M. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertation Series, Cambridge University Press, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....PVS Co Let us first mention an implementation specific point. Both approaches HOL LCF and PVS Co are essentially based on a single axiom. For PVS Co it determines the property of being a final coalgebra, whereas in HOL LCF it ensures that initial algebras and final coalgebras coincide (see [AJ94, Fio96] Interestingly, both approaches have alternatively been developed in a definitional way as well: PVS Co by Paulson [Pau97] and HOL LCF by Agerholm [Age94b] But whereas Paulson succeeded in hiding the additional overhead efficiently, Agerholm had to turn to an unmanageable, HOL Sum like ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge University Press, 1996.

..... 74 1 Introduction In his work on algebraic compactness, Freyd [9, 10] identi ed the categorical structure required to model recursive types. Many examples of algebraically compact categories are known. Domain theory provides the classical example of the category of 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, ....

....recursive types. Many examples of algebraically compact categories are known. Domain theory provides the classical example of the category of 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 ....

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M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertation Series, CUP, 1996.

....find examples of models. Nevertheless, several sources of such models are known. Domain theory provides the classical example of the category of #cpos [24] More generally, axiomatic domain theory has successfully abstracted the idiosyncracies of domains to provide a host of neo classical models [2, 4]. A quite different type of model is given by gametheoretic semantics [18] Finally, while the structure has not previously been exhibited in the form above, it has long been known that there should be a variety of models based on realizability semantics [9, 20, 21, 22, 17] What has been missing ....

....for PCF in [30] However, the extension of the result to FPC is non trivial, see Section 8. Finally, in Section 10, we present applications of our work across the range models discussed earlier. The classical domain theoretic models, such as the category of #cpos, and their generalizations [2, 4], all embed in Grothendieck toposes [3, 5] and hence, by [15, Ch. IV] in categories with class structure. Moreover, under mild conditions, Axiom N is satisfied. Also, by their very definition, realizability models [9, 20, 21, 22, 16, 17] embed in realizability toposes [10, 12] and hence in ....

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M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertation Series, CUP, 1996.

....semantics [32] so it could be said that any category proposed for denotational semantics should at least be able to model FPC . A further indication of the importance of this language is that it appears in two well known modern textbooks [20, 35] It has recently been studied by Fiore and Plotkin [12, 13], who provide an axiomatisation of sound domain theoretic models of the call by value variant, and by Andrew Gordon [19] who develops an operationally based theory of program equivalence for it. Here we provide the first fully abstract denotational semantics of a language as rich as FPC . 2 A ....

....to demonstrate computational adequacy of models of recursively typed programming languages. We will make use of this later. 3 The language FPC Here we give the definition of the metalanguage FPC . This language, and similar ones, has appeared in [20, 32, 35] A detailed treatment can be found in [12]. It is a type theory with products, exponentials, sums and recursive types; we consider it as a typed functional programming language in its own right, as has been done by Plotkin, Fiore, Winskel and Gordon [18, 19] 3.1 Syntax There are two syntactic classes of variables: TypeVar for type ....

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M. P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science. Cambridge University Press, 1996.

....of making concurrency less separate a study. Through presheaf models we are trying to bring concurrency theory within domain theory, though with the proviso that this should be understood liberally enough to include generalisations of domain theory like those envisaged in axiomatic domain theory [11, 5]. The paper [14] is a further step in this programme. 2 Traditional models We focus on three traditional models for concurrency: transition systems, synchronisation trees and event structures (see [15] for more background) A transition system is a structure (S; i; L; tran) where ffl S is a ....

Fiore, M.P., Axiomatic Domain Theory in Categories of Partial Maps. Ph.D Thesis. Distinguished Dissertations in Computer Science, Cambridge University 15

....of making concurrency less separate a study. Through presheaf models we are trying to bring concurrency theory within domain theory, though with the proviso that this should be understood liberally enough to include generalisations of domain theory like those envisaged in axiomatic domain theory [24, 9, 11]. The use of presheaves as models for concurrency is not confined to languages expressible in Proc. For instance, following the work in [28] a treatment of presheaf categories as domains has been devised [7, 4] and applied The paper [8] shows that any P factorisable functor preserves open maps ....

