## Semantic Domains for Combining Probability and Non-Determinism (2005)

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Venue: | ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE |

Citations: | 18 - 6 self |

### BibTeX

@INPROCEEDINGS{Tix05semanticdomains,

author = {Regina Tix and Klaus Keimel and Gordon Plotkin},

title = {Semantic Domains for Combining Probability and Non-Determinism},

booktitle = {ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE},

year = {2005},

pages = {2005},

publisher = {Elsevier}

}

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### Abstract

### Citations

3410 | A theory of communicating sequential processes
- BROOKES, HOARE, et al.
- 1984
(Show Context)
Citation Context ... (or Hoare) powerdomain. Unlike the convex powerdomains, the lower powerdomain cannot be attributed to a single person. It received its name from its connection to Hoare's work on partial correctness =-=[19]-=-. The upper powerdomain can also be used to model total correctness, while the convex powerdomain combines both approaches. Under certain conditions on the underlying space there exist nice topologica... |

1404 |
A Discipline of Programming
- Dijkstra
- 1976
(Show Context)
Citation Context ...ible set of states one can start with, in order to end up in a final state which satisfies a certain predicate. ThissSOME CONNECTIONS WITH SEMANTICS 97 corresponds to Dijkstra's weakest preconditions =-=[8]-=-: as g \Gamma 1 gives the greatest possible set, it gives the weakest condition a state must fulfill such that the program transforms it to a state satisfying the desired predicate. Recall that Scott-... |

1068 | Introduction to Lattices and Order - Davey, Priestley - 1990 |

924 | Categories for the working mathematician - Lane - 1971 |

457 | Domain theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ... theory; however, a complete introduction to this topic by far exceeds the scope of this work. Thus, we present selected items only and omit all proofs as we go along to fix our notation. We refer to =-=[1, 7, 14, 15, 33]-=- for more details. 1.1 Dcpos and Scott-Continuous Functions We shall use the term ordered set in the sense of partially ordered set, that is, it denotes a set X with a reflexive, antisymmetric and tra... |

321 |
A Compendium of Continuous Lattices
- Gierz, Hoffmann, et al.
- 1980
(Show Context)
Citation Context ...fixed points of continuous functions. Domain theory has since attracted many researchers and evolved in various directions. It owes much to the theory of continuous lattices and domains, most notably =-=[14, 15]-=-. An important problem in domain theory is the modelling of non-deterministic features of programming languages and of parallel features treated in a non-deterministic way. If a non-deterministic prog... |

212 | A powerdomain construction
- Plotkin
- 1976
(Show Context)
Citation Context ...n a non-deterministic way. If a non-deterministic program runs several times with the same input, it may produce different outputs. To describe this behaviour, powerdomains were introduced by Plotkin =-=[40, 41]-=- and Smyth [52]. A powerdomain over a domain X is a subset of the power set of X. Which subsets of X constitute the powerdomain depends on the kind of non-determinism that is be modelled. There are th... |

191 |
Outline of a mathematical theory of computation
- Scott
- 1970
(Show Context)
Citation Context ...The semantics of programming languages has been intensively studied by mathematicians and computer scientists. In the late sixties Dana S. Scott invented appropriate semantic domains for that purpose =-=[51, 49, 50]-=-. Continuous domains are directed complete partially ordered sets together with an order of approximation, the so called way-below relation. As they allow one to represent `ideal objects' and their `f... |

136 |
Probabilistic Non-determinism
- Jones
- 1990
(Show Context)
Citation Context ...domains, often referred to as Plotkin, Smyth, and Hoare powerdomains. viisviii INTRODUCTION Probabilistic non-determinism has also been studied and has led to the probabilistic powerdomain as a model =-=[47, 42, 24, 23]-=-. Different runs of a probabilistic program with the same input may again result in different outputs. In this situation, it is also known how likely these outputs are. Thus, a probability distributio... |

125 |
A probabilistic powerdomain of evaluations
- Jones, Plotkin
- 1989
(Show Context)
Citation Context ...e functions form a basis. Lower semicontinuous functions have an integral with respect to every continuous valuation. Three different approaches to integration have been developed and can be found in =-=[24, 23, 29]-=-, in [54], and in [18]. Integration plays an important role in Jones and Plotkin's model for probabilistic non-determinism. There, integrals arise if one looks at the probabilistic powerdomain functor... |

