## Operational domain theory and topology of a sequential language (2005)

### Cached

### Download Links

- [www.cs.bham.ac.uk]
- [www.cs.bham.ac.uk]
- [www.cs.bham.ac.uk]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proceedings of the 20th Annual IEEE Symposium on Logic In Computer Science |

Citations: | 12 - 8 self |

### BibTeX

@INPROCEEDINGS{Escardó05operationaldomain,

author = {Martín Escardó and Weng Kin Ho},

title = {Operational domain theory and topology of a sequential language},

booktitle = {In Proceedings of the 20th Annual IEEE Symposium on Logic In Computer Science},

year = {2005},

pages = {427--436},

publisher = {IEEE Computer Society Press}

}

### OpenURL

### Abstract

A number of authors have exported domain-theoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact set and show that total programs with values on certain types are uniformly continuous on compact sets of total elements. We apply this and other conclusions to prove the correctness of non-trivial programs that manipulate infinite data. What is interesting is that the development applies to sequential programming languages, in addition to languages with parallel features. 1

### Citations

499 | A.: Domain Theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...y side-stepping denotational semantics and reformulating domain-theoretic and topological notions directly in terms of programming concepts, interpreted in an operational way. Regarding domain theory =-=[3, 11]-=-, we replace directed sets by rational chains, which we observe to be equivalent to programs defined on a “vertical natural numbers” type. Many of the classical definitions and theorems go through wit... |

414 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ...t model of PCF [32]. As is well known, for a sequential language like this, the match of the model with the operational semantics is imprecise: computational adequacy holds but full abstraction fails =-=[29]-=-. The main achievement of the present work is a reconciliation of a good deal of domain theory and topology with sequential computation. This is accomplished by side-stepping denotational semantics an... |

284 |
Foundations of constructive mathematics
- Beeson
- 1985
(Show Context)
Citation Context ..., that, when seen from the point of view of the data language, map programmable total elements to total elements, but diverge at some non-programmable total inputs. The construction uses Kleene trees =-=[6]-=-, and can be found in [10, Chapter 3.11]. This is analogous to the fact that totality with respect to P also disagrees with totality with respect to denotational models. A proof for the Scott model ca... |

279 |
Computable Analysis
- Weihrauch
- 2000
(Show Context)
Citation Context ... idea of invoking a data language to formulate higher-type program specifications in a sequential operational setting is already developed in [10] and is related to relative realizability [5] and TTE =-=[37]-=-. 1.2 Organization Section 2: Language, oracles, extensionality, monotonicity and rational chains. Section 3: Rational chains, open sets and continuity principles. 3sSection 4: Finite elements, contin... |

250 | The lazy lambda calculus
- Abramsky
- 1990
(Show Context)
Citation Context ...y (f == g) = ∀ total s ∈ Cantor.f(s) == g(s). 7.3 Simpson’s functional Simpson [33] applied Corollary 7.4 to develop surprising sequential programs for computing integration and supremum functionals (=-=[0, 1]-=- → R) → R, with real numbers represented as infinite sequences of digits. The theory developed here copes with that, again allowing a direct operational translation of the original denotational develo... |

238 | Domain theory in logical form
- Abramsky
- 1991
(Show Context)
Citation Context ...above proposition is the definition of the Scott topology. Then one argues, informally, that (Scott) open sets correspond to semi-decidable properties (Smyth [37]), or observable properties (Abramsky =-=[1, 3]-=-), or affirmable properties (Vickers [40]). Here we have defined open sets to be semi-decidable sets and then mathematically proved that they are (rationally) Scott open. However, when the language un... |

232 |
Data types as lattices
- Scott
- 1976
(Show Context)
Citation Context ...n, we explicitly include a Sierpinski base type Σ and a vertical-natural-numbers base type ω, although such types can be easily encoded in other existing types if one so desires (e.g. via retractions =-=[31]-=-). The type Σ will have elements ⊥ (non-terminating computation) and ⊤ (terminating computation). Intuitively, we think of programs of type ω as clocks that either tick for ever or else tick finitely ... |

226 |
C.H.L.: On full abstraction for PCF
- Hyland, Ong
- 2000
(Show Context)
Citation Context ...gram equivalence defined by ground program contexts, but the notion of totality changes. It is worth mentioning that the resulting data language for PCF defines precisely the elements of games models =-=[2, 17]-=-, with the programming language capturing the effective parts of the models. Similarly, the resulting data language for PCF extended with parallel-or and Plotkin’s existential quantifier defines preci... |

