exact real ....

C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings, fourth Annual IEEE Symposium on Logic In Computer Science, 1989.

....upper A) 1 (A) 2 (A) 1 v 2 ] Lawson compactness is stable under inverse limits: Lemma 4.4 (A.Jung) continuous Lawson compact dcpo s are closed under inverse limits. 4.2. The domain Proc We fix a (countable) set Labels of labels and use the Jones Plotkin probabilistic powerdomain [29, 28]. For notational convenience, we write Labels D for the product Y Labels D. Processes are given by the recursive domain equation: Proc = Labels P Prob (Proc) We will write v for the partial order in the domain. Proposition 4.5 The above domain equation can be solved in the category of ....

C. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual IEEE Symposium On Logic In Computer Science, pages 186--195, 1989.

....with, and can be represented using our form of specification. Section 5 considers the problem of preventing unwanted probabilistic information flows in programs. We show how this can be solved in the same framework by utilising a probabilistic semantics based on the probabilistic powerdomain [18]. Section 6 shows how the probabilistic security specification satisfies compositionality properties which facilitates straightforward proofs of correctness for compositional analyses. We illustrate the usefulness of the compositional nature of the security specifications by presenting a ....

....to be secure, and the security condition implicit in their correctness argument is not directly comparable due to the fact that they consider parallel deterministic threads and a non compositional semantics. To counter this problem indicated by the example we consider probabilistic powerdomains [18] which allow the probabilistic nature of choice to be reflected in the semantics of programs, and hence enable us to capture the fact that varying the value of h causes a change in the probability distribution of values of l. 5.1 Probabilistic Powerdomain of Distributions In the possibilistic ....

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Jones, C., and Plotkin, G. D. A probabilistic powerdomain of evaluations. In Proceedings, Fourth Annual Symposium on Logic in Computer Science (Asilomar Conference Center, Pacific Grove, California, 5--8 June 1989), IEEE Computer Society Press, pp. 186--195.

....introduced above. There are non continuous dcpo s where they do not produce the free semilattice. Summarising, one can say that no simple concrete representation of the Plotkin power domain was found so far. The situation is quite different with the probabilistic power domain of Jones and Plotkin [9]. It was already defined with a satisfactory concrete representation: the probabilistic power domain of D is the set of all probabilistic valuations on D. These are continuous, strict, and modular functions from Omega D, the lattice of open sets of D, to the unit interval [0: 1] of the real ....

C. J. Jones and G. D. Plotkin. A probabilistic powerdomain of evaluations. In LICS '89, pages 186--195. IEEE Computer Society Press, 1989.

....in a functional language. A quite rich language could contain a construct [p 1 : x 1 ; p n : x n ] where p i are real numbers between 0 and 1 whose sum is 1. The semantics would be to select one of the possibilities x i with probability p i . Graham [1] Jones [3] and Jones Plotkin [4] consider an intermediate language with a construct x p y, which is written as p x; y by Graham and x p y by Jones, where p is a real number between 0 and 1. The semantics of this construct is to select x with probability p and y with probability 1 Gamma p. The notion of an (abstract) ....

....as p x; y by Graham and x p y by Jones, where p is a real number between 0 and 1. The semantics of this construct is to select x with probability p and y with probability 1 Gamma p. The notion of an (abstract) probabilistic domain was introduced by Graham [1] and further elaborated by Jones [3, 4] to describe the denotational semantics of a probabilistic language with the construct x p y. A probabilistic domain is a dcpo together with a continuous operation satisfying several axioms, which is used to model the choice construct semantically. For reasons of simplicity, we denote the ....

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C.J. Jones and G.D. Plotkin. A probabilistic powerdomain of evaluations. In LICS '89, pages 186--195. IEEE Computer Society Press, 1989.

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C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, Proc. LICS 4 (1989) 186--195.

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C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, Proc. LICS '89, pp. 186-195, IEEE Press, 1989.

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C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, in Proc. 4th LICS, Asilomar, pp. 186--195, Washington: IEEE Press, 1989.

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C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, in Proc. 4th LICS, Asilomar, pp. 186--195, Washington: IEEE Press, 1989.

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C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, in Proc. LICS '89, pp. 186--195, Washington: IEEE Press, 1989.

