### BibTeX

@MISC{Abramsky_domaintheory,

author = {Samson Abramsky},

title = {Domain Theory in Logical Form},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

... Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. The theory of concurrency and systems behaviour developed by Milner, Hennessy based on operational semantics. Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of of processes). Moreover, the underlying logic is, which can be computationally interpreted as the logic of properties--i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme: 1. A metalanguage is introduced, comprising types = universes of discourse for various computational situations. terms = programs = syntactic intensions for models or points. 2. A standard denotational interpretation of the metalanguage is given, assigning do-mains to types and domain elements to terms. 3. The metalanguage is also given a interpretation, in which types are interpreted as propositional theories and terms are interpreted a program logic, which ax- iomatizes the properties they satisfy. 4. The two interpretations are related by showing that they are Stone duals of each other. Hence, semantics and logic are guaranteed to be in harmony with each other, and in fact each determines the other up to isomorphism. 5. This opens the way to a whole range of applications. Given a denotational description of a computational situation in our meta-language, we can turn the handle to obtain a logic for that situation.

### Citations

1407 |
A Discipline of Programming
- Dijkstra
- 1976
(Show Context)
Citation Context ...f the open sets as properties or propositions, we can think of spaces as logical theories; continuous maps act on these theories under inverse image as predicate transformers in the sense of Dijkstra =-=[Dij76]-=-, or modal operators as studied in dynamic logic [Pra81, Har79]. There is also an important theme in Computer Science which emerges as confluent with these mathematical developments; namely, the use o... |

1372 | An axiomatic basis for computer programming
- Hoare
- 1969
(Show Context)
Citation Context ...s are embedded in formulas as modal operators. Notation: [P ]φ where P is now a program denoting a function or relation. Examples: dynamic logic [Har79, Pra81], including as special cases Hoare logic =-=[Hoa69]-=-, since “Hoare triples” {φ}P {ψ} can be represented by φ → [P ]ψ, 48sand Dijkstra’s wlp-calculus [Dij76], since wlp(P, ψ) can be represented as [P ]ψ. (Total correctness assertions can also be catered... |

1337 | A Calculus of Communicating Systems - Milner - 1980 |

1219 |
The temporal logic of programs
- Pnueli
- 1977
(Show Context)
Citation Context ...n programming language theory which has received intensive development over the past 15 years has been logics of programs, e.g. Hoare logic [Hoa69, dB80], dynamic logic [Pra81, Har79], temporal logic =-=[Pnu77]-=-, etc. However, to date there has not been a satisfactory integration of this work with domain theory. For example, dynamic logic deals with sets and relations, which from the perspective of domain th... |

923 |
Categories for the Working Mathematician
- Lane, S
- 1971
(Show Context)
Citation Context ...e theory of partial orders and lattices, we refer to [GHK + 80, Joh82]. Finally, we shall assume a modicum of familiarity with elementary category theory and general topology; suitable references are =-=[ML71]-=- and [Dug66] respectively. 2.2 Domains We shall assume some familiarity with [Plo81], and use it as our main reference for Domain theory. We shall also refer to [Gun85, Gun87]. For the reader’s conven... |

496 | Algebraic laws for nondeterminism and concurrency
- Hennessey, Milner
- 1985
(Show Context)
Citation Context ...ting this correspondence, we may hope to obtain a rapprochement between domain theory and denotational semantics, on the one hand, and operationally formulated notions such as observation equivalence =-=[HM85]-=- on the other. Denotational vs. Axiomatic Another area in programming language theory which has received intensive development over the past 15 years has been logics of programs, e.g. Hoare logic [Hoa... |

452 |
Model Theory
- Chang, Keisler
- 1990
(Show Context)
Citation Context ...organisational purposes. What we need is Definition 3.3.2 A � B iff (s1) A ⋐ B (s2) ≤A = ≤B ∩ |A| 2 (s3) CA = CB ∩ |A| (s4) TA = TB ∩ |A| Note that this is just the usual notion of submodel (cf. e.g. =-=[CK73]-=-). Proposition 3.3.3 The class of domain prelocales under � is an ω-chain complete partial order. Proof. The verification that � is a partial order is routine. Let {An} be a �-chain. Set A∞ ≡ ( � |An|... |

