Smyth, M.B., Power domains, Journal of Computer and System Sciences, vol. 16, pp. 23-36, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we consider U , while in Section 4, L is considered. In Section 5, we consider some variations of the story. 3 Characterising FS Domains by Upper Power Domains The upper power domain UD of a domain (continuous dcpo) D consists of all compact upper subsets of D, ordered by opposite inclusion [8,9]. For our purpose, it does not matter whether ; is included in UD or not. For every domain D, UD is again a domain. There is a continuous function s = s D : D UD defined by sx = D x for every x in D, called the singleton function. Actually, it is an order embedding (x v y iff sx v sy) and ....

M. B. Smyth. Power domains. Journal of Computer and System Sciences, 16:23--36, 1978.

....on labeled objects in order to define approximate answers to queries formulated against databases of labeled objects, we define partial pre ordering relation on these objects. We will use three well known partial ordering relations, which have been used in the study of the semantics of programs [ 18], feature structures [6] and value oriented complex objects [3,5,13,16] The orderings are: partial ordering for disjunctive sets, s (Smyth ordering) partial ordering for conjunctive sets, Hoare s ordering) and partial ordering for convex sets, Plotkin s ordering) which combines ....

Smyth M.B., Power Domains, Journal of Computer and System Sciences, Vol 16, pp. 23-36, 1978. 10

.... was thought to be more straightforward than the pre order (partial order) approach, which would model the non determinism typically occurring in the context of concurrent languages with one of the power domain constructions in use, the Smyth, Plotkin or Hoare power domain (see e.g. Plotkin 76] Smyth 78] and [Apt Plotkin 81] It was easy to introduce a distance between two sequences, following the paradigm that the longer you had to watch them to detect a difference, the closer they were. This is also at the root of the metric concept of approximation: an element approximates another to the ....

Smyth, M.B., Power domains, Journal of Computer and System Sciences, vol. 16, pp. 23-36, 1978.

....finite antichains in , ordered by: n 1 v n 2 ( 8r 2 n 1 : 9s 2 n 2 : r v s: For example, f0; 01g is not a valid element of N because 0 v 01, and we have f00; 011g v f000; 011; 1g and f00; 10g 6v f000; 011; 1g as well. As the order defined on N is the classic order in lower powerdomains [16], at first sight looks like we are in the presence of a pre order. However, N was defined in a way so that it is a partial order and not merely a pre order. More specifically: Proposition 4.2 N is a partial order; moreover it is a join semillatice with join given by: n 1 t n 2 = fs 2 n 1 [ n 2 ....

M. B. Smyth. Power domains. Journal of Computer and System Sciences, 16:23--36, 1978.

....of processes is denoted by the cpo P and the domain of values by V . An output capability of p on c, on the other hand, is modelled as an element of V Theta P labelled by c. A process is modelled as a set of communication capabilities or more precisely as an element of the Smyth Power Domain [Smy78] over the domain of communication capabilities. The empty set is embedded into the domain in such a way that it becomes the top element of the domain. This leads to a recursive domain equation over a suitable class of domains. The process domain is then defined as the initial solution to this ....

....value passing based on the late approach using the Plotkin Power Domain with the empty set adjoined as an isolated element. Here the main difference is that we use a different representation for the Plotkin Power Domain to the one used in [Abr91] The representation we use is the one due to Smyth, [Smy78] and will be described below. In the definition of the domain we use the following operations on cpos: Cartesian product Theta: Plo81] x2 and x6) Let hA; v A i and hA ; v 0 i be two pos. We define the partial order A ThetaA on A Theta A by: a; a A ThetaA 0 (b; b ) ....

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M. Smyth. Powerdomains. Journal of Computer and System Sciences, 16(1), 1978.

....of all nite antichains in , ordered by: n 1 v n 2 ( 8r 2 n 1 : 9s 2 n 2 : r v s: For example, f0; 01g is not a valid element of N because 0 v 01, and we have f00; 011g v f000; 011; 1g and f00; 10g 6v f000; 011; 1g as well. As the order de ned on N is the classic order in lower powerdomains [16], at rst sight looks like we are in the presence of a pre order. However, N was de ned in a way so that it is a partial order and not merely a pre order. More speci cally: Proposition 4.2 N is a partial order; moreover it is a join semillatice with join given by: n 1 t n 2 : fs 2 n 1 [ n 2 j ....