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C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, in Proc. LICS '89, pp. 186--195, Washington: IEEE Press, 1989.

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C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the Fourth Annual Symposium on Logic in Computer Science, pages 186-195. IEEE Computer Society Press, 1989.

....theoretic account of computational e#ects, which he called notions of computation. He modelled each e#ect by means of a strong monad T on a base category C with finite products. The monads corresponding to the e#ects listed above are given by a powerdomain [20] a probabilistic powerdomain [11, 12], and the monads (S ) S , E, TX = Y. O Y Y I X) and R R respectively [17 19] assuming C has appropriate additional structure; the set S of states is typically analysed as V Loc where V is a set of values and Loc is a set of locations. Moggi s unified approach has proved useful, ....

....given by sum. But that is not always the case: the combination with side e#ects is given by taking the commutative combination, which we define in the next section. Another non trivial example of a computationally natural countable Lawvere #Cpo theory is given by probabilistic nondeterminism [5, 11, 12]. More detail appears in [24] albeit in the mathematical terms of [14] 3 The commutative combination of e#ects In this section, we define the commutative combination L# L # of countable Lawvere theories L and L # and develop mathematical theory in support of the definition of this tensor ....

C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, in Proc. LICS '89, pp. 186--195, Washington: IEEE Press, 1989.

....y# Tx y# # x # f # T z # z # commutes. For some examples of algebraic operations where C = V = Set, let T be the nonempty finite power set monad with the binary choice operation [12, 1] alternatively, let T be the monad for probabilistic nondeterminism with a probabilistic choice operation [4, 5]; or take T to be the monad for printing with printing operations [13] Observe the non commutativity in the latter example. One can, of course, generalise from Set to categories such as that of # Cpo, for instance considering the various power domains together with binary choice operators. ....

C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, in Proc. 4th LICS, Asilomar, pp. 186--195, Washington: IEEE Press, 1989.

....special features of the category # Cpo and the monad L. In general, if all operations of an algebraic structure on # Cpo are of the form P n # P and satisfy the equation f(x, x, x) x then such a distributive law exists, necessarily uniquely. Example 4. The probabilistic power domain [5, 6] can be treated algebraically in a number of equivalent ways as described in [3] where three probabilistic choice operators are considered. Of the three, if studied on the category of # cpo s rather than that of # dcpo s, two fit within the framework of [8] while the other does not seem to do ....

C. Jones and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, in Proc. LICS '89, pp. 186--195, Washington: IEEE Press, 1989.

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C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In IEEE Symp. on Logic in Computer Science, pp. 186-195, 1989.

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Claire Jones and Gordon D. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of 4th LICS, pages 186--195, 1989.

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Jones, C., and G. D. Plotkin, A Probabilistic Powerdomain of Evaluations, Proc. LICS 4 (1989) 186--195.

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C. Jones and G.D. Plotkin. A probabilistic powerdomain of evaluations. In Proc. 4th IEEE Int. Symp. on Logic in Computer Science (LICS), pages 186-195, 1989.

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C. Jones and G. D. Plotkin, A probabilistic powerdomain of evaluations, in Proc. of 4th Ann. IEEE Symp. Logic in Computer Science, LICS'89 (Pacific Grove, CA, June 1989.

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C. Jones and G.D. Plotkin. A probabilistic powerdomain of evaluations. In Proc. 4th IEEE Int. Symp. on Logic in Computer Science (LICS), pages 186195, 1989.

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C. Jones and G. Plotkin. A probabilistic powerdomain of evaluations. In Proceedings of the 4th Annual Symposium on Logic in Computer Science, pages 186--195. IEEE Computer Society Press, 1989.

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C. Jones, G. Plotkin, A probabilistic powerdomain of evaluations. In: LICS '89, IEEE Computer Society Press, 186 - 195, 1998. 17

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C. Jones, G. Plotkin, A probabilistic powerdomain of evaluations. In: LICS '89, IEEE Computer Society Press, 186 - 195.

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C. Jones and G. Plotkin. A probabilistic power domain of evaluations, In Proceedings of 4th IEEE Symposium on Logic in Computer Science, 186-195, Cambridge, Mass., (1989)

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