212 | Full abstraction for PCF
- Abramsky, Jagadeesan, et al.
(Show Context)
Citation Context ...gram equivalence defined by ground program contexts, but the notion of totality changes. It is worth mentioning that the resulting data language for PCF defines precisely the elements of games models =-=[2, 17]-=-, with the programming language capturing the effective parts of the models. Similarly, the resulting data language for PCF extended with parallel-or and Plotkin’s existential quantifier defines preci... |

130 |
Bisimilarity as a theory of functional programming
- Gordon
- 1999
(Show Context)
Citation Context ...usly developed in [21]. Regarding the above description of the elements of the vertical-natural-numbers type, a denotational proof using adequacy is easy, and operational proofs are obtained applying =-=[12]-=- or [27] (see [16]). 3 Rational chains and open sets We begin by developing fundamental order-theoretic and topological properties of the types of our language. 3.1 Order By the results recalled in th... |

119 | Topology via Logic - Vickers - 1989 |

78 | Operationally-based theories of program equivalence
- Pitts
- 1997
(Show Context)
Citation Context ... order fail to have suprema in general. A counter-example is given for type level 3. On the other hand, it is known that rational chains always have suprema, even in the absence of oracles — see e.g. =-=[27]-=-. Regarding topology [23, 35], we define open sets of elements via programs with values on a “Sierpinski” type, and compact sets of elements via Sierpinskivalued universal-quantification programs. The... |

66 | Continuous Lattices and Domains - Gierz, Hofmann, et al. - 2003 |

42 | Domain Theory and the Logic of Observable Properties
- Abramsky
- 1987
(Show Context)
Citation Context ...above proposition is the definition of the Scott topology. Then one argues, informally, that (Scott) open sets correspond to semi-decidable properties (Smyth [37]), or observable properties (Abramsky =-=[1, 3]-=-), or affirmable properties (Vickers [40]). Here we have defined open sets to be semi-decidable sets and then mathematically proved that they are (rationally) Scott open. However, when the language un... |

39 | From operational semantics to domain theory
- Mason, Smith, et al.
- 1996
(Show Context)
Citation Context ... manipulations with terms. 1.1 Related work The idea that order-theoretic techniques from domain theory can be directly understood in terms of operational semantics goes back to Mason, Smith, Talcott =-=[21]-=- and Sands (see Pitts [27] for references). Already in [21], one can find, in addition to rational-chain principles, two equivalent formulations of an operational notion of finiteness. One is analogou... |

36 | Operational semantics and program equivalence
- Pitts
- 2002
(Show Context)
Citation Context ...odel fails for sequential languages, proofs exploiting computational adequacy are possible [13] (see [18]). Proofs using game semantics can be found in [1, 12], and operational proofs can be found in =-=[19, 20]-=- (where an earlier operational proof of the rational-chains principle is attributed to Sands). For a call-by-value untyped language, an operational proof of the rational-chains principle was previousl... |

35 | On the computational content of the axiom of choice
- Berardi, Bezem, et al.
- 1998
(Show Context)
Citation Context ...for the Scott model can be found in [30]. For the intriguing relationship between totality in the Scott model with sequential computation, see [24]. 6.3 Higher-type oracles Berardi, Bezem and Coquand =-=[7]-=- work with a seemingly more expressive language. They have the following term-formation rule: if ti : σ is any sequence of terms, 28sthen λi.ti : Nat → σ is a term. When σ = Nat, this amounts to the c... |

31 | Synthetic topology of data types and classical spaces
- Escardó
- 2004
(Show Context)
Citation Context ...a language for PCF extended with parallel-or and Plotkin’s existential quantifier defines precisely the 2selements of the Scott model, again with the programming language capturing the effective part =-=[29, 10]-=-. But we don’t rely on these facts. We illustrate the scope and flexibility of the theory by applying our conclusions to prove the correctness of various non-trivial programs that manipulate infinite ... |

30 |
Fully Abstract Models of Typed λ-Calculi
- Milner
- 1977
(Show Context)
Citation Context ...xn = h(g n (⊥)). 6s2.8 Proofs The facts stated in this background section are all well known. The extensionality, monotonicity and rational-chain principles follow directly from Milner’s construction =-=[22]-=-. Even though full abstraction of the Scott model fails for sequential languages, proofs exploiting computational adequacy are possible [18] (see [26]). Proofs using game semantics can be found in [2,... |