M. B. Smyth. Power domains. Journal of Computer and System Sciences, 16:23-36, 1978.

....applications in Computer Science. In domain theory a great variety of data types and programming concepts can be modelled and analysed. This includes nondeterminism, computations on continuous and higher type data, and the operational semantics and expressive power of functional languages [51, 48, 53, 28, 1]. On the other hand proof theory provides formal methods for program development. For example, the so called proofs as programs paradigm [2, 50, 59] according to which a constructive proof of a speci cation corresponds to a satisfying program, has led to a method of program synthesis from ....

M.B. Smyth. Powerdomains. Journal of Computer and Systems Science 16, 23-36, 1978.

....a complete partial order of state transformations (cf. Definitions 5.1, 5.2, and 5. 3 of [Plo79] We postulate a denumerable set (oe 2) Sigma of states (cf. page 212 of [Wan77] and page 529 of [Plo79] State transformations are defined as mappings from Sigma to the so called Smyth power domain ([Smy78]) of Sigma in Definition 1.1 The partial order (ff 2) ST of state transformations is the set Sigma P S ( Sigma ) where P S ( Sigma ) f S Sigma j S is nonempty and finite g [ f Sigma g and Sigma denotes the disjoint union of Sigma and f g. This set is ordered by ff v ff 0 if ....

M.B. Smyth. Power Domains. Journal of Computer and System Sciences, 16(1):23--36, February 1978.

....can be 1 8, 1 16 or 1 32. In the gure, a directed edge indicates the partial ordering relationship, e.g. 12,1 32) f (12,1 16) The ordering f de ned on fascicles can be extended to give an ordering s on sets of fascicles. To do so, we rely on the well known Smyth ordering for power sets [20]: S 1 s S 2 i 8F 2 2 S 2 ; 9F 1 2 S 1 : F 1 f F 2 (2) For the example in Figure 2, the sets S 1 = f(12; 1=8) 10; 1=16)g and S 2 = f(11; 1=32)g are mutually incomparable. But the set S 3 = f(12; 1=32)g dominates both, i.e. S 3 s S 1 , S 3 s S 2 . The Smyth ordering induces equivalence ....

....are in this fascicle. These are players who have very limited Fascicles (12,1 16) 10,1 12) 8,1 8) Population 10 12.5 12.5 Games played [1,82] 1,5] 1,79] 1,82] Goals Scored [0,50] 0,0] 0,3] 0,3] Assists [0,80] 0,2] 0,5] 0,5] Points [0,130] 0,2] 0,7] 0,7] PlusMinus [ 20,40] [ 2,1] 2,1] 13,8] Penalty Mins [0,400] 0,18] 0,22] 0,233] Power Play Goals [0,20] 0,0] 0,1] 0,1] Short Handed Goals [0,9] 0,0] 0,0] 0,1] Game winning Goals [0,12] 0,0] 0,1] 0,1] Game tying Goals [0,8] 0,0] 0,0] 0,1] Shots [0,400] 0,8] 0,23] 0,25] Percent Goals ....

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M. Smyth. Power Domains. Journal of Computer and System Sciences, 16, 1, pp. 23-36, 1978.

....trace refinement is an adequate semantics to the ACSR refinement. In addition, an interresting result is to check completeness of the ACSR process refinement with respect to the extended trace refinement. We next elaborate on these steps. Extensions of Trace Refinement. Powerdomain theory [Smy78, Gun92] gives us the possibility to extend our trace refinement (Definition 4.2) to several relations over sets of traces. Three most commonly used extensions are the Hoare ordering, Smyth ordering and Egli Milner ordering. For sets of traces R and S and a function ae : events(S) Gamma ....

M. Smyth. Power domains. Journal of Computer and System Science, 16(1):23-- 36, 1978.

....antichains of A, and P fin (A) the family of finite subsets of A. A similar problem arises in the semantics of programming languages, most notably in the semantics of concurrency, cf. Gun92] Three orderings, called the Hoare, the Smyth and the Plotkin ordering have been proposed ( Gun92, Smy78, Plo76] Hoare) X v [ B , 8x 2 X9y 2 Y : x y (Smyth) X v ] Y , 8y 2 Y 9x 2 Y : x y (Plotkin) X v Y , X v [ Y and X v ] Y All of them have been used for databases with partial information: the Hoare ordering in [Bis81, IL84, Lib91] the Smyth ordering in [BJO91, Oho90] the 10 ....