124 |
Continuous lattices
- Scott
(Show Context)
Citation Context ...The semantics of programming languages has been intensively studied by mathematicians and computer scientists. In the late sixties Dana S. Scott invented appropriate semantic domains for that purpose =-=[51, 49, 50]-=-. Continuous domains are directed complete partially ordered sets together with an order of approximation, the so called way-below relation. As they allow one to represent `ideal objects' and their `f... |

108 | Probabilistic predicate transformers
- Morgan, McIver, et al.
- 1996
(Show Context)
Citation Context ...ombine both kinds of nondeterminism? A research group in Oxford tackled various aspects of this problem. From this group, Morgan, McIver, and Seidel introduced probabilistic predicate transformers in =-=[38]-=- where non-determinism coexists with probabilistic choice. In [36], Morgan and McIver built a Plotkin style powerdomain over the space of probability distributions on a dicrete state space. We modify ... |

69 |
Powerdomains and predicate transformers: a topological view
- Smyth
- 1983
(Show Context)
Citation Context ...h kinds of assertions taken together. Heckmann studies these and other powerdomain constructions in a general algebraic framework [17]. The topological concept of powerdomains is stressed by Smyth in =-=[53]-=-. There, he highlights that open sets should be viewed as computable properties. A different kind of non-determinism is given by probabilistic choice. We already mentioned that a suitable model is giv... |

46 |
Local compactness and continuous lattices
- Hofmann, Mislove
(Show Context)
Citation Context ...ty compact saturated subsets of a topological space is denoted by Sc(X) and will be ordered by reverse inclusion. An important property of sober spaces X is the so called Hofmann-Mislove Theorem (see =-=[20, 26]-=-, [15, Theorem II-1.20]). The following proposition (see [15, TheoremII-1.21, Corollary II-1.22]) is a consequence of this theorem. It will be used extensively in Section 4.2. Proposition 1.3 Let X be... |

40 | The troublesome probabilistic powerdomain
- Jung, Tix
- 1998
(Show Context)
Citation Context ...pter 3. It still is an open problem whether there is a cartesian closed category of continuous domains which is closed under the construction of probabilistic powerdomains. This issue is discussed in =-=[25]-=-. Cartesian closure is essential in the denotational semantics of functional languages. There is a new challenge: What happens if non-deterministic choice coexists with probabilistic choice? And how c... |

32 | An extension result for continuous valuations
- Alvarez-Manilla, Edalat, et al.
- 1997
(Show Context)
Citation Context ...der to obtain an extension to a Borel measure. And this is indeed the crucial step that could be performed for various spaces using quite different techniques. Using ideas from [47], the authors from =-=[2]-=- could verify countable subadditivity in the following situation: Proposition 2.15 Let X be a dcpo with the Scott topology and let (_i)i2I be a directed set of simple valuations on X with _ = W" i2I _... |

30 | Power domains and iterated function systems
- Edalat
- 1996
(Show Context)
Citation Context ...werdomain originated in denotational semantics, more recently Edalat found applications inside mathematics, e.g. the the generalised Riemann integral [9], iterated function systems with probabilities =-=[11]-=-, dynamical systems and fractals [10]. We now recall some special valuations. For any element x 2 X, the point valuation jx : O(X) ! R+ is defined by jx(U ) := ( 1; if x 2 U 0; if x 62 U: A point valu... |

26 |
The versatile continuous order
- Lawson
- 1987
(Show Context)
Citation Context ... theory; however, a complete introduction to this topic by far exceeds the scope of this work. Thus, we present selected items only and omit all proofs as we go along to fix our notation. We refer to =-=[1, 7, 14, 15, 33]-=- for more details. 1.1 Dcpos and Scott-Continuous Functions We shall use the term ordered set in the sense of partially ordered set, that is, it denotes a set X with a reflexive, antisymmetric and tra... |

25 | Nondeterminism and probabilistic choice: obeying the laws - Mislove |

23 | Power domain constructions
- Heckmann
- 1991
(Show Context)
Citation Context ...aviour of a process, while the convex powerdomain is built up from both kinds of assertions taken together. Heckmann studies these and other powerdomain constructions in a general algebraic framework =-=[17]-=-. The topological concept of powerdomains is stressed by Smyth in [53]. There, he highlights that open sets should be viewed as computable properties. A different kind of non-determinism is given by p... |

22 |
Valuations on continuous lattices
- Lawson
- 1982
(Show Context)
Citation Context ...al variant of a measure or a probability distribution. In connection with order theory it is quite natural to impose Scott continuity as an additional condition on valuations as was done by Lawson in =-=[32]-=-. From a measure theoretical background, one might only impose the weaker condition of countable continuity, that is, for any increasing sequence of open sets, _ \GammasS" n2N Un \Deltas= W" n2N _(Un)... |