28 | Finitary PCF is not decidable
- Loader
- 1996
(Show Context)
Citation Context ...χ↑ b(c) = ⊤ = χ↑ c(b). As all elements of finitary PCF are finite, and contextual equivalence is co-semidecidable for finitary PCF, this would give a decision procedure for equivalence, contradicting =-=[19]-=-. Proposition 4.19. If an open set U has finite characteristic then U = ↑ F def = � {↑ b | b ∈ F } for some set F of finite cardinality consisting of finite elements. Proof. By Lemma 4.15, if U has fi... |

26 | Lazy functional algorithms for exact real functionals
- Simpson
- 1450
(Show Context)
Citation Context ...We illustrate the scope and flexibility of the theory by applying our conclusions to prove the correctness of various non-trivial programs that manipulate infinite data. We take one such example from =-=[33]-=-. In order to avoid having exact real-number computation as a prerequisite, as in that reference, we consider modified versions of the program and its specification that retain their essential aspects... |

25 | Local realizability toposes and a modal logic for computability
- Awodey, Birkedal, et al.
- 1999
(Show Context)
Citation Context ...toposes. The idea of invoking a data language to formulate higher-type program specifications in a sequential operational setting is already developed in [10] and is related to relative realizability =-=[5]-=- and TTE [37]. 1.2 Organization Section 2: Language, oracles, extensionality, monotonicity and rational chains. Section 3: Rational chains, open sets and continuity principles. 3sSection 4: Finite ele... |

25 |
Totale Objekte und Mengen in der Bereichstheorie
- Berger
- 1990
(Show Context)
Citation Context ...ion of Theorem 7.1 is not invoked for the purposes of this section, because we only apply compactness to get uniform continuity. 7.2 The Gandy–Berger functional The following theorem is due to Berger =-=[8]-=-, with domain-theoretic denotational specification and proof, and it was known to Gandy, according to M. Hyland. As discussed in the introduction, the purpose of this section is to illustrate that suc... |

24 | domains and predicate transformers: a topological view , Techincal monograph 126 - Power - 1983 |

21 |
A type theoretical alternative to CUCH
- Scott
- 1993
(Show Context)
Citation Context ...ith parallel features. 1 Introduction Domain theory and topology in programming language semantics have been applied to manufacture and study denotational models, starting with the Scott model of PCF =-=[32]-=-. As is well known, for a sequential language like this, the match of the model with the operational semantics is imprecise: computational adequacy holds but full abstraction fails [29]. The main achi... |

17 |
domain theory and theoretical computer science
- Topology
- 1998
(Show Context)
Citation Context ...ma in general. A counter-example is given for type level 3. On the other hand, it is known that rational chains always have suprema, even in the absence of oracles — see e.g. [27]. Regarding topology =-=[23, 35]-=-, we define open sets of elements via programs with values on a “Sierpinski” type, and compact sets of elements via Sierpinskivalued universal-quantification programs. Then 1. the open sets of any typ... |

15 | Logical full abstraction and PCF
- Longley, Plotkin
- 1997
(Show Context)
Citation Context ...oracles and then behaves as g, so that x = h(Ω). Now, for every type τ there is an “enumerator” Eτ : Nat → τ such that Eτ (�t�) = t for any program t: τ with Gödel number �t�. See Plotkin and Longley =-=[20]-=- for a purely operational proof that works with and without parallel features in the language. Hence if we define evσ(n, f) = EBaire→σ(n)(f) then we get an “evaluator” evσ : Nat × Baire → σ such that ... |

13 | Computability over the partial continuous functionals
- Normann
(Show Context)
Citation Context ...th totality with respect to denotational models. A proof for the Scott model can be found in [30]. For the intriguing relationship between totality in the Scott model with sequential computation, see =-=[24]-=-. 6.3 Higher-type oracles Berardi, Bezem and Coquand [7] work with a seemingly more expressive language. They have the following term-formation rule: if ti : σ is any sequence of terms, 28sthen λi.ti ... |

12 | Domain-theoretic foundations of functional programming
- Streicher
(Show Context)
Citation Context ...pes Nat for natural numbers and Bool for booleans. We regard this as a programming language under the call-by-name evaluation strategy. In summary, we work with PCF extended with finite-product types =-=[31, 15, 29, 39]-=-. Other possibilities are briefly discussed in Section 9. 2.2 Inessential, but convenient, extensions of the base language For clarity of exposition, we explicitly include a Sierpinski base type S and... |

11 | A note on logical relations between semantics and syntax
- Pitts
- 1997
(Show Context)
Citation Context ...ciples follow directly from Milner’s construction [22]. Even though full abstraction of the Scott model fails for sequential languages, proofs exploiting computational adequacy are possible [18] (see =-=[26]-=-). Proofs using game semantics can be found in [2, 17], and operational proofs can be found in [27, 28] (where an earlier operational proof of the rational-chains principle is attributed to Sands). Fo... |

