## Universal Profinite Domains (1987)

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Venue: | Information and Computation |

Citations: | 15 - 1 self |

### BibTeX

@ARTICLE{Gunter87universalprofinite,

author = {Carl A. Gunter},

title = {Universal Profinite Domains},

journal = {Information and Computation},

year = {1987},

volume = {72},

pages = {1--30}

}

### Years of Citing Articles

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### Abstract

. We introduce a bicartesian closed category of what we call profinite domains. Study of these domains is carried out through the use of an equivalent category of pre-orders in a manner similar to the information systems approach advocated by Dana Scott and others. A class of universal profinite domains is defined and used to derive sufficient conditions for the profinite solution of domain equations involving continuous operators. As a special instance of this construction, a universal domain for the category SFP is demonstrated. Necessary conditions for the existence of solutions for domain equations over the profinites are also given and used to derive results about solutions of some equations. A new universal bounded complete domain is also demonstrated using an operator which has bounded complete domains as its fixed points. 1 Introduction. For our purposes a domain equation has the form X ¸ = F (X) where F is an operator on a class of semantic domains (typically, F is an endof...

### Citations

1115 |
The Lambda Calculus: Its Syntax and Semantics
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(Show Context)
Citation Context ...east element, we know that V V 1 1 / ~ V 1 . Hence V V 1 1 is a retract of V 1 . Since PLT is a concrete cartesian closed category, we may conclude that V 1 is a model of the untyped fi-calculus (see =-=[2]-=- and [10]). But there is something more which is true. It is possible to prove that N(V 1 ) / ~ V V 1 1 (see [3]), so the theory described in [22] and [23] for U applies also to V = jV 1 j. In particu... |

212 | A Powerdomain Construction
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- 1976
(Show Context)
Citation Context ... involved in obtaining solutions to equations over the category of profinite domains which will be defined below. This is a rather natural, and in a sense inevitable, category which contains SFP (see =-=[17]-=-) as a full subcategory. It has the unusual property of being bicartesian closed, i.e. it is cartesian closed and has a coproduct. Such categories have a rich type structure and form models of the typ... |

210 |
Data types as lattices
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(Show Context)
Citation Context ...attice of subsets of !, ordered by set Universal Profinite Domains 17 ? tt ff ffi ffi ffi J J J J\Omega \Omega \Omega \Omega Figure 2: The truth value dcpo. inclusion. It receives a detailed study in =-=[21]-=- where it is proved that any countably based algebraic lattice is a retract of P!. 5 Some domain theorists felt, however, that for applications in denotational semantics of programming languages it wo... |

191 |
Outline of a mathematical theory of computation
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(Show Context)
Citation Context ...ns is the primary goal of the paper. As a secondary theme we show how to extend the neighborhood or information system approach to categories (such as SFP) which are larger than the one considered in =-=[22]-=- and [24]. Section two gives some of the basic definitions and explains the equivalence defined by the ideal completion functor. In section three the category of Plotkin orders is introduced and shown... |

169 |
The category-theoretic solution of recursive domain equations
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- 1982
(Show Context)
Citation Context ...red sets). Techniques for solving such equations have been worked out for specific categories (see any of the references by Scott or Plotkin) and in rather general category-theoretic settings as well =-=[28]-=-. Computability has been successfully incorporated into many of these treatments ([29], [8], [9]). All of these approaches use one of three techniques. The most general is the inverse limit constructi... |

140 |
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Citation Context ...bcategory. It has the unusual property of being bicartesian closed, i.e. it is cartesian closed and has a coproduct. Such categories have a rich type structure and form models of the typed - calculus =-=[11]-=-. Obtaining profinite solutions for domain equations involving the coproduct can be problematic, however. There are categorical impediments to the solution of some equations. For example, the equation... |

90 | Full abstraction for a simple parallel programming language
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- 1979
(Show Context)
Citation Context ...allowing MA to include the emptyset, then the resulting operator does not even preserve the property of having a least element. Further discussion of the properties of these operators can be found in =-=[16]-=- and [15]. The precise relationship between the bounded complete algebraic dcpo's and the profinites is not well understood. Although the join completion operator does provide some connection, it does... |

53 |
Using Information Systems to Solve Recursive Domain Equations
- Larsen, Winskel
- 1991
(Show Context)
Citation Context ...egories (see any of the references by Scott or Plotkin) and in rather general category-theoretic settings as well [28]. Computability has been successfully incorporated into many of these treatments (=-=[29], [8-=-], [9]). All of these approaches use one of three techniques. The most general is the inverse limit construction used by Scott [20] to solve the domain equation D �� = D ! D (where \Delta ! \Delta... |

49 |
T ω as a universal domain
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(Show Context)
Citation Context ...rists felt, however, that for applications in denotational semantics of programming languages it would be easier to use a class which did not require the existence of a largest (top) element. Plotkin =-=[19]-=- showed that the poset T ! of functions from ! into the truth value dcpo T (see figure 2) is universal in the sense that every coherent !-algebraic dcpo is a retract of T ! . Since T ! is itself algeb... |

