B. Reus and T. Streicher, "General Synthetic Domain Theory---a Logical Approach", Mathematical Structures in Computer Science, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... cpos [3] More generally, axiomatic domain theory has successfully abstracted the particularities of domains to provide a host of neo classical models [3, 6] A very di erent type of model is given by game theoretic semantics [25] Finally, there are a variety of models based on realizability [11, 28, 29, 30, 21, 22, 35]. What has been missing hitherto is a single unifying treatment accounting for the existence of all these types of model. In this paper, we provide the axiomatic basis for such a treatment. In a follow up paper [44] we shall demonstrate how the various types of model are incorporated within our ....

....argument using classical logic, cannot be full subcategories of the category of sets. In [38] Dana Scott showed that such categories can nonetheless live as full subcategories of models of intuitionistic set theory, an observation that led to the subsequent development of synthetic domain theory [36, 14, 28, 46, 22, 40, 35, 27, 7]. In this paper, we exploit this idea to obtain algebraically compact categories in a uniform way. Roughly speaking, we start o with a category S of intuitionistic sets that satis es one simple axiom, Axiom 1 of Section 2. From any such category S, we extract a full subcategory of predomains, ....

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B. Reus and Th. Streicher. General synthetic domain theory | a logical approach. Math. Struct. in Comp. Sci., 9:177-223, 1999.

....also be Both authors acknowledge the PIONIER project The Geometry of Logic , led by Professor I. Moerdijk, which employs the first author, and enabled the second author to visit Utrecht in the period February June 1998. 1 possible to use an (impredicative) intuitionistic type theory, as in [22], or even intuitionistic Zermelo Fraenkel set theory ( 27] without changing the nature of the mathematics (only the metamathematics) An interesting challenge would be to attempt an axiomatic development in a predicative type theory. It seems that the best way of isolating a full subcategory of ....

.... or open subset ) As in [15] our main axiom, Axiom 1, asks for Sigma to satisfy a certain completeness property. In the presence of this axiom alone, it is possible to identify a number of different notions of predomain. Amongst these, the replete sets [8, 30] and the well complete sets [15, 22, 26] form two extreme choices. The former form the smallest full reflective subcategory of sets containing Sigma, and the latter form what appears to be the largest full subcategory of sets supporting an adequate treatment of recursion. Although it is not known if well complete sets form a reflective ....

B. Reus and Th. Streicher. General synthetic domain theory --- a logical approach. In Category Theory and Computer Science, Proceedings of CTCS '97, pages 293--313. Springer LNCS 1290, 1997.

....class structure by putting C = S and stipulating that every map be small. The remaining goal of this section is to isolate a full subcategory of S to act as a category of predomains. This will require imposing further axioms on C. Many axiomatizations have been proposed for this purpose, see e.g. [26, 11, 20, 33, 17, 30, 25, 19]. Here, we follow [30] As first proposed in [26] the definition of predomain is predicated on a notion of partiality. To implement this, we require a distinguished subobject # . Intuitively # corresponds to the subobject of those propositions in# that express the termination of programs. ....

B. Reus and Th. Streicher. General synthetic domain theory --- a logical approach. Math. Struct. in Comp. Sci., 9:177-- 223, 1999.

....these objects any map is automatically continuous. It was suggested by Dana Scott. Rosolini ( 81] at the time Scott s student, was the first who made real progress in setting up the theory; later work was done by, among others, Hyland ( 42] Phoa ( 72] Taylor ( 91] and Streicher Reus ([76]) In [101] the force of a truly axiomatic and rigorously internal approach is advocated. Algebraic Set Theory. In their elegant little book ( 49] Joyal and Moerdijk present a novel way of looking at set theory. They point to a model in Eff , which needs to be further investigated. ....

B. Reus and T. Streicher. General synthetic domain theory---a logical approach. Mathematical Structures in Computer Science, 9:177--223, 1999.

....programming languages with recursively defined (mixed variance) datatypes. The classic example of such a setting is the category of continuous functions between directed complete partial orders. However, many other types of model are also known to support recursive datatypes; see, for example, [25, 15, 19, 3, 16]. 1 The goal of axiomatic domain theory is to axiomatize the structure common to such models. In his notion of algebraic compactness, Peter Freyd isolated the crucial universal property of (possibly contravariant) recursive types [6, 7] In one modern formulation, a model supporting the ....

....correct general approach to axiomatic domain theory. The axioms of synthetic domain theory have been well studied. Since [20] axiomatizations have been based on a distinguished subobject # of the subobject classifier in the ambient category C. The axiomatization we follow is close to those of [15, 24, 19, 17]. The particular category of predomains we choose is the category of well complete separated objects (see Section 5) but it seems likely that our results apply to any of the smaller categories often considered. Our main result, Theorem 1, states that the category, pPredom, of (# )partial ....