22 |
CPO’s of measures for nondeterminism
- Saheb-Djahromi
- 1980
(Show Context)
Citation Context ...domains, often referred to as Plotkin, Smyth, and Hoare powerdomains. viisviii INTRODUCTION Probabilistic non-determinism has also been studied and has led to the probabilistic powerdomain as a model =-=[47, 42, 24, 23]-=-. Different runs of a probabilistic program with the same input may again result in different outputs. In this situation, it is also known how likely these outputs are. Thus, a probability distributio... |

18 |
A direct proof of the Hofmann-Mislove theorem
- Keimel, Paseka
- 1994
(Show Context)
Citation Context ...ty compact saturated subsets of a topological space is denoted by Sc(X) and will be ordered by reverse inclusion. An important property of sober spaces X is the so called Hofmann-Mislove Theorem (see =-=[20, 26]-=-, [15, Theorem II-1.20]). The following proposition (see [15, TheoremII-1.21, Corollary II-1.22]) is a consequence of this theorem. It will be used extensively in Section 4.2. Proposition 1.3 Let X be... |

17 |
is not always sober
- Johnstone
- 1981
(Show Context)
Citation Context ...her property of sober spaces that the specialisation order yields a dcpo, with the original topology being coarser than the Scott topology. However, a dcpo with the Scott topology is not always sober =-=[22]-=-. In the next section we introduce special dcpos, called continuous domains, which are always sober spaces with respect to the Scott topology [31]. 1.4 Continuous Domains On a (conditionally) directed... |

17 |
Probabilistic Systems Group: Collected reports. //web.comlab.ox.ac.uk/oucl/research/areas/probs/bibliography.html
- PSG
(Show Context)
Citation Context ...h probabilistic choice? And how can the classical powerdomain constructions together with the probabilistic powerdomain be used for modelling such situations? The Programming Research Group in Oxford =-=[43]-=- hassINTRODUCTION ix tackled various aspects of this problem. Out of this group, McIver and Morgan have chosen a subdomain of the Plotkin powerdomain over the space of subprobability distributions on ... |

17 |
Stetige Bewertungen auf topologischen Räumen
- Tix
- 1995
(Show Context)
Citation Context ... POWERDOMAIN 25 from the category of continuous domains to the category of continous dcones. The algebras of this monad are the continuous d-cones by the following universal property (see [23], [29], =-=[54]-=-, [15, Theorem IV-9.24]): Theorem 2.11 Let X be a continuous domain. The map jX : X ! V(X) that to every x 2 X assigns the point valuation jx is a topological embedding and, for every Scott-continuous... |

16 | Continuous D-Cones: Convexity and Powerdomain Constructions - Tix - 1999 |

16 |
A complete proof system for SCCS with modal assertions
- Winskel
- 1986
(Show Context)
Citation Context ...s of the whole power set. We will recall these representations later when we modify them within the context of continuous d-cones. Winskel describes the classical powerdomains via modal assertions in =-=[61]-=-. The lower powerdomain is built up from assertions about possible behaviour of a process, the upper powerdomain is built up from assertions 55s56 CHAPTER 4. POWER CONSTRUCTIONS about the inevitable b... |

15 | Probabilistic models for the guarded command language
- Jifeng, McIver, et al.
- 1997
(Show Context)
Citation Context ...ining non-deterministic and probabilistic choice within a single framework. A Smyth style semantics of non-determinism together with probabilistic non-determinism can also be found in their work: see =-=[21, 38]-=-. Probabilistic choice between executing given programs P and P 0 with probability p in [0; 1], written P p+ P 0 , means that program P is executed with probability p and program P 0 is executed with ... |

14 | Spaces of valuations
- Heckmann
- 1996
(Show Context)
Citation Context ...ture of the probabilistic powerdomain quite well. Most of their results carry over to the extended probabilistic powerdomain (see [29]). Various classes of valuations are also surveyed by Heckmann in =-=[18]-=-. Although the interest in the probabilistic powerdomain originated in denotational semantics, more recently Edalat found applications inside mathematics, e.g. the the generalised Riemann integral [9]... |

