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Convex powerdomains II

Karel Hrbacek
<span title="">1989</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/joe2ngto45hbnl3pncnesnq344" style="color: black;">Information and Computation</a> </i> &nbsp;
X is called convex if whenever s < y < z and x, z E X then y E X. Clearly full sets are convex, but the converse may fail.  ...  Finally, some results on the relationship of this powerdomain to the classical Plotkin powerdomain are presented. ' 198') Academic Press.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0890-5401(89)90034-5">doi:10.1016/0890-5401(89)90034-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qwd5d23jwbcblmklvc7twjpg7u">fatcat:qwd5d23jwbcblmklvc7twjpg7u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171003043907/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/c47/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wODkwNTQwMTg5OTAwMzQ1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/20/69/2069df12a5061deb93c67cbbe2e7dbfa75fb7875.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0890-5401(89)90034-5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Relating total and partial correctness interpretations of non-deterministic programs

Carl A. Gunter
<span title="">1990</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2qaxtqe2rfgjnfefqcokcnzelq" style="color: black;">Proceedings of the 17th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL &#39;90</a> </i> &nbsp;
correctness) powerdomain on the other.  ...  The mixed powerdomain is closely related to new powerdomains which have been recently investigated as mathematical models of partial information in databases.  ...  It is shown that M is distinct from the convex (Plotkin) powerdomain but, like the convex powerdomain, does not preserve the bounded completeness property of Scott domains.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/96709.96741">doi:10.1145/96709.96741</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/popl/Gunter90.html">dblp:conf/popl/Gunter90</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6e5wxbtaqnbtzdvzupmwqablgq">fatcat:6e5wxbtaqnbtzdvzupmwqablgq</a> </span>
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The mixed powerdomain

Carl A. Gunter
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
., The mixed power domain, Theoretical Computer Science 103 (1992) 311-334 This paper introduces an operator Ju called the mixed powerdomain which generalizes the convex (Plotkin) powerdomain.  ...  This provides a richer family of meaningful partial descriptions than are available in the convex powerdomain and also makes it possible to include the empty set in a satisfactory way.  ...  That is(1) if4$$, then II~IGUICIII; Proof.  ... 
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Probabilistic Completion of Nondeterministic Models

Guy Beaulieu
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
The completion construction relies upon a new closure operation on convex sets, dependant on the given semilattice.  ...  Finally, we show that building a free mixed choice model is equivalent to applying the convex completion functor to its corresponding free nondeterministic model.  ...  (ii) Monads (a) P * : Set → Set -non-empty powerset monad. (b) D : Set → Set -distributions monad. (c) G cvx : Set → Set -geometrically convex powerset monad.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2007.02.028">doi:10.1016/j.entcs.2007.02.028</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eldiddhlybb2dm55mqiqh4lrfu">fatcat:eldiddhlybb2dm55mqiqh4lrfu</a> </span>
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A Monad for Randomized Algorithms

Tyler Barker
<span title="">2016</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
Then, in these categories, there is a distributive law between this monad and the convex powerdomain.  ...  In order to work with the convex powerdomain, an alteration to the monad of random choice is made, so that the Cartesian closed categories RB and FS are closed under this construction.  ...  Distributive Law With the Convex Powerdomain Let Γ C denote the convex powerdomain functor. Recall that the convex powerdomain of a coherent domain D consists of Lens(D), the nonempty lenses of D.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2016.09.031">doi:10.1016/j.entcs.2016.09.031</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ff642kxm5gu7g67ienctz6khq">fatcat:5ff642kxm5gu7g67ienctz6khq</a> </span>
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The Troublesome Probabilistic Powerdomain

Achim Jung, Regina Tix
<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
In 12] it is shown that the probabilistic powerdomain of a continuous domain is again continuous.  ...  In this paper, we g i v e a c o u n terexample to Graham's proof and describe our own attempts at proving a closure result for the probabilistic powerdomain construction.  ...  Acknowledgements A numberof people have been subjected to our (mostly awed) attempts to prove a general closure theorem for the probabilistic powerdomain construction, and we gratefully acknowledge the  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s1571-0661(05)80216-6">doi:10.1016/s1571-0661(05)80216-6</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fdmae4p3cneldf6g4h73radoqi">fatcat:fdmae4p3cneldf6g4h73radoqi</a> </span>
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Probabilistic Observations and Valuations

Matthias Schröder, Alex Simpson
<span title="">2006</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
We give a universal property for an "abstract probabilistic powerdomain" based on an analysis of observable properties of probabilistic computation.  ...  In the category of topological spaces, the abstract probabilsitic powerdomain is given explicitly as the space of continuous probability valuations with weak topology.  ...  Using (S, ∨) One should obtain a "lower powerdomain" using (S, ∨) for observations (recall that S is Sierpinski space), an "upper powerdomain" using (S, ∧), and a "convex powerdomain" using ({{⊥}, {⊥,  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2005.11.075">doi:10.1016/j.entcs.2005.11.075</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5f2a7oafhvhshpzbrwplqjzt64">fatcat:5f2a7oafhvhshpzbrwplqjzt64</a> </span>
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The Extended Probabilistic Powerdomain Monad over Stably Compact Spaces [chapter]

Ben Cohen, Martin Escardo, Klaus Keimel
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We mainly want to determine the algebras of this powerdomain monad and the algebra homomorphisms.  ...  It is the aim of this work to obtain similar results for the (extended) probabilistic powerdomain monad over stably compact spaces.  ...  For every stably compact space Y , the cone C = VY has property (U). 8, II-3.7/8/9] and [12, Theorem 5.1].Theorem 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11750321_54">doi:10.1007/11750321_54</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/s3gjcsnqifgddic3g2e4d2kwsa">fatcat:s3gjcsnqifgddic3g2e4d2kwsa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170705123033/http://www.mathematik.tu-darmstadt.de/~keimel/Papers/keimeltamc06.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/7c/90/7c906424595ed90c2494dc6f42e4ca6f0bae6402.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11750321_54"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

RETRACTED: Semantic Domains for Combining Probability and Non-Determinism

Regina Tix, Klaus Keimel, Gordon Plotkin
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
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.  ...  The results are variants of the upper, lower and convex powerdomains over the probabilistic powerdomain (see Chapter 4).  ...  This leads to powerdomains carrying a convex structure.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2004.06.063">doi:10.1016/j.entcs.2004.06.063</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kcsyh7rsgncadlrlnlonoaraau">fatcat:kcsyh7rsgncadlrlnlonoaraau</a> </span>
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Semantic Domains for Combining Probability and Non-Determinism

Regina Tix, Klaus Keimel, Gordon Plotkin
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
The results are variants of the upper, lower and convex powerdomains over the probabilistic powerdomain (see Chapter 4).  ...  Our construction combines the well-known domains modelling nondeterminism -the lower, upper and convex powerdomains, with the probabilistic powerdomain of Jones and Plotkin [24] modelling probabilistic  ...  This leads to powerdomains carrying a convex structure.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2009.01.002">doi:10.1016/j.entcs.2009.01.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d25ftdfq6rbv7ivv6f7olglarq">fatcat:d25ftdfq6rbv7ivv6f7olglarq</a> </span>
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Guards, Failure, and Partiality: Dijkstra's Guarded-Command Language Formulated Topologically [chapter]

David A. Schmidt
<span title="2015-12-25">2015</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We reexamine Dijkstra's language, redefining its denotational semantics with powerdomains formulated in topological terms.  ...  P X's convex powerspace, Ω C P X, is the topology generated from a subbase consisting of sets of form (i) and (ii) above.  ...  -possessing all possible properties -which is unacceptable for failure); and an isolated element in the convex powerdomain (meaning that logical negation in the powerdomain is neither set complement nor  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-27810-0_13">doi:10.1007/978-3-319-27810-0_13</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kwxmwiq56bfiheww34izrd2vkm">fatcat:kwxmwiq56bfiheww34izrd2vkm</a> </span>
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Fractals and domain theory