M. P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science. Cambridge University Press, 1996.

....finite words of Act regarded as a category and there is an equivalence of categories: Open map bisimulation in this case corresponds to Park Milner bisimulation. For these intuitions to make precise sense one needs to develop a suitably general theory of domains. The axiomatic approach [105, 30, 35] to the theory of domains paves the way. There one defines axiomatically classes of categories that can be thought of as categories of domains (generally order enriched ones) and that by the axioms are guaranteed to provide uniform solution to recursive domain equations [125, 30] From this ....

....approach [105, 30, 35] to the theory of domains paves the way. There one defines axiomatically classes of categories that can be thought of as categories of domains (generally order enriched ones) and that by the axioms are guaranteed to provide uniform solution to recursive domain equations [125, 30]. From this perspective presheaf categories can be thought of as categories of non deterministic domains and the operation of forming the presheaf category as analogous to a powerdomain construction. This thesis builds on the above hopes and intuitions about presheaves. We analyse properties of ....

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Marcelo P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science. Cambridge University Press, 1996.

....a dependently typed calculus could be modelled by two fibrations one to handle the dependency and the other the logic. Underdeterminism should be a separate feature on top of this set up. We should be able to characterise # # semantically, using the specialisation order for example, as in [Fio94] Whether derived or assumed, there is a posetenriched structure where the ordering of morphisms corresponds to refinement. More generally, we could envisage a 2 categorical structure, where the 2 cells correspond to proof of refinement. The let congruence rules give a 2 categorical structure. ....

Marcelo Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, Department of Computer Science, University of Edinburgh, 1994.

....(based on the associated notions of dominion, see Section 3, and dominance) and proved representation results into categories of partial functions on presheaf toposes. In computer science, this categorical approach to partiality has proved its value through applications in axiomatic domain theory [5, 6] and synthetic domain theory [8] In spite of the above, there are reasons to look for more general approaches to partiality. In particular, the notion of dominion requires every partial map to have its domain of definition represented as an object of the category. There are cases in which it is ....

....I) 3 Dominical lifting monads In this section we review the standard categorical approach to partiality and the notion of lifting monad it induces. All the definitions and results are contained (at least implicitly) in [17] A good range of computationally motivated examples can be found in [5]. 5 Definition 3.1 A dominion on a category C is given by a collection D of monomorphisms in C that is closed under composition, contains every isomorphism, and is closed under pullback along arbitrary morphisms. Let C be a category and let D be a dominion on it. We use the symbol # # to ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertation Series, Cambridge University Press, 1996.

....for information systems. Given a type constructor (i.e. an operation on games) F which is continuous with respect to #, we can find a game D = F (D) in the usual way by setting D = # # n=0 F n (I ) In fact this can be generalised and strengthened to obtain parameterized minimal invariants [11, 10] for a large class of functors F : I op I) n # I, including all the functors built out of the product, sum and function space constructions described here. 3.9. A model of FPC. We have seen that I is a cartesian closed category, and that sums can be modelled in I using # and Girard s ....

M. P. Fiore, Axiomatic domain theory in categories of partial maps, Distinguished Dissertations in Computer Science, Cambridge University Press, 1996.

....associated partial category is algebraically compact. This formulation leaves two questions unanswered. First, which endofunctors should algebraic compactness apply to Second, although the universal property of recursive types is identified, how are their solutions constructed In his PhD thesis [1], Fiore answered these questions in an order enriched setting: the functors are the enriched ones, and there are easily verifiable conditions for algebraic compactness to hold. This work was later extended by Fiore, Plotkin and Power to cover more general enrichments [3] However, a basic ....

....in many realizability models (even when expressed internally) see [17] for a counterexample. The aim of this paper is to provide a general axiomatic account of the construction of solutions to recursive domain equations, one that is applicable equally well both to the neo classical examples of [1, 3], and also to the whole range of realizability models studied in, for example, 15, 17] The idea behind our approach is that categories of predomains do not in general have su#cient structure themselves to account for the principles underlying the construction of recursive types within them. ....

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M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertation Series, Cambridge University Press, 1996.

....is the best we can do in the category of cppos. Note that this does not contradict the fact that stable models etc. are cppos as we have built the continuous function space into our notion of model. We conjecture this pleasant situation should extend further, certainly to a fragment of, say, FPC [Fio96] with recursion but restricting recursive types X.# so that # contains no occurrence of lifting or function space (it may even be possible to allow lifting and function space, asking only that all occurrences of X are positive) In contrast we would expect the same phenomena as arose for our ....