13 |
Ordered Cones and Approximation
- Keimel, Roth
- 1992
(Show Context)
Citation Context ...d-cones. Other results about ordered cones, of which we will take advantage, are due to W. Roth [45]. He deals with ordered cones equipped with a quasiuniform structure proposed by Keimel and Roth in =-=[27]-=-. In the context of harmonic analysis and abstract potential theory, cones have been studied by Boboc, Bucur and Cornea [5]. Rauch has shown in [44] that a special class of their ordered cones, their ... |

12 |
The duality of continuous posets
- Lawson
- 1979
(Show Context)
Citation Context ...r, a dcpo with the Scott topology is not always sober [22]. In the next section we introduce special dcpos, called continuous domains, which are always sober spaces with respect to the Scott topology =-=[31]-=-. 1.4 Continuous Domains On a (conditionally) directed complete partial order X we introduce a binary relation o/ as follows: Let x and y be elements of X. We say that x approximates y or x is way-bel... |

11 | The way-below relation of function spaces over semantic domains
- Erker, Escardó, et al.
- 1998
(Show Context)
Citation Context ...y compact and if the intersection of any two compact saturated subsets is compact. In stably locally compact spaces we can apply the characterisation of the way-below relation on function spaces from =-=[12]-=-. We denote by suppf := fx 2 X j f (x) 6= ?g the support of a continuous function f : X ! L, where L is a bounded complete continuous domain and ? the least element of L. Obviously, suppf is open. Usi... |

10 |
Algebras for Generalized Power Constructions
- Schalk
- 1993
(Show Context)
Citation Context .... The sets conv F for nonempty finite F ` C form a basis of the continuous lattice H(C). Proof Let C be a continuous d-cone. To show the continuity of H(C), we imitate the proof of Proposition 6.5 in =-=[48]-=-. First, we show that x o/ y in C implies #x o/H #y in H(C). Let #y ` S" Ai for some directed family (Ai)i2I in H(C). Since S" Ai = f W" S j S directed; S ` S" Aig by Lemma 1.4, there exist a directed... |

10 | The powerdomain of indexed valuations
- Varacca
- 2002
(Show Context)
Citation Context ... ones. Accordingly, for the new powerdomains we speak of the convex lower, convex upper, and biconvex powercones, rather than the convex Hoare, convex Smyth, and convex Plotkin powercones. D. Varacca =-=[57, 58, 59]-=- took a related approach to combining probability and nondeterminism via indexed valuations. His equational theory is weaker; he weakens one natural equation, but the theory becomes more flexible. M. ... |

9 |
Domain theory and integration. Theoretical Computer Science, 151:163–193
- Edalat
- 1995
(Show Context)
Citation Context ...18]. Although the interest in the probabilistic powerdomain originated in denotational semantics, more recently Edalat found applications inside mathematics, e.g. the the generalised Riemann integral =-=[9]-=-, iterated function systems with probabilities [11], dynamical systems and fractals [10]. We now recall some special valuations. For any element x 2 X, the point valuation jx : O(X) ! R+ is defined by... |

6 |
Hahn-Banach type theorems for locally convex cones
- Roth
- 2000
(Show Context)
Citation Context ...tional analytic setting. In Chapter 3 we will show that some of their results still hold for continuous d-cones. Other results about ordered cones, of which we will take advantage, are due to W. Roth =-=[45]-=-. He deals with ordered cones equipped with a quasiuniform structure proposed by Keimel and Roth in [27]. In the context of harmonic analysis and abstract potential theory, cones have been studied by ... |

6 |
A type-theoretical alternative to ISWIM, CUCH, OWHY. Theoretical Computer Science, 121:422–440
- Scott
- 1993
(Show Context)
Citation Context ...The semantics of programming languages has been intensively studied by mathematicians and computer scientists. In the late sixties Dana S. Scott invented appropriate semantic domains for that purpose =-=[51, 49, 50]-=-. Continuous domains are directed complete partially ordered sets together with an order of approximation, the so called way-below relation. As they allow one to represent `ideal objects' and their `f... |

5 |
Some results on Hahn-Banach type theorems for continuous d-cones
- Tix
(Show Context)
Citation Context ...tructions and submitted to the Department of Mathematics of Technische Universit"at Darmstadt. Only a small part of this thesis, namely three sections of Chapter 3, has previously been published (see =-=[56]-=-). Since then, the main body of the thesis, Chapter 4 on powerdomains for modelling non-determinism, has become of increasing interest: indeed the main goal of the thesis was to provide semantic domai... |