KEYE MARTIN
<span title="">2004</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jdpukxfslnfmradqb65llq43na" style="color: black;">Mathematical Structures in Computer Science</a> </i> &nbsp;
We show that a measurement µ on a continuous dcpo D extends to a measurementμ on the convex powerdomain CD iff it is a Lebesgue measurement.  ...  (ii) The canonical extension of µ to the convex powerdomain µ : CD → [0, ∞) * is a measurement. In either case, kerμ = {K * : K ∈ P com (ker µ)}. Proof. (i) ⇒ (ii) Let A I in CD withμ(I) = 0.  ...  powerdomain CD models the Vietoris hyperspace of X.8 Contractions on the convex powerdomainWe now show that contractions on domains extend to contractions on the convex powerdomain.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0960129504004384">doi:10.1017/s0960129504004384</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/urfgjrrs7bfhpfbt74gf5lg7ey">fatcat:urfgjrrs7bfhpfbt74gf5lg7ey</a> </span>
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Predicate transformers for extended probability and non-determinism

KLAUS KEIMEL, GORDON D. PLOTKIN
<span title="2009-03-17">2009</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jdpukxfslnfmradqb65llq43na" style="color: black;">Mathematical Structures in Computer Science</a> </i> &nbsp;
The relationship between suitable notions of predicate transformer for the lower and order-convex powerdomains was considered in Bonsangue (1998).  ...  As with standard powerdomains for non-deterministic choice, these come in three flavourslower, upper and (order-)convex -so there are also three kinds of predicate transformers.  ...  The convex powercone and the order-convex conical powerdomain Let C be a coherent continuous d-cone.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0960129509007555">doi:10.1017/s0960129509007555</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/szyaxvnvrbgjzkstoz32pi3g3q">fatcat:szyaxvnvrbgjzkstoz32pi3g3q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808200956/http://www.research.ed.ac.uk/portal/files/7944748/Predicate_transformers.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c2/fd/c2fd4809b399a3afbbfb134945238ef31b79e4f4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/s0960129509007555"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> cambridge.org </button> </a>

Algebras of the extended probabilistic powerdomain monad [article]

Jean Goubault-Larrecq, Xiaodong Jia
<span title="2019-03-22">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad $\mathcal V_w$ over the category $\mathbf{TOP}_0$ of $T_0$ topological spaces and continuous maps.  ...  In $\mathbf{TOP}_0$ their algebras are fully characterised as weakly locally convex topological cones and weakly locally convex sober topological cones, respectively.  ...  Equations (i) and (ii) are immediate.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1903.07472v3">arXiv:1903.07472v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/t2o52wcnozaalebypptqvxbqxu">fatcat:t2o52wcnozaalebypptqvxbqxu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200927055339/https://arxiv.org/pdf/1903.07472v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/06/40/06409fde2e90906aa92cfe2ea0b6d39b422b5b70.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1903.07472v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Observationally-induced Algebras in Domain Theory

Ingo Battenfeld
<span title="">2014</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uy5mv2ncw5eahkdx47hkrglxmm" style="color: black;">Electronical Notes in Theoretical Computer Science</a> </i> &nbsp;
However, the Plotkin powerdomain turns out to be more problematic. Here we show that with the obvious prototype algebra, Heckmanns algebra A, one does not get the classical Plotkin powerdomain.  ...  We furthermore investigate the classical powerdomain constructions in the observationally-induced approach.  ...  Several different definitions of the traditional convex powerdomain (or Plotkin powerdomain) can be found in the literature.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2014.01.003">doi:10.1016/j.entcs.2014.01.003</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tafw7qgpubab7n2767pco6lmqe">fatcat:tafw7qgpubab7n2767pco6lmqe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190228150549/https://core.ac.uk/download/pdf/82372989.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/20/41/20411de9766aef75faec2e589a54629d82ac25d5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.entcs.2014.01.003"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> elsevier.com </button> </a>
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