10 |
Semantics of Programming Languages—Structures and Techniques
- Gunter
- 1992
(Show Context)
Citation Context ...pes Nat for natural numbers and Bool for booleans. We regard this as a programming language under the call-by-name evaluation strategy. In summary, we work with PCF extended with finite-product types =-=[13, 27]-=-. Other possibilities are briefly discussed in Section 8.2. For clarity of exposition, we explicitly include a Sierpinski base type Σ and a vertical-natural-numbers base type ω, although such types ca... |

10 |
Interdefinability of Parallel Operations
- Stoughton
- 1991
(Show Context)
Citation Context ...gence or weak parallel-or. Abramsky showed that parallel-or on the booleans is not definable from this [1], Stoughton showed that parallel-or is equivalent to the parallel conditional at ground types =-=[36]-=-, and Plotkin showed that the parallel conditional is not PCF-definable but that the Scott model 35sis fully abstract for PCF extended with the parallel conditional [29]. On the other hand, it is easy... |

9 | Full abstraction, totality and PCF
- Plotkin
- 1999
(Show Context)
Citation Context ...ound in [10, Chapter 3.11]. This is analogous to the fact that totality with respect to P also disagrees with totality with respect to denotational models. A proof for the Scott model can be found in =-=[30]-=-. For the intriguing relationship between totality in the Scott model with sequential computation, see [24]. 6.3 Higher-type oracles Berardi, Bezem and Coquand [7] work with a seemingly more expressiv... |

8 | On sequential functionals of type 3
- Normann
(Show Context)
Citation Context ...s, a Kleene-Kreisel density theorem for total elements, and a number of continuity principles based on finite elements. We work with a restricted kind of increasing chain because we must: Dag Normann =-=[25]-=- has shown that, even in the presence of oracles (see below), increasing chains in the contextual order fail to have suprema in general. A counter-example is given for type level 3. On the other hand,... |

5 |
Computability and totality in domains
- Berger
(Show Context)
Citation Context ...(⊤, ⊥) and V = ↑(⊥, ⊤) and observe that for F = {(⊤, ⊥), (⊥, ⊤)} we have ↑ F = U ∪ V . 16s4.3 Density of the total elements We now develop an operational version of the Kleene–Kreisel density theorem =-=[9]-=-. Definition 4.21. (Hereditary) totality is defined by induction on types as follows: 1. An element of ground type is total iff it is maximal in the contextual order. 2. An element f ∈ (σ → τ) is tota... |

3 | An Operational Domain-theoretic Treatment of Recursive Types
- Ho
- 2006
(Show Context)
Citation Context ... lists. There is no difficulty in developing our results in a callby-value setting. An operational domain theory of recursive types, which is built 37supon ideas developed here, has been developed in =-=[15, 16]-=- by the second-named author, where well known denotational algebraic-compactness results are established with respect to contextual equivalence. But computational features such as state, control and c... |

2 |
The role of compactness in analysis
- Hewitt
- 1960
(Show Context)
Citation Context ... y =⇒ f(x) = f(y). The intuition behind the classical topological notion of compactness is that a compact set behaves, in many important respects, as if it were a set of finite cardinality — see e.g. =-=[14]-=-. The official definition, which is more obscure, says that a subset Q of a topological space is compact iff it satisfies the Heine–Borel property: any collection of open sets that covers Q has a fini... |

2 |
1 Introduction 1 1.1 Related work
- Soc
- 1994
(Show Context)
Citation Context ...owever, p(s) = ⊤ because s ∈ � j≥k Qj by construction. We observe that this proof can be seen as a special case of that of the topological Tychonoff theorem for a well-ordered set of indices given in =-=[38]-=-. 8 End 8.1 Remarks on parallel convergence A function (∨) ∈ (Σ × Σ → Σ) such that p ∨ q = ⊤ ⇐⇒ p = ⊤ or q = ⊤ is known as parallel convergence or weak parallel-or. Abramsky showed that parallel-or on... |

1 |
Talk at the Workshop on Full abstraction
- Jung
- 1995
(Show Context)
Citation Context ...chain principles follow directly from Milner’s construction [22]. Even though full abstraction of the Scott model fails for sequential languages, proofs exploiting computational adequacy are possible =-=[18]-=- (see [26]). Proofs using game semantics can be found in [2, 17], and operational proofs can be found in [27, 28] (where an earlier operational proof of the rational-chains principle is attributed to ... |