428 |
P.: Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ... our logic. 4 Programs As Morphisms: Exogenous Logic Now we introduce a second extension of our denotational metalanguage, which is based on the algebraic metalanguage for cartesian closed categories =-=[19, 13]-=-, just as the language of the previous section is an extended typed λ-calculus. Terms are sorted on morphism types (σ, τ), with formation rules exemplified by f : (σ × τ, υ) Λ(f) : (σ, τ → υ) Ap : ((σ... |

395 | LCF considered as a programming language - Plotkin - 1977 |

321 | A Compendium of Continuous Lattices - Gierz, Hofmann, et al. - 1980 |

315 |
Calculi for Synchrony and Asynchrony
- Milner
- 1983
(Show Context)
Citation Context ...nal reconstruction of it. Furthermore, we automatically get a compositional proof theory for this logic, along the lines indicated above. Since one can define a denotational semantics for, e.g., SCCS =-=[Mil83]-=- in our denotational metalanguage, we get a compositional proof system along the lines of those developed by Stirling and Winskel [Sti87, Win85]. Moreover, this proof system is guaranteed to be in har... |

233 |
Principal Type Schemes for Functional Programs
- Damas, Milner
- 1982
(Show Context)
Citation Context ...pertaining to the “world” of a single program. Notation: P |= φ where P is a program and φ is a formula. Examples: temporal logic as used e.g. in [Pnu77]; Hennessy-Milner logic [HM85]; type inference =-=[DM82]-=-. Exogenous logic Here, programs are embedded in formulas as modal operators. Notation: [P ]φ where P is now a program denoting a function or relation. Examples: dynamic logic [Har79, Pra81], includin... |

214 | A powerdomain construction - Plotkin - 1976 |

189 | Toposes, Triples, and Theories - Barr, Wells - 1985 |

176 |
An Introduction to Inductive Definitions
- Aczel
- 1977
(Show Context)
Citation Context ...i i∈I j∈Ji lij ⇒ l which is equivalent to � [ � lij ⇒ l] and hence to a finite set of rules of the standard form. Thus our axiomatization generates a monotone inductive definition in the standard way =-=[Acz77]-=-. Moreover, it is presented by a finite set of schemes, the set of instances of each of which is readily seen to be recursive; thus the set of theorems of the logic is recursively enumerable. In fact,... |

169 |
The category-theoretic solution of recursive domain equations
- Smyth, Plotkin
- 1982
(Show Context)
Citation Context ...in equation t = σ(t), i.e. as the initial fixpoint of an endofunctor over SDom. Other constructions (e.g. strict function space, smash product) can be added to the list. So far, all this is standard (=-=[17, 23]-=-). Now we begin our alternative approach. For each type expression σ, we shall define a propositional theory L(σ) = (L(σ), ≤σ, =σ), where: • L(σ) is a set of formulae • ≤σ, =σ are the relations of log... |

156 |
The Lambda Calculus, its Syntax and Semantics. NorthHolland, 2nd edition
- Barendregt
- 1984
(Show Context)
Citation Context ...ty) subset of Y . We write substitution of N for x in M, where M, N are expressions and x is a variable, as M[N/x]. We shall assume the usual notions of free and bound variables, as expounded e.g. in =-=[Bar84]-=-. We shall always take expressions modulo α-conversion, and treat substitution as a total operation in which variable capture is avoided by suitable renaming of bound variables. Our notations for sema... |

155 |
The theory of representations of Boolean algebras
- Stone
- 1936
(Show Context)
Citation Context ...spondence is obtained. Some applications to logics for specific computational situations (e.g. concurrency) are mentioned. 1 Introduction The classic Stone Representation Theorem for Boolean algebras =-=[25]-=- is the prototype for a wide class of “Stone-type duality theorems” [11]. The general form of these theorems is to assert an equivalence between a category of topological spaces and (the opposite of) ... |

