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Convex powerdomains I

Karel Hrbacek
1987 Information and Computation  
Let (D, < ) be an algebraic lattice; we define its powerdomain PAL( D, < ) as the Frink completion of (P(K(D)), < EM). By Theorem 1, PAL(D) is again an algebraic lattice.  ...  to P"(D) as PAL(D) bears to PAP(D) for algebraic lattices.  ... 
doi:10.1016/0890-5401(87)90021-6 fatcat:o77u7acbdfhlbbtirev72qcghi

Generalized metric spaces: Completion, topology, and powerdomains via the Yoneda embedding

M.M. Bonsangue, F. van Breugel, J.J.M.M. Rutten
1998 Theoretical Computer Science  
Combining Lawvere's (1973 ) enriched-categorical and Smyth's (1988 , 1991 topological view on generalized metric spaces, it is shown how to construct (1) completion, (2) two topologies, and (3) powerdomains  ...  ordinary metric spaces, these constructions yield, respectively: (1) chain completion and Cauchy completion; (2) the Alexandroff and the Scott topology, and the s-ball topology; (3) lower, upper, and convex powerdomains  ...  We have been able to correct all errors but one, for which indeed a counterexample was provided in the referee report: the converse of Lemma 4.4, which we believed to hold as well.  ... 
doi:10.1016/s0304-3975(97)00042-x fatcat:z3xewd6tzzf63nmkgklayqu63u

Linear types and approximation

MICHAEL HUTH, ACHIM JUNG, KLAUS KEIMEL
2000 Mathematical Structures in Computer Science  
FS-lattices are studied in the presence of distributivity and algebraicity.  ...  We study continuous lattices with maps that preserve all suprema rather than only directed ones.  ...  Hence in our domain-theoretic model of Linear Logic the two modalities are just the two fundamental powerdomains. Algebraicity The category FS has plenty of algebraic lattices as objects.  ... 
doi:10.1017/s0960129500003200 fatcat:cobuytswjfhxtehapjp37mohti

Book Introduction by the Author: Semantics of Probabilistic Processes An Operational Approach

Yuxin Deng
2015 Bulletin of the European Association for Theoretical Computer Science  
The semantic comparison of different systems is based on appropriate behavioural relations such as bisimulation equivalences and testing preorders. This book mainly consists of two parts.  ...  Acknowledgements Most of the work reported in this book was carried out during the last few years with a number of colleagues including Rob van Glabbeek, Matthew Hennessy, Carroll Morgan, Chenyi Zhang,  ...  [1] , Sangiorgi [15] , as well as Sangiorgi and Rutten [16] . They are mainly about classical (nonprobabilistic) process algebras.  ... 
dblp:journals/eatcs/Deng15 fatcat:wardvn6yr5ah3ahdxmmlhvypu4

Domain Theory and the Logic of Observable Properties [article]

Samson Abramsky
2011 arXiv   pre-print
Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes).  ...  framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: - Domain Theory, the mathematical theory of computation introduced by Scott as  ...  ) and Boolean algebras, i.e. complemented distributive lattices.  ... 
arXiv:1112.0347v1 fatcat:dtnbwwmcrret5lgh2yusnmtq4m

All cartesian closed categories of quasicontinuous domains consist of domains

Xiaodong Jia, Achim Jung, Hui Kou, Qingguo Li, Haoran Zhao
2015 Theoretical Computer Science  
It is Scott-continuous because the chain C ∪ { } is an algebraic lattice as we argued in Proposition 2.6, and hence meet-continuous.  ...  Preliminaries We use the standard definitions of domain theory as can be found in [1] or [2] .  ... 
doi:10.1016/j.tcs.2015.05.014 fatcat:4q4huwdaafbepbdhmq4b3gmuk4

The Powerdomain of Indexed Valuations

Daniele Varacca
2002 BRICS Report Series  
In domain theory, nondeterminism is modeled using the notion of powerdomain, while probability is modeled using the powerdomain of valuations. Those two functors do not combine well, as they are.  ...  We define the notion of powerdomain of indexed valuations, which can be combined nicely with the usual nondeterministic powerdomain. We show an equational characterization of our construction.  ...  If we omit it we get a theory whose free domain-algebra distributes over the Plotkin powerdomain.  ... 
doi:10.7146/brics.v9i38.21753 fatcat:5mlx23srnbe3rpxb2ch5jp7nai

Book report

2004 Computers and Mathematics with Applications  
Topological Algebra and Lattice Theory: Applications. VII-1 One-Sided Topological Semilattices. Exercises. Old Notes. VII-2 Topological Lattices. Exercises. Old notes. New notes.  ...  IV-8 powerdomains. The Hoare powerdomaln. The Smyth powerdomain. The Plotkin powerdomain. Exercises. New notes. IV-9 The Extended Probabilistic powerdomaln. Exercises. New notes. V.  ... 
doi:10.1016/s0898-1221(04)84016-1 fatcat:szz7sksvhnentlcyspkplxuaya