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B. Reus and Th. Streicher. General synthetic domain theory --- a logical approach. Math. Struct. in Comp. Sci., 9:177--223, 1999.

.... to be known as the effective topos (Hyland 1982) Although motivated by applications in constructive analysis, this topos turned out to have some intriguing properties (Hyland 1988; Rosolini 1990) of use to the related fields of type theory and programming language semantics; see (Phoa 1990) and (Reus and Streicher 1999), for example. But back in 1979, the personal significance of Hyland s lectures was that they led me to formulate the notion of tripos and were the catalyst for the research that formed my PhD thesis. The description Hyland gave of his topos was analogous to Higg s version of the category of ....

Reus, B. and T. Streicher (1999). General synthetic domain theory---a logical approach. Mathematical Structures in Computer Science 9, 177--223.

....such that between these objects any map is automatically continuous. Suggested by Dana Scott. Scott s student Rosolini ( 76] was the first who made real progress in setting up the theory; later work was done by, among others, Hyland ( 39] Phoa ( 67] Taylor ( 85] and Streicher Reus ([71]) In [95] the force of a truly axiomatic and rigorously internal approach is advocated. Algebraic Set Theory. In their elegant little book ( 45] Joyal and Moerdijk present a novel way of looking at set theory. They point to a model in Eff , which needs to be further investigated. 18 ....

B. Reus and T. Streicher. General synthetic domain theory---a logical approach. Mathematical Structures in Computer Science, 9:177--223, 1999.

....All these classes are closed under subspace, product, and exponentiation. Suitable choices of S and 1 T yield classes whose intersections with TOP yield the classes of T 0 , T 1 , Hausdorff, completely Hausdorff, and totally disconnected spaces. S; S) separated spaces correspond to the S spaces [11] or S posets [13] in Synthetic Domain Theory. These are also characterised by the fact that the canonical map X : X [ X S] S] is injective. People working with cartesian closed topological categories usually are more interested in the class of spaces X where X is initial [5, 8] In ....

....of S. An assembly X is (S; T ) separated if for all x 6= x 0 in X , there is a realisable function f : X S such that fx 6= fx 0 and fx; fx 0 2 T . In case of T = S, one need not refer to T at all. The resulting notion of (S; S) separated assemblies is analogous to the notion of S spaces [11] or S posets [13] in Synthetic Domain Theory. Since TA is left adjoint to AT and S is topological, we immediately obtain: Proposition 3.1 X is (S; T) separated iff AT(TAX ) is (S; T ) separated. Thus, S; T ) separation of X really is a property of the induced topological space TAX . In the ....

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B. Reus and T. Streicher. General synthetic domain theory -- a logical approach. Submitted to MSCS, October 1997.

....properties of partial map classifiers (an equational lifting monad [4] partial function spaces (Kleisli exponentials [22, 28] and a (parameterized) natural numbers object. These conditions are always satisfied by the categories of predomains that arise in axiomatic and synthetic domain theory [10, 12, 20, 11, 26, 30]. Theorem 4 states that such categories support at most one uniform recursion operator (a T fixed point operator) and moreover it determines a unique parametrically uniform Conway operator on the associated category of domains. Thus, in the presence of a lifting monad and a parameterized natural ....

B. Reus and T. Streicher. General synthetic domain theory --- a logical approach. Math. Struct. in Comp. Sci., 9:177-- 223, 1999.

....: are exactly those which are definable in PCF extended with the datatype R and operator 9. Preliminary investigations by Scott and Birkedal suggest that PER(P) can be the types, or sets, portion of a model for a suitable version of calculus of constructions, such as used by Reus and Streicher [25]. Possibly this formalization could serve both as a core programming language for PER(P) and as a proof system for program specification and verification. I hope to obtain a good understanding of this subject from Lars Birkedal s thesis work [7] and focus my attention to the following problem. ....

B. Reus and T. Streicher, "General Synthetic Domain Theory---a Logical Approach", Mathematical Structures in Computer Science, 1998.

....1988; Rosolini 1990) of use to the related fields of type theory and programming language semantics. For example, see (Phoa 1990) This is a preliminary version. The final version will be published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl locate entcs Pitts and (Reus and Streicher 1999). But back in 1979, the personal significance of Hyland s lectures was that they led me to formulate the notion of tripos and were the catalyst for the research that formed my PhD thesis. The description Hyland gave of his topos was analogous to Higg s version of the category of sheaves on a ....

Reus, B. and T. Streicher (1999). General synthetic domain theory---a logical approach. Mathematical Structures in Computer Science 9, 177--223.

....own internal logic. In other words, realizability toposes can be seen as universes within which one can perform various kinds of constructive set theoretic reasoning. Although the precise practical significance of this fact is perhaps not yet clear, the ongoing work of Reus and Streicher [28] explores the possibility of using this logic as a basis for program verification. 3. Many of the combinatory algebras we shall describe are in some way inspired by the fully abstract models mentioned above; however, it sometimes turns out that the realizability models are technically simpler to ....

B. Reus and T. Streicher. General synthetic domain theory---a logical approach. In Category Theory in Computer Science '97, pages 293--313. Springer LNCS 1290, 1997.

....also be Both authors acknowledge the PIONIER project The Geometry of Logic , led by Professor I. Moerdijk, which employs the first author, and enabled the second author to visit Utrecht in the period February June 1998. possible to use an (impredicative) intuitionistic type theory, as in [22], or even intuitionistic Zermelo Fraenkel set theory ( 27] without changing the nature of the mathematics (only the metamathematics) An interesting challenge would be to attempt an axiomatic development in a predicative type theory. It seems that the best way of isolating a full subcategory of ....

.... or open subset ) As in [15] our main axiom, Axiom 1, asks for Sigma to satisfy a certain completeness property. In the presence of this axiom alone, it is possible to identify a number of different notions of predomain. Amongst these, the replete sets [8, 30] and the well complete sets [15, 22, 26] form two extreme choices. The former form the smallest full reflective subcategory of sets containing Sigma, and the latter form what appears to be the largest full subcategory of sets supporting an adequate treatment of recursion. Although it is not known if well complete sets form a reflective ....

B. Reus and Th. Streicher. General synthetic domain theory --- a logical approach. In Category Theory and Computer Science, Proceedings of CTCS '97, pages 293--313. Springer LNCS 1290, 1997.

....oe sets. This work was followed by work of Hyland [32] and Taylor [77] who both gave axioms for domain theory, and by Phoa [47] More recent work in this direction includes work by Longley [35] Rosolini [63] Reus [53] Longley and Simpson [36] Fiore and Rosolini [17] Reus and Streicher [54]. 3 Partial Combinatory Algebras We shall recall here the notion of a partial combinatory algebra. 4 A partial combinatory algebra (PCA) consists of a set A together with a partial application function Delta : A Theta A A such that there exists two distinct elements K;S 2 A satisfying, for ....

B. Reus and T. Streicher. General synthetic domain theory --- a logical approach. Mathematical Structures in Computer Science, 1998.

....in [15] where its consequences were worked out in detail in the specific case of realizability toposes. This axiomatization has since proved to be applicable also to: Grothendieck toposes [5, 4] models of intuitionistic ZermeloFraenkel set theory [30] and models of intuitionistic type theory [25]. In this paper we consider the same general axiomatic approach in the setting of an elementary topos. Given an elementary topos with a natural numbers object and a distinguished dominance (determining a lift monad) we isolate a full subcategory of predomains, the well complete objects, as in ....

....3 Completeness and Fixed points The mono : I F plays a fundamental role in developing a basic notion of chain completeness sufficient for establishing fixed points for endomorphisms on suitable objects. For further motivation see [15] The results in this section are, by now, standard [15, 27, 25]. Definition 1. An object X is said to be complete if the induced morphism X : X F X I is an isomorphism. If X is complete and ff : LX X is a monad algebra, then, for any morphism f : X X let h : I X be the unique algebra homomorphism given by Proposition 1. Because X is ....

[Article contains additional citation context not shown here]

B. Reus and Th. Streicher. General synthetic domain theory --- a logical approach. In Proceedings of CTCS '97, pages 293--313. Springer LNCS 1290, 1997.

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Draft. Reus, B., & Streicher, Th. (1997). General Synthetic Domain Theory -- a logical approach (extended abstract). Pages 293--313 of: Moggi, E., & Rosolini, G. (eds), 7th Conf. Category Theory in Computer Science. Lecture Notes in Computer Science, vol. 1290. Berlin: Springer.

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B. Reus and T. Streicher, "General Synthetic Domain Theory---a Logical Approach", Mathematical Structures in Computer Science, 1998.

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B. Reus and T. Streicher, "General Synthetic Domain Theory---a Logical Approach", Mathematical Structures in Computer Science, 1998.

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B. Reus and T. Streicher. General synthetic domain theory --- a logical approach. Math. Struct. in Comp. Sci., 9:177-- 223, 1999.

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