5 | nondeterminism and concurrency: Two denotational models for probabilistic computation
- Varacca, “Probability
- 2003
(Show Context)
Citation Context ... ones. Accordingly, for the new powerdomains we speak of the convex lower, convex upper, and biconvex powercones, rather than the convex Hoare, convex Smyth, and convex Plotkin powercones. D. Varacca =-=[57, 58, 59]-=- took a related approach to combining probability and nondeterminism via indexed valuations. His equational theory is weaker; he weakens one natural equation, but the theory becomes more flexible. M. ... |

4 |
A.,Keimel, K.: The probabilistic powerspace for stably compact spaces. Theoretical Computer Science 328
- Manilla, Jung
- 2004
(Show Context)
Citation Context ...hen Tix's thesis was submitted: it was not known then whether the extended probabilistic powerdomain over a Lawson-compact continuous domain is Lawson-compact. Extending slightly a recent result from =-=[3]-=-, we now know that the extended probabilistic powerdomain is Lawson-compact over any stably locally compact space. For continuous domains the converse also holds. This allows us in particular to inclu... |

4 |
Order and Convexity in Potential Theory: HCones
- Boboc, Bucur, et al.
- 1981
(Show Context)
Citation Context ...cones equipped with a quasiuniform structure proposed by Keimel and Roth in [27]. In the context of harmonic analysis and abstract potential theory, cones have been studied by Boboc, Bucur and Cornea =-=[5]-=-. Rauch has shown in [44] that a special class of their ordered cones, their standard H-cones, can also be viewed as continuous lattice-ordered d-cones with addition and scalar multiplication being La... |

4 |
Free constructions of powerdomains
- Main
- 1985
(Show Context)
Citation Context ...classical powerdomains that they are universal with respect to an additional semilattice operation. Developing this point of view, Main related free constructions of powerdomains with semiring modules=-=[35]-=-. Abramsky and Jung studied free continuous domainalgebras in [1]. In this chapter we will also show our convex d-cones to be universal in a suitable setting. Note that even if applied to subsets of a... |

3 |
Partial correctness for probablistic demonic programs
- McIver, Morgan
(Show Context)
Citation Context ...ackled various aspects of this problem. Out of this group, McIver and Morgan have chosen a subdomain of the Plotkin powerdomain over the space of subprobability distributions on discrete state spaces =-=[36]-=-. The subsets they allow are the convex ones. Our approach to convex powercones was motivated by theirs. We modify and generalize their construction to continuous Lawson-compact d-cones. Therefore, we... |

2 |
Probabilistic powerdomains
- Plotkin
- 1982
(Show Context)
Citation Context ...domains, often referred to as Plotkin, Smyth, and Hoare powerdomains. viisviii INTRODUCTION Probabilistic non-determinism has also been studied and has led to the probabilistic powerdomain as a model =-=[47, 42, 24, 23]-=-. Different runs of a probabilistic program with the same input may again result in different outputs. In this situation, it is also known how likely these outputs are. Thus, a probability distributio... |

2 |
Stetige Verb"ande in der axiomatischen Potentialtheorie, in Continuous Lattices and Related Topics
- Rauch
- 1982
(Show Context)
Citation Context ...asiuniform structure proposed by Keimel and Roth in [27]. In the context of harmonic analysis and abstract potential theory, cones have been studied by Boboc, Bucur and Cornea [5]. Rauch has shown in =-=[44]-=- that a special class of their ordered cones, their standard H-cones, can also be viewed as continuous lattice-ordered d-cones with addition and scalar multiplication being Lawson continuous. Most of ... |

2 |
Bewertungen auf stetigen Bereichen. Master's thesis, Technische Hochschule Darmstadt
- Weidner
- 1995
(Show Context)
Citation Context ...a regular Borel measure on the locally compact pospace X in a unique way. For the compact case, this result is due to Lawson [32]. It has been extended to the properly locally compact case by Weidner =-=[60]-=- and Keimel [28]. In particular, Proposition 2.17 can be applied to locally compact Hausdorff spaces: Equipped with the trivial order, Hausdorff spaces can be viewed as ordered topological spaces. The... |

1 |
29] O. Kirch. Bereiche und Bewertungen
- Valuations, unpublished
- 1996
(Show Context)
Citation Context ...ntinuous. In the remainder of this section we present some answers to the question: when can a continuous valuation be extended to a Borel measure? We present a brief summary of some unpublished work =-=[28]-=-. Notation and results on the measure theoretical background are taken from [4, 16]. For metric spaces an affirmative answer can be given using outer measures and Carath'eodory's condition: Propositio... |

1 | Functional Analysis. McGraw Hill Series in Higher Mathematics - Rudin - 1973 |