32 |
Continuous lattices, Toposes, Algebraic Geometry and Logic
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(Show Context)
Citation Context ...as been successfully incorporated into many of these treatments ([29], [8], [9]). All of these approaches use one of three techniques. The most general is the inverse limit construction used by Scott =-=[20] to -=-solve the domain equation D �� = D ! D (where \Delta ! \Delta is the exponential functor). The second uses the Tarski Fixed Point Theorem, which says: if D is a poset with joins for !-chains and a... |

31 |
A note on inconsistencies caused by fixpoints in a cartesian closed category
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(Show Context)
Citation Context ...diments to the solution of some equations. For example, the equation D �� = 1 + (D ! D) (where 1 is the terminal object) has no solution in a any non-trivial bicartesian closed category (see [12] =-=and [6]-=-). Moreover, there are equations which have a non-trivial solution in a bicartesian closed category but have no non-trivial solution over the profinites. We provide a condition which, in effect, reduc... |

22 |
Th,e Largest Cartesian Closed Category of Domains, Theoretical Computer Science 27
- Smyth
- 1983
(Show Context)
Citation Context ... which will be our primary technical tool for studying the profinite domains. Plotkin orders are less abstract than profinite domains and in many ways they are easier to work with. For example, Smyth =-=[27]-=- proves many facts about strongly algebraic domains by taking a detailed look at the particular class of Plotkin orders which correspond to such domains. Their use makes some arguments more algebraic ... |

20 |
The finitary projection model for second order lambda calculus and solutions to higher order domain equations
- Amadio, Bruce, et al.
- 1986
(Show Context)
Citation Context ... N(V 1 ) / ~ V V 1 1 (see [3]), so the theory described in [22] and [23] for U applies also to V = jV 1 j. In particular, V is a finitary projection model of the polymorphic -calculus in the sense of =-=[1]. It seems-=- unlikely that the theory of V is much different from that of U, but it is a "bigger" model in the sense that there is a projection from V onto U. Moreoever, the powerdomain operators mentio... |

20 |
Diagonal Arguments and Cartesian Closed Categories
- Lawvere
- 1969
(Show Context)
Citation Context ...ical impediments to the solution of some equations. For example, the equation D �� = 1 + (D ! D) (where 1 is the terminal object) has no solution in a any non-trivial bicartesian closed category (=-=see [12]-=- and [6]). Moreover, there are equations which have a non-trivial solution in a bicartesian closed category but have no non-trivial solution over the profinites. We provide a condition which, in effec... |

17 | Some ordered sets in computer science - Scott - 1982 |

16 |
Models of the Lambda Calculus
- Koymans
- 1982
(Show Context)
Citation Context ...ment, we know that V V 1 1 / ~ V 1 . Hence V V 1 1 is a retract of V 1 . Since PLT is a concrete cartesian closed category, we may conclude that V 1 is a model of the untyped fi-calculus (see [2] and =-=[10]-=-). But there is something more which is true. It is possible to prove that N(V 1 ) / ~ V V 1 1 (see [3]), so the theory described in [22] and [23] for U applies also to V = jV 1 j. In particular, V is... |

14 | Profinite Solutions for Recursive Domain Equations
- Gunter
- 1985
(Show Context)
Citation Context ...domains; two of these were mentioned at the beginning of the section. The definition above was chosen because it is the best suited for the constructions in the next section. The reader is refered to =-=[4]-=- for a full discussion. 5 Universal Domains. We now investigate the mathematical problem of the existence of a profinite universal domain. In the literature there are three primary examples of univers... |

12 |
The category of complete partial ord.ers: ø tool of mo,lci,ng meaning
- Plotkin
(Show Context)
Citation Context ... binary relation, we write X f Y g Z for X f Y and Y g Z. When the relation f is being considered as an arrow in a category, we write f : A ! B for f ` A \Theta B. The following definition appears in =-=[18]-=- and [25]. Definition: An approximable relation f : A ! B is a subset of A \Theta B which satisfies the following axioms for any X;X 0 2 A and Y; Y 0 2 B: 1. for every X 2 A, there is a Z 2 A such tha... |

8 |
Domains for denotational semantics. In Automata, languages and programming
- Scott
- 1982
(Show Context)
Citation Context ... primary goal of the paper. As a secondary theme we show how to extend the neighborhood or information system approach to categories (such as SFP) which are larger than the one considered in [22] and =-=[24]-=-. Section two gives some of the basic definitions and explains the equivalence defined by the ideal completion functor. In section three the category of Plotkin orders is introduced and shown to be bi... |

7 |
Fully Effective Solutions of Recursive Domain Equations
- Kanda
- 1979
(Show Context)
Citation Context ...s (see any of the references by Scott or Plotkin) and in rather general category-theoretic settings as well [28]. Computability has been successfully incorporated into many of these treatments ([29], =-=[8], [9-=-]). All of these approaches use one of three techniques. The most general is the inverse limit construction used by Scott [20] to solve the domain equation D �� = D ! D (where \Delta ! \Delta is t... |

2 |
Finitely continuous posets
- Kamimura, Tang
- 1984
(Show Context)
Citation Context ...able Plotkin poset with rt(B) �� = A, then B / ~ VA . A fairly detailed outline of one technique of construction is offered here and we mention a second (closely related) technique. Kamimura and T=-=ang [7]-=- use a different approach to get a retraction universal model for the !-profinite domains having bottoms. Their model, like P! and T ! , is locally finite but is somewhat less natural than either of t... |

2 | E#ective solutions of recursive domain equations - Kanda - 1979 |

1 |
F.� Continuously Generated Fixed�Points. Doctoral Dissertation� Ox� ford University
- Bracho�
- 1983
(Show Context)
Citation Context ...nown results on universal domains. Posets in the left column are assumed to be countable; their ideal completions are countably based. Elementary proofs of the universality of U appear in [22] and in =-=[3]-=-. A less elementary proof which uses results from the previous section can be carried out as follows. Let B be the countable atomless boolean algebra and suppose A is a countable bounded complete pose... |

1 |
Powerdomain construction in the category of algebraic lattices
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- 1985
(Show Context)
Citation Context ...l for bounded complete algebraic dcpo's, it is not isomorphic to Scott's universal domain U. Universal Profinite Domains 25 A variant on the join completion operator has been studied independently in =-=[5] for-=- a different purpose. The Frink completion kAk of a pre-order A is defined there. This operation is related to the join completion by the isomorphism kAk �� = jJ(A) ? j were (\Delta) ? is the oper... |

1 |
An ideal model for recursive polymorhpic types
- MacQueen, Plotkin, et al.
- 1984
(Show Context)
Citation Context ...int Theorem, which says: if D is a poset with joins for !-chains and a least element then any function f : D ! D which preserves such joins has a least fixed point. The third---which is introduced in =-=[13]-=----uses the Banach Fixed Point Theorem, which says: a uniformly contractive function f : X ! X on a non-empty complete metric space X has a unique fixed point. These last 1 Information and Computation... |

1 |
Full Abstraction and Semantic Equivalence. Doctoral Disserta� tion� Carnegie�Mellon University� 1985� 133 pp. �15� Poign�e� A.� A note on distributive laws and power domains
- Mulmuley�
(Show Context)
Citation Context ...ot just as a retract). There are instances in which a "retraction universal" domain does not have all of the desired properties so that a "projection universal" domain is needed. F=-=or example Mulmuley [14]-=- requires a projection universal domain to prove some of his results on the existence of inclusive predicates (for showing equivalence of semantics). Table 1 lists some of the known results on univers... |

1 |
A note on distributive laws and power domains
- Poign', A
(Show Context)
Citation Context ...MA to include the emptyset, then the resulting operator does not even preserve the property of having a least element. Further discussion of the properties of these operators can be found in [16] and =-=[15]-=-. The precise relationship between the bounded complete algebraic dcpo's and the profinites is not well understood. Although the join completion operator does provide some connection, it does not seem... |

1 |
Some ordered sets in computer science. In: Ordered Sets, edited by I
- Scott
- 1981
(Show Context)
Citation Context ... D. In this section we look at the relationship between normal substructures of pre-orders and pe-pairs from the point of view of approximable relations. We thereby generalize the theory exposited in =-=[23]-=- to the category of algebraic dcpo's. These results will be used to derive a universal domain technique for the Plotkin orders. Let A and B be pre-orders. Write A / ~ B if there is an A 0 / B such tha... |

1 |
Notes on cpo's/ SFP-objects and the like
- Scott
(Show Context)
Citation Context ...elation, we write X f Y g Z for X f Y and Y g Z. When the relation f is being considered as an arrow in a category, we write f : A ! B for f ` A \Theta B. The following definition appears in [18] and =-=[25]-=-. Definition: An approximable relation f : A ! B is a subset of A \Theta B which satisfies the following axioms for any X;X 0 2 A and Y; Y 0 2 B: 1. for every X 2 A, there is a Z 2 A such that X f Z; ... |

1 | Poign�e� A.� A note on inconsistencies caused by �xpoints in a carte� sian closed category - Huwig� - 1986 |

1 | A.� E�ective Solutions of Recursive Domain Equations. Doctoral Dis� sertation - Kanda� - 1980 |

1 | School on Foundations of Arti�cial Intelligence and Computer Science. Instituto di Scienze dell - Summer - 1978 |

1 | Notes on cpo�s� SFP�objects and the like. Manuscript� unpublished - Scott� - 1982 |