M. P. Fiore, Axiomatic Domain Theory in Categories of Partial Maps, Cambridge: Cambridge University Press, 1985.

....of programming languages, it is useful to keep account of a category in which to model standard features. For instance, to model partiality, one analyses a category D of partial maps in terms of a category C of total maps. See, for Premonoidal categories and notions of computation 3 instance, (Fiore 1996; Fiore and Plotkin 1994) for an account of partiality treated as a special case of the general approach we will take here. There is a structure preserving faithful functor j : C Gamma D that is the identity on objects, exhibiting the total maps as being among the partial maps and representing ....

Fiore, M. (1996) Axiomatic domain theory in categories of partial maps. Cambridge University Press Distinguished Dissertations in Computer Science.

....A of C. We assume that the reader is able to verify, or believe, various routine calculations in models, especially those just involving products, sums, monads and exponentials. 1.2 Call by Value FPC We describe the version of Plotkin s FPC [Plo85] that we use. We follow the presentation of [Fio94] as far as is convenient. Our notation is slightly terser, and we include in the calculus a type 0 and constants A of each type. These additions are useful for our arguments, but as they are definable in the unextended calculus, they make no difference to our results. FPC has a countable set V of ....

....judgements, of FPC is given in table 1. The type decorations on various constructs above guarantee uniqueness of typing: if Gamma r : A and Gamma r : A 0 , then A = A 0 . We shall usually omit type decorations. 1 INTRODUCTION 4 1. 3 Models Categorical models of FPC are discussed in [Fio94] and [Sim92] In [Fio94] chapter 8 and section 9.1) models of FPC are carefully developed in a categorical axiomatisation of partial map structures. We use the more general monadic setting of [Sim92] the details of the interpretation of FPC generalise more or less verbatim to this setting. ....

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Marcelo P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, Department of Computer Science, University of Edinburgh, October 1994. REFERENCES 19

....within domain theory, which is so successful in handling higher order and recursive features. Here the term domain theory should be understood liberally enough to include generalisations of domain theory like those envisaged in axiomatic domain theory , being developed by Fiore and Plotkin (Fiore 1994). The paper (Winskel 1996) is a step in this programme; it shows how presheaf models can be extended to higher order process languages, where processes are transmitted along channels, and along similar lines, incorporating ideas of (Stark 1996; Fiore, Moggi, and Sangiori 1996) a presheaf ....

Fiore, M. P. (1994). Axiomatic Domain Theory in Categories of Partial Map.

....cpo s. 2 However, since there is no unsolvable term in m , we should assign a nonpointed cpo to each type. The decades of study on domain theory and on axiomatic domain theory shows that the existence of the bottom element plays an essential role in solving a domain equation ( SP82] Fre91] [Fio94]) and therefore the standard theories are not applicable directly to this problem. In this study, we solve this by considering an op bration D obtained by gathering all the domains, and expressing the above equation as an equation between op brations. By considering a simultaneous construction of ....

....of 1 2 3 : J J Int . Since there is no term whose meaning is Int , this semantics does not re ect the structure of m , and is far from sucient. However, in standard theories, the existence of the bottom element plays an essential role in solving a domain equation ( SP82] Fre91] [Fio94]) and therefore they are not applicable directly to this problem. We solve this problem by considering a pair (I, E) of a poset I and a functor E from I to C as an object and construct both T and D simultaneously. Then, the pair consisting of the one point poset for I and one point cpo for the ....

Marcelo P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994.

....f(a) x) 1 ( if 8n 2 N: fn (a) x) 1 ( f (a) x) else, where is the least n with fn (a) x) 6= 1 ( 9) Kleisli composition is continuous (i.e. preserves the order and least upperbounds) in both its arguments. This means that the Kleisli category Kl(J) is cpo enriched, see [4]. We summarise what we have found so far. Proposition 1. The functor J from (3) describing the outputs of Java statements and expressions is a strong monad on the category of sets. Its Kleisli composition and extension correspond to composition and extension in Java. And its Kleisli category ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge Univ. Press, 1996.

....are equipped with cpo structure, with maps respecting such structure. More generally, that work should and probably soon will be incorporated into axiomatic domain theory, requiring study of the 2 category V Cat for a symmetric monoidal closed V subject to some domain theoretic conditions [3]. Moreover, 3 our de nitions and analysis naturally live at the level of 2 categories, so that level of generality makes the choices clearest and the proofs simplest. Mathematically, this puts our analysis exactly at the level of generality of the study of monads by Street in [15] but see also ....