A Categorical View on Algebraic Lattices in Formal Concept Analysis [article]

Pascal Hitzler, Markus Krötzsch, Guo-Qiang Zhang
2004 arXiv   pre-print
At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott  ...  The latter are commonly studied in denotational semantics and domain theory and captured most prominently by the notion of algebraicity, e.g. of lattices.  ...  A very preliminary report on some of the results in this paper has appeared as [HZ04] .  ... 
arXiv:cs/0410065v1 fatcat:jzjasw55kvhcnkelclvzkxlphe

Full Abstractness for a Functional/Concurrent Language with Higher-Order Value-Passing

Chrysafis Hartonas, Matthew Hennessy
1998 Information and Computation  
Based on standard domain constructors we define a model, a prime algebraic lattice, which is fully abstract with respect to this behavior preorder; expressions are related in the model if and only if they  ...  addition to the usual types for higher-order functions contains an extra type called proc, for processes; the constructors for terms of this type are similar to those found in standard process calculi such as  ...  are prime algebraic lattices.  ... 
doi:10.1006/inco.1998.2723 fatcat:dajyp5kkrzbndkz2qfrv3wa7wu

Complementing logic program semantics [chapter]

Roberto Giacobazzi, Francesco Ranzato
1996 Lecture Notes in Computer Science  
We apply this idea to the case of logic programming, characterizing in a uniform algebraic setting, the interaction between a number of well known declarative semantics for logic programs.  ...  We consider abstract interpretation, and in particular the basic operators of reduced product and complementation of abstract domains, as a tool to systematically derive denotational semantics by composition  ...  Theorem 5.3 If L is an algebraic complete lattice and 2 uco(L) then, id = = t x2 (L)\MI (L) ' x .  ... 
doi:10.1007/3-540-61735-3_16 fatcat:iqencjkujjd3veenxgibufdttm

Semantics of weakening and contraction

Bart Jacobs
1994 Annals of Pure and Applied Logic  
It will be shown that on the category of complete lattices one hasyomonads 4 and j with A,! ='r= ! !. cw But there is a crucial operator !  ...  which restores weakening and contraction in annotated form: 1 The work reported here was done during '91-'92 at Department of Pure Mathematics, Cambridge UK.  ...  first emerged in the semantics of "substructure" versions of the untyped A-cal&lus ("as reported in [ 161). Soon it became clear that ! and !  ... 
doi:10.1016/0168-0072(94)90020-5 fatcat:ufkx5vm2vzfcfpzkj7dxkeht5e

Probabilistic Event Structures and Domains

Daniele Varacca, Hagen Völzer, Glynn Winskel
2004 BRICS Report Series  
Traditionally, in domain theory a probabilistic process is represented as a continuous valuation on the open sets of a domain, i.e., as an element of the probabilistic powerdomain of Jones and Plotkin  ...  The set of tests, endowed with the Egli-Milner order has an interesting structure: the set of all tests is a complete lattice, while finitary tests form a lattice.  ...  A DCPO is an algebraic domain if or every x ∈ D, x is the directed least upper bound of ↓ x ∩ Cp(D). It is ω-algebraic if Cp(D) is countable.  ... 
doi:10.7146/brics.v11i10.21835 fatcat:4zwxe3q3yrcm3mck2lcf6tqauu

Probabilistic Event Structures and Domains [chapter]

Daniele Varacca, Hagen Völzer, Glynn Winskel
2004 Lecture Notes in Computer Science  
Traditionally, in domain theory a probabilistic process is represented as a continuous valuation on the open sets of a domain, i.e., as an element of the probabilistic powerdomain of Jones and Plotkin  ...  The set of tests, endowed with the Egli-Milner order has an interesting structure: the set of all tests is a complete lattice, while finitary tests form a lattice.  ...  A DCPO is an algebraic domain if or every x ∈ D, x is the directed least upper bound of ↓ x ∩ Cp(D). It is ω-algebraic if Cp(D) is countable.  ... 
doi:10.1007/978-3-540-28644-8_31 fatcat:73hx5cze75crbpeanelys7xade

Domain theory, testing and simulation for labelled Markov processes

Franck van Breugel, Michael Mislove, Joël Ouaknine, James Worrell
2005 Theoretical Computer Science  
The main results characterize similarity as a testing preorder and bisimilarity as a testing equivalence.  ...  One of the main technical contributions is to show that a final object in a suitable category of LMPs can be constructed by solving a domain equation D ∼ = V(D) Act , where V is the probabilistic powerdomain  ...  An issue that is as yet unresolved is how to define a suitable norm in order to get a C * -algebra.  ... 
doi:10.1016/j.tcs.2004.10.021 fatcat:ej6wdbof75czpcvxi3rjxkvyga
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