117 | editors. Abstract Interpretation of Declarative Languages. Computers and their Applications. Ellis Horwood - Abramsky, Hankin - 1987 |

109 | Fully abstract models of the typed lambda calculus - Milner - 1977 |

96 |
A Probabilistic PDL
- Kozen
- 1985
(Show Context)
Citation Context ...al groups, Gabriel-Ulmer duality for locally finitely presented categories, etc. [Joh82]. Particularly promising for Computer Science applications are the measure-theoretic dualities studied by Kozen =-=[Koz83]-=- as a basis for the semantics and logic of probabilistic programs. A very interesting feature of these dualities is that whereas the purely topological dualities have the Sierpinski space O as their “... |

92 | Abstract Interpretation and Optimising Transformations for Applicative Programs - MYCROFT - 1981 |

90 | Full Abstraction for a Simple Parallel Programming Language ", LNCS 74
- Hennessy, Plotkin
- 1979
(Show Context)
Citation Context ...m the adjunction defining the powerdomain construction; ⊎ is the operation of the free algebras for this adjunction; while ⊗ is the universal map for the tensor product with respect to this operation =-=[HP79]-=-. We now introduce an endogenous program logic with assertions of the form M, Γ ⊢ φ 60swhere M : σ, φ ∈ L(σ), and Γ ∈ � σ {Var(σ) → L(σ)} gives assumptions on the free variables of M. Notation Γ ≤ ∆ ≡... |

84 | A Theory of Programming Language Semantics - Milne, Strachey - 1976 |

81 | M.J.C.,The Denotational Description of Programming Languages - Gordon - 1979 |

78 |
J.I.Zucker: Processes and the Denotational Semantics of Concurrency
- Bakker
- 1982
(Show Context)
Citation Context ...of [Abr87a] can be carried through. Moreover, there is a satisfying relationship between the Stone space of synchronisation trees (which is the metric topology on the ultrametric space constructed in =-=[dBZ82]-=-), and the corresponding domain studied in [Abr87a]; namely, the former is the 78ssubspace of maximal elements of the latter. This is in fact an instance of a general relationship, as set out in [Abr]... |

76 |
On observing nondeterminism and concurrency
- Milner
- 1979
(Show Context)
Citation Context ...mulas describe properties pertaining to the “world” of a single program. Notation: P |= φ where P is a program and φ is a formula. Examples: temporal logic as used e.g. in [18]; Hennessy-Milner logic =-=[8]-=-; type inference [5]. Exogenous logic Here, programs are embedded in formulas as modal operators. Notation: [P]φ where P is now a program denoting a function or relation. Examples: dynamic logic [7], ... |

69 |
Powerdomains and predicate transformers: a topological view
- Smyth
- 1983
(Show Context)
Citation Context ...ality for computer science is that it provides the right framework for understanding the relationship between denotational semantics and program logic. (As far as I know, this point was first made in =-=[22]-=-, and also in unpublished work of Gordon Plotkin.) Denotational semantics is always based, more or less explicitly, on a typed functional metalanguage. The types are interpreted as topological spaces ... |

61 |
Events in computation
- Winskel
- 1980
(Show Context)
Citation Context ...onstruction of “points”, i.e. denotations of computational processes, from the properties they satisfy is very compatible with work currently being done in a mainly operational setting in concurrency =-=[8, 26]-=- and elsewhere [4], and offers a promising approach to unification of this work with denotational semantics. 2 Domains As Propositional Theories The setting we take for our work in this paper is SDom,... |

60 |
Post-graduate lecture notes in advanced domain theory
- Plotkin
- 1981
(Show Context)
Citation Context ...anguages will be presented elsewhere [Abr88, Abr87a]. 1.1 Background Domain Theory has been extensively studied since it was introduced by Scott [Sco70], both as regards the basic mathematical theory =-=[Plo81]-=-, and the applications, particularly in denotational semantics [MS76, Sto77, Gor79, Sch86], and more recently in static program analysis [Myc81, Nie84, AH87]. In the course of this development, a numb... |

60 |
First order dynamic logic
- Harel
- 1984
(Show Context)
Citation Context ...c [8]; type inference [5]. Exogenous logic Here, programs are embedded in formulas as modal operators. Notation: [P]φ where P is now a program denoting a function or relation. Examples: dynamic logic =-=[7]-=-, including as special cases Hoare logic [10], since “Hoare triples” {φ}P {ψ} can be represented by φ → [P]ψ, and Dijkstra’s wlp-calculus [6], since wlp(P, ψ) can be represented as [P]ψ. (Total correc... |

54 |
Using Information Systems to Solve Recursive Domain Equations Effectively", LNCS 173
- Larsen
- 1984
(Show Context)
Citation Context ...and proved equivalent to SFP. This is the category in which, implicitly, all the work of Chapter 4 is set. In section 3, following the ideas of a number of authors, particularly Larsen and Winskel in =-=[LW84]-=-, a large cpo of domain prelocales is defined, and used to reduce the solution of domain equations to taking least fixpoints of continuous functions over this cpo. In section 4, a number of type const... |

53 |
Full Abstraction for sequential languages: the state of the art
- Berry, Curien, et al.
- 1985
(Show Context)
Citation Context ...”, i.e. denotations of computational processes, from the properties they satisfy is very compatible with work currently being done in a mainly operational setting in concurrency [8, 26] and elsewhere =-=[4]-=-, and offers a promising approach to unification of this work with denotational semantics. 2 Domains As Propositional Theories The setting we take for our work in this paper is SDom, the category of S... |

52 | Extended Type Structures and Filter Lambda Models", Logic Colloquium '82 - Coppo, Dezani-Ciancaglini, et al. - 1984 |

41 | Domain Theory and the Logic of Observable Properties - Abramsky - 1987 |

36 | Process logic - Pratt - 1979 |

35 |
Spaces, volume 3 of Cambridge
- Stone
- 1982
(Show Context)
Citation Context ... confirm this suggestion, but also show how it may be put to use. Points vs. Properties An important recent development in mathematics has been the rise of locale theory, or “topology without points” =-=[Joh82]-=-, in which the open-set lattices rather than the spaces of points become the primary objects of study. That these mathematical developments have direct bearing on Computer Science was emphasised by Sm... |

32 |
Theory and practice of sequential algorithms: the kernel of the applicative language CDS0
- Berry, Curien
- 1985
(Show Context)
Citation Context .... denotations of computational processes, from the properties they satisfy is very compatible with work currently being done in a mainly operational setting in concurrency [HM85, Win80] and elsewhere =-=[BC85]-=-, and offers a promising approach to unification of this work with denotational semantics. The remainder of the Chapter is organised as follows. In section 2, we interpret the types of our denotationa... |

30 |
A framework for intuitionistic modal logic
- Plotkin, Stirling
- 1986
(Show Context)
Citation Context ... [Dij76, Smy83b]; dualities for the Vietoris construction provide a natural setting for intuitionistic modal logic, with interesting differences to the approach recently taken by Plotkin and Stirling =-=[PS86]-=-; while there are some remarkable self-dualities arising from the Smyth powerdomain [Vic87]. These turn out, quite unexpectedly, to provide a model for Girard’s classical linear logic [Gir87]; more sp... |

28 |
Lectures on Predomains and Partial Functions. Notes for a course at CSLI
- Plotkin
- 1985
(Show Context)
Citation Context ...set of terms of type σ. Note the systematic presentation of these constructs as introduction and elimination rules for each of the type constructions, following ideas of Martin-Löf [ML83] and Plotkin =-=[Plo85]-=-. Note that λ, let, cases, lift, extend, µ are all variable binding operations in the obvious way. Also, note that {|.|}, extend arise from the adjunction defining the powerdomain construction; ⊎ is t... |

26 |
On Semantic Foundations for Applicative Multiprogramming
- Abramsky
- 1983
(Show Context)
Citation Context ...om these constructions. Almost all the domains needed in denotational semantics to date can be defined from these constructions by composition and recursion (some exceptions of three different kinds: =-=[Abr83]-=-, [Ole85], [Plo82]). Now algebraic domains are freely constructed from their bases, i.e. D ∼ = Idl(K(D)) where Idl(P ), for any poset P , is the ideal completion formed by taking all directed, left-cl... |

22 |
Interpretation of analysis by means of functionals of finite type
- Kreisel
- 1959
(Show Context)
Citation Context ...heory was taken by Martin-Löf, in his Domain Interpretation of Intuitionistic Type Theory [ML83]. His formulation also traces a line of descent from Kreisel’s definition of the continuous functionals =-=[Kre59]-=-, via [ML70, Ers72]. The general tendency of these developments is to suggest that domains may as well be viewed in terms of theories as of models. Our work should not only confirm this suggestion, bu... |

20 | A Powerdomain for countable non-determinism
- Plotkin
- 1982
(Show Context)
Citation Context ...ions. Almost all the domains needed in denotational semantics to date can be defined from these constructions by composition and recursion (some exceptions of three different kinds: [Abr83], [Ole85], =-=[Plo82]-=-). Now algebraic domains are freely constructed from their bases, i.e. D ∼ = Idl(K(D)) where Idl(P ), for any poset P , is the ideal completion formed by taking all directed, left-closed subsets of P ... |

16 |
Metric spaces, generalised logic, and closed categories. In Rendiconti del Seminario Matematico e Fisico di Milano, volume 43. Tipografia Fusi
- Lawvere
- 1973
(Show Context)
Citation Context ...3]. The richer mathematical structure of these dualities should deepen our understanding of the framework. Furthermore, there are intriguing connections with Lawvere’s concept of “generalised logics” =-=[Law73]-=-. 3. The logics of compact-open sets considered in this paper are very weak in expressive power, and are clearly inadequate as a specification formalism. For example, we cannot specify such properties... |

16 |
Infinite words, infinite trees, infinite computations
- Nivat
- 1979
(Show Context)
Citation Context ...based, more or less explicitly, on a typed functional metalanguage. The types are interpreted as topological spaces (usually domains in the sense of Scott [20, 21], but sometimes metric spaces, as in =-=[3, 16]-=-), while the terms denote elements of or functions between these spaces. A program logic comprises an assertion language of formulas for expressing properties of programs, and an interface between the... |

16 | A complete proof system for SCCS with modal assertions - Winskel - 1986 |

15 | Universal profinite domains - Gunter - 1987 |

15 |
Vietoris locales and localic semilattices
- Johnstone
- 1985
(Show Context)
Citation Context ...f (infinitary) algebra. In particular, constructions of universal objects of various kinds by “generators and relations” are possible. Two highly relevant examples in the locale theory literature are =-=[Joh85]-=- and [Hyl81]. This provides a link with the information systems approach to domain theory as in [Sco82, LW84]. Some of our work in Chapters 3 and 4 can be seen as a systematization of these ideas in a... |

14 | Profinite Solutions for Recursive Domain Equations
- Gunter
- 1985
(Show Context)
Citation Context ...g the denotational semantics of terms as approximable mappings. Most of it can be found, couched in the language of information systems, in [Sco82], and for neighbourhood systems in [Sco81]. See also =-=[Gun85]-=-. We shall just give a couple of the less familiar cases for illustration. 65s(xi). where (xii). • Con(Z) ⊑ [M ⊗ N ]ρ • Con(X) ⊑ [M ⊎ N ]ρ ⇐⇒ X ⊑EM ([M ]ρ ∪ [N ]ρ) ⇐⇒ Con(Y ) ⊑ [M ]ρ & Con(Z) ⊑ [N ]ρ ... |

13 | On the duality of dynamic algebras and Kripke models - Kozen - 1981 |