M.P. Fiore, Axiomatic domain theory in categories of partial maps, Cambridge University Press Distinguished Dissertations in Computer Science, 1996

....(based on the associated notions of dominion, see Section 3, and dominance) and proved representation results into categories of partial functions on presheaf toposes. In computer science, this categorical approach to partiality has proved its value through applications in axiomatic domain theory [4, 5] and synthetic domain theory [7] In spite of the above, there are reasons to look for more general approaches to partiality. In particular, the notion of dominion requires every partial map to have its domain of definition represented as an object of the category. There are cases in which it is ....

....I) 3 Dominical lifting monads In this section we review the standard categorical approach to partiality and the notion lifting monad it induces. All the definitions and results are contained (at least implicitly) in [16] A good range of computationally motivated examples can be found in [4]. Definition 3.1 A dominion on a category C is given by a collection D of monomorphisms in C that is closed under composition, contains every isomorphism, and is closed under pullback along arbitrary morphisms. Let C be a category and let D be a dominion on it. We use the symbol # # to ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertation Series, Cambridge University Press, 1996.

.... of fij equality in the simply typed calculus [31] The remainder of the paper is devoted to providing conditions for establishing the existence (and uniqueness) of parametrically uniform Conway operators (hence iteration operators) In one common setting, which arises in axiomatic domain theory [13, 10, 12], one has that the category D of domains is obtained as the co Kleisli category of a comonad on the category of strict maps S. For example, Cppo is the co Kleisli category of the lifting comonad on Cppo . In axiomatic domain theory, S satisfies a curious property, first identified by Freyd ....

....class of endofunctors on S have initial algebras whose inverses are final coalgebras (in Freyd s terminology, S is algebraically compact) Following [7] we call such initial final algebras coalgebras bifree algebras. In the example of Cppo , every Cppo enriched endofunctor has a bifree algebra [10]. In Section 5, we give a quick overview of initial algebras, final coalgebras and bifree algebras, including a couple of minor new propositions. Then, in Section 6, we show how bifree algebras in S can induce properties of fixed point operators in D. This programme was begun by Freyd and others ....

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M. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertation Series, Cambridge University Press, 1996.

....in the analysis of a much wider class of languages. When some success has been achieved with concrete constructions, it becomes important to identify the key properties of these constructions at a more abstract level; thus the trend towards axiomatic and synthetic domain theory, for example [Fio96, Hyl91]. There has also been considerable progress in axiomatizing sufficient conditions for computational adequacy [FP94, Bra97, McC96a] In another vein, the work on action structures [MMP95] can be seen as an axiomatics for process calculi and other computational formalisms. In the present paper we ....

M. Fiore. Axiomatic domain theory in categories of partial maps. Cambridge University Press, 1996.

....the 2 category of small categories with filtered colimits, functors preserving these, and natural transformations. In some respects his work is a precursor to ours; however, we take a step further and develop an axiomatic theory in accordance with the approach to Axiomatic Domain Theory adopted in [6, 26, 8, 9]. Conceptually, the categorical theory of domains that we put forward maybe seen as the traditional theory of Smyth and Plotkin [30] where cpos ( complete partial orders) are replaced with their categorical analogue (viz. small categories with colimits of chains) Technically, this is not ....

....Algebraic compactness is a universal property due to Freyd [10] that provides canonical interpretations of recursive types. In this section we show this property for socalled Kcats; these may be seen as a 2 categorical analogue of cppos ( complete pointed partial orders) Following [6], our approach is to obtain the result from the LocalCharacterisation and Limit Colimit Coincidence Theorems, together with the Basic Lemma [30] Recall that the Basic Lemma provides conditions under which an initial algebra (and hence a fixed point, by a lemma due to Lambek) of an endofunctor can ....

[Article contains additional citation context not shown here]

M. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science. Cambridge University Press, 1996.

....are equipped with cpo structure, with maps respecting such structure. More generally, that work should and probably soon will be incorporated into axiomatic domain theory, requiring study of the 2 category V Cat for a symmetric monoidal closed V subject to some domain theoretic conditions [3]. Moreover, our definitions and analysis naturally live at the level of 2 categories, so that level of generality makes the choices clearest and the proofs simplest. Mathematically, this puts our analysis exactly at the level of generality of the study of monads by Street in [11] but see also ....

M.P. Fiore, Axiomatic domain theory in categories of partial maps, Cambridge University Press Distinguished Dissertations in Computer Science, 1996

....19 The definitions of bisimulation and of invariant are determined by the functor T of the coalgebras that one is considering. The notion of simulation is determined in a similar manner if there is an order on the objects T (X) It is described as an ordered bisimulation in [Fio96] Our notion of simulation for sequences is an instance of an ordered bisimulation, using the flat order on the functor T (X) 1 (A Theta X) 20 We could have used as well infinite sequences of sequences defining the ascending chain property as a greatest invariant similarly to ordered in ....

....in [Reg95] often leads to easier proofs (than in the present coalgebraic approach) since induction can be used for admissible predicates (see the end of Subsection 4. 2) Typically in domain theory initial algebras and final coalgebras coincide as a solution of a domain equation (see e.g. AJ94, Fio96] Thus, their central induction proof principle is restricted to such frameworks. Of course, the domain theoretic approach is perfect if the problem (involving sequences) fits into the context of the Logic of Computable Functions. However, the domain theoretic setting requires that all functions ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge Univ. Press, 1996.

....subcategory of Cpo, the symmetric monoidal closed category of cpos and continuous functions. In the sequel, the lifting functor ( Gamma) will denote both the left adjoint, the induced monad (on Cpo) and the induced comonad (on Cppo ) of the reflection. Since Cppo is a bipolar Cpo category [10] we can try to apply Theorem 4.3 to its endofunctors, and to a Cpo enriched version of B in particular. There are several endofunctors which one can associate to (strong) deterministic behaviour depending on the use that one makes of the lifting endofunctor. The following proposition takes care of ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations Series. Cambridge University Press, 1996.

....semantics [22] so it could be said that any category proposed for denotational semantics should at least be able to model fpc. A further indication of the importance of this language is that it appears in two well known modern textbooks [11, 24] It has recently been studied by Fiore and Plotkin [6, 7], who provide an axiomatisation of sound domain theoretic models of the call by value variant, and by Andrew Gordon [10] who develops an operationally based theory of program equivalence for it. Here we provide the first fully abstract denotational semantics of a language as rich as fpc. 2 The ....

....as for information systems. Given a type constructor (i.e. an operation on games) F which is continuous with respect to E, we can find a game D = F (D) in the usual way by setting D = F 1 n=0 F n (I) In fact this can be generalised and strengthened to obtain parameterized minimal invariants [6, 8] for a large class of functors F : E op Theta E) n E , including all the functors built out of the product, sum and function space constructions described here. Similar results in a different category of games are presented in detail in [4] 3.10 A model of fpc We have seen that E is a ....

Marcelo P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994.

....semantics [32] so it could be said that any category proposed for denotational semantics should at least be able to model FPC . A further indication of the importance of this language is that it appears in two well known modern textbooks [20, 35] It has recently been studied by Fiore and Plotkin [12, 13], who provide an axiomatisation of sound domain theoretic models of the call by value variant, and by Andrew Gordon [19] who develops an operationally based theory of program equivalence for it. Here we provide the first fully abstract denotational semantics of a language as rich as FPC . 2 A ....

....to demonstrate computational adequacy of models of recursively typed programming languages. We will make use of this later. 3 The language FPC Here we give the definition of the metalanguage FPC . This language, and similar ones, has appeared in [20, 32, 35] A detailed treatment can be found in [12]. It is a type theory with products, exponentials, sums and recursive types; we consider it as a typed functional programming language in its own right, as has been done by Plotkin, Fiore, Winskel and Gordon [18, 19] 3.1 Syntax There are two syntactic classes of variables: TypeVar for type ....

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M. P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science. Cambridge University Press, 1996.

....of programming languages (see e.g. 23, 37] It also opens the possibility of studying dynamics of programs at a level close to implementation using game semantics. Central themes of this paper are a further investigation in this direction, and an axiomatisation of a category of games following [9, 10]. Among other things, we aim to advance the work of Hyland and Ong [18] in a couple of points. ffl Many programming languages adopt a call by value evaluation strategy and have more intricate type structure like sums and recursive types, cf. 26] Can game semantics handle these features, and ....

....represents the original strategies. The resulting code exