## Domain Theory in Logical Form (1991)

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Venue: | Annals of Pure and Applied Logic |

Citations: | 233 - 10 self |

### BibTeX

@ARTICLE{Abramsky91domaintheory,

author = {Samson Abramsky},

title = {Domain Theory in Logical Form},

journal = {Annals of Pure and Applied Logic},

year = {1991},

volume = {51},

pages = {1--77}

}

### Years of Citing Articles

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### Abstract

The mathematical 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 a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. 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 properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable 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:

### Citations

1533 |
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... |

1484 | 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... |

1417 | A calculus of communicating systems - Milner - 1980 |

982 |
Categories for the Working Mathematician
- Lane
- 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... |

516 | Algebraic laws for nondeterminism and concurrency
- Hennessy, 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... |

497 |
Model Theory
- Chang, Keisler
- 1973
(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|... |

448 |
Introduction to higher order categorical logic, volume 7 of Camb
- 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... |

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

332 |
Calculi for synchrony and asynchrony
- Milner
- 1993
(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... |

244 |
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... |

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

188 |
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,... |

166 |
Lambda Calculus: its Syntax and Semantics (North-Holland
- 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... |

119 | editors. Abstract Interpretation of Declarative Languages - Abramsky, Hankin - 1987 |

112 | Fully abstract models of typed lambda-calculus - Milner - 1977 |

108 |
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 “... |

94 | Full abstraction for a simple parallel programming language
- 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 ≤ ∆ ≡... |

93 | Abstract Interpretation a d Optimising Transformation f rAp- plicative programs - Mycroft - 1981 |

87 |
Processes and the denotational semantics of concurrency
- Bakker, Zucker
- 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]... |

87 | A Theory of Programming Language Semantics - MILNE, STRACHEY - 1976 |

84 | M.,The Denotational description of programming languages - Gordon - 1979 |

57 | Extended type structures and filter lambda models - Coppo, Dezani-Ciancaglini, et al. - 1984 |

57 |
Using information systems to solve recursive domain equations effectively
- Winskel, 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... |

54 |
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... |

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

38 |
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... |

31 |
Theory and practice of sequential algorithms: the kernel of the applicative language CDS
- 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... |

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... |

23 |
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... |

17 |
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... |

16 | Universal profinite domains - Gunter - 1985 |

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... |

15 | 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 |

12 |
Function space in the category of locales
- Hyland
- 1981
(Show Context)
Citation Context ...y) 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 an explicitly... |

12 | A representation theorem for models of *-free PDL - Kozen - 1980 |

11 |
The lazy λ-calculus
- Abramsky
- 1990
(Show Context)
Citation Context ...he handle”, and generate a logic for this situation in a quite mechanical way. Two substantive case studies of this kind have been carried out, in the areas of concurrency [Abr87a] and the λ-calculus =-=[Abr88]-=-. For example, in [Abr87a] we define a domain equation for synchronisation trees, and generate a logic which can be applied to the whole class of labelled transition systems. This logic subsumes Henne... |

11 | Computable functionals of finite types - Ershov - 1972 |

10 |
Lecture note on the domain interpretation of type theory
- Martin-Löf
- 1983
(Show Context)
Citation Context ...deliberately suggestive of proof theory. A further step towards explicitly syntactic presentations of domain theory 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 domain... |

9 | de Bakker. Mathematical Theory of Program Correctness - W |

9 | Computability con cepts for programming language semantics - Constable |

9 | Metric Domains for Completeness - Matthews - 1986 |

7 |
Fully Effective Solutions of Recursive Domain Equations
- Kanda
- 1979
(Show Context)
Citation Context ...aning of the process. Extensionality in this sense is obviously relative to our choice of interface; it is orthogonal to the notion being discussed in the main text. 17sgiven domains, as advocated in =-=[Kan79]-=-. It should also be said that the link between observables and open sets in domain theory was clearly (though briefly!) stated in [Plo81, Chapter 8 p. 16], and used there to motivate the definition of... |

6 |
The largest first-order axiomatizable cartesian closed category of domains
- Gunter
- 1986
(Show Context)
Citation Context ...entary (first-order); in particular, (p2)–(p4) use quantification over finite subsets of |A| (i.e. “weak second order logic” in the terminology of [Bar77]). This is in fact inevitable; the results of =-=[Gun86]-=- can be adapted to show that the lattices of compact-open subsets arising from SFP domains are not first-order definable. Proposition 3.2.2 Let A be a coherent algebraic prelocale, a ∈ |A|. Then [∀u ⊆... |

5 | A Logic for the Specification and Proof of Regular Controllable Processes of CCS - Graf, Sifakis - 1986 |

4 |
Total vs. partial objects in denotational semantics
- Abramsky
(Show Context)
Citation Context ...85]. These spaces in their metric topologies are Stone spaces, and indeed the category of compact ultrametric spaces and continuous maps is equivalent to the category of second-countable Stone spaces =-=[Abr]-=-. A restricted denotational metalanguage comprising product, (disjoint) sum and powerdomain (the Vietoris construction [Joh85, Smy83b], which in this context is induced by the Hausdorff metric [Niv81,... |

4 |
An Introduction to First Order Logic
- Barwise
- 1977
(Show Context)
Citation Context ...onstruction. Note that our axiomatization is not elementary (first-order); in particular, (p2)–(p4) use quantification over finite subsets of |A| (i.e. “weak second order logic” in the terminology of =-=[Bar77]-=-). This is in fact inevitable; the results of [Gun86] can be adapted to show that the lattices of compact-open subsets arising from SFP domains are not first-order definable. Proposition 3.2.2 Let A b... |

3 | Mariangiola Dezani-Ciancaglini. A filter lambda model and the completeness of type assignment - Barendregt, Coppo - 1983 |

2 |
First order dynamic logic. Volume 68
- Harel
- 1979
(Show Context)
Citation Context ...oare 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 for; see =-=[Har79]-=-.) Extensionally, formulas denote sets of points in our denotational domains, i.e. φ is a syntactic description of {x : x satisfies φ}. Then P |= φ can be interpreted as x ∈ U, where x is the point de... |

1 |
A domain equation for bisimulation. Chapter 5
- Abramsky
(Show Context)
Citation Context ...ause the empty set is excluded from P( Â). (In fact, dropping (✷ − 0) and removing the side-condition I �= ∅ in (C − ✷ − ✸) corresponds exactly to retaining the empty set. For further discussion, see =-=[Abr87a]-=-.) Proposition 3.4.10 (T3) For all a, b ∈ C(P(A)): [a] P(A) ⊆ [b] P(A) =⇒ a ≤ P(A) b. Proof. By induction on the proofs of CP(A)(a), CP(A)(b). The non-trivial case is when these are both inferred by (... |