M. B. Smyth. Power domains. Journal of Computer and System Sciences, 16(1):23--36, 1978.

.... In fact, the Plotkin powerdomain is the most discriminating of the classical powerdomain constructions, since it combines the lower preorder ( L) above, associated with the Hoare powerdomain construction [Smy83] and the upper preorder ( U) above, associated with the Smyth powerdomain construction [Smy78]) There are other powerdomain constructions [Hec90, Gun92] based on the classic powerdomains, and they su er from similar over identi cation. Another problem arises because amb leads to countable non determinism. As shown in [Plo82] it is possible to use amb to de ne terms that have countably ....

M. B. Smyth. Power domains. Journal of Computer and System Sciences, 16(23-26):23-35, 1978.

....it. Fitting s work suggests that ordinary logic programming can be considered as a way of computing partial truth assignments. We show that this view continues to hold domain theoretically, when we introduce the Smyth powerdomain, a way to express the domain theoretic semantics of nondeterminism [Smy78]. We proved in [RZ] that the set of logically closed consistent theories in three valued logic, ordered by inclusion, is isomorphic to the Smyth powerdomain of the domain of three valued truth assignments. With our state transition semantics, the TP operator is thus seen to express a fixpoint in a ....

....program Q has exactly the same models as does P . 8 WILLIAM C. ROUNDS AND GUO QIANG ZHANG 3.3. The Smyth powerdomain. We can generalize disjunctive logic programming over the domain of partial truth assignments to programming over a Scott domain. In fact, the Smyth powerdomain was invented [Smy78] to specify a kind of non deterministic computation. The definition is based on a standard way to lift the order in a given domain to subsets of that domain. For this we need the definitions of directed set and finite element. Definition 3.6. A directed set in a domain D is a subset L of D ....

M. B. Smyth, Powerdomains, Journal of Computer and System Science 16 (1978), no. 1, 23--36.

....in Haggis, and a theory for concurrent Haskell has been outlined in [PJAF96] This suggests that a theory of fudgets could be developed using this route. 2. 3 Non Determinism in Functional Languages One usually gives meaning to non deterministic constructs via powerdomain constructions [Plo76,Smy78,Smy83,Bro86,SS92,Cli82] domain theoretic analogues of the powerset operator. The Plotkin, or convex, powerdomain [Plo76] models erratic choice very well, but like all powerdomain semantics, ignores the issue of sharing. An alternative to erratic choice is McCarthy s amb, a bottom avoiding ....

M. B. Smyth, Power domains, Journal of Computer and System Sciences 16 (1978), no. 23-26, 23-35.

....tree for N = 2. uniqueness of the fixed point. This can however be proved using the Plotkin power domain which was originally constructed to capture the semantics of non determinism [96] The Plotkin power domain of an # continuous dcpo can be defined using the notion of finitely generable sets [111, 114]. Suppose (D, #) is any pointed # continuous dcpo and B # D a countable basis for it. Consider any finitely branching tree such that each node is an element of B and each child y of any parent node x satisfies x # y. Repetitions of elements are allowed so that an element can appear in more ....

, Power domains, Journal of Computer and System Sciences, vol. 16 (1978), no. 1, pp. 23--36.

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Smyth, M.B., Power domains, Journal of Computer and System Sciences, vol. 16, pp. 23-36, 1978.

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Smyth, M. B. (1978) Powerdomains. Journal of Computer and Systems Sciences 16 23--36.

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Michael B. Smyth. Powerdomains. Journal of Computer and Systems Sciences, 16(1):23--36, February 1978.

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M. B. Smyth. Power domains. Journal of Computer and System Sciences, 16:23--36, 1978.

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M. B. Smyth. Powerdomains. Journal of Computer and Systems Sciences, 16:23--36, 1978.

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M.B. Smyth. Powerdomains. Journal of Computer and System Sciences, 16:23-36, 1978. 34

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M. B. Smyth. Powerdomains. Journal of Computer and Systems Sciences, 16:23-36, 1978.

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M.B. Smyth. Power domains. Journal of Computer and System Sciences, 16:23--36, 1978.

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M. Smyth. Powerdomains. Journal of Computer and System Sciences, 16(1), 1978.

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Smyth M.B., Power Domains, Journal of Computer and System Sciences, Vol 16, pp. 23-36, 1978.

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