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Convex powerdomains I

1987
*
Information and Computation
*

Let (D, < ) be an

doi:10.1016/0890-5401(87)90021-6
fatcat:o77u7acbdfhlbbtirev72qcghi
*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*. ...##
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Generalized metric spaces: Completion, topology, and powerdomains via the Yoneda embedding

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)

doi:10.1016/s0304-3975(97)00042-x
fatcat:z3xewd6tzzf63nmkgklayqu63u
*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. ...##
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Linear types and approximation

2000
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Mathematical Structures in Computer Science
*

FS-

doi:10.1017/s0960129500003200
fatcat:cobuytswjfhxtehapjp37mohti
*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. ...##
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Book Introduction by the Author: Semantics of Probabilistic Processes An Operational Approach

2015
*
Bulletin of the European Association for Theoretical Computer Science
*

The semantic comparison of different systems is based on appropriate behavioural relations such

dblp:journals/eatcs/Deng15
fatcat:wardvn6yr5ah3ahdxmmlhvypu4
*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*. ...##
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Domain Theory and the Logic of Observable Properties
[article]

2011
*
arXiv
*
pre-print

Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (

arXiv:1112.0347v1
fatcat:dtnbwwmcrret5lgh2yusnmtq4m
*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*. ...##
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All cartesian closed categories of quasicontinuous domains consist of domains

2015
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Theoretical Computer Science
*

It is Scott-continuous because the chain C ∪ { } is an

doi:10.1016/j.tcs.2015.05.014
fatcat:4q4huwdaafbepbdhmq4b3gmuk4
*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] . ...##
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The Powerdomain of Indexed Valuations

2002
*
BRICS Report Series
*

In domain theory, nondeterminism is modeled using the notion of

doi:10.7146/brics.v9i38.21753
fatcat:5mlx23srnbe3rpxb2ch5jp7nai
*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*. ...##
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Book report

2004
*
Computers and Mathematics with Applications
*

Topological

doi:10.1016/s0898-1221(04)84016-1
fatcat:szz7sksvhnentlcyspkplxuaya
*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. ...##
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A Categorical View on Algebraic Lattices in Formal Concept Analysis
[article]

2004
*
arXiv
*
pre-print

At the same time, the paper provides a relatively comprehensive account of the representation theory of

arXiv:cs/0410065v1
fatcat:jzjasw55kvhcnkelclvzkxlphe
*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] . ...##
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Full Abstractness for a Functional/Concurrent Language with Higher-Order Value-Passing

1998
*
Information and Computation
*

Based on standard domain constructors we define a model, a prime

doi:10.1006/inco.1998.2723
fatcat:dajyp5kkrzbndkz2qfrv3wa7wu
*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*. ...##
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Complementing logic program semantics
[chapter]

1996
*
Lecture Notes in Computer Science
*

We apply this idea to the case of logic programming, characterizing in a uniform

doi:10.1007/3-540-61735-3_16
fatcat:iqencjkujjd3veenxgibufdttm
*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 . ...##
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Semantics of weakening and contraction

1994
*
Annals of Pure and Applied Logic
*

It will be shown that on the category of complete

doi:10.1016/0168-0072(94)90020-5
fatcat:ufkx5vm2vzfcfpzkj7dxkeht5e
*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 ! ...##
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Probabilistic Event Structures and Domains

2004
*
BRICS Report Series
*

Traditionally, in domain theory a probabilistic process is represented

doi:10.7146/brics.v11i10.21835
fatcat:4zwxe3q3yrcm3mck2lcf6tqauu
*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. ...##
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Probabilistic Event Structures and Domains
[chapter]

2004
*
Lecture Notes in Computer Science
*

Traditionally, in domain theory a probabilistic process is represented

doi:10.1007/978-3-540-28644-8_31
fatcat:73hx5cze75crbpeanelys7xade
*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. ...##
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Domain theory, testing and simulation for labelled Markov processes

2005
*
Theoretical Computer Science
*

The main results characterize similarity

doi:10.1016/j.tcs.2004.10.021
fatcat:ej6wdbof75czpcvxi3rjxkvyga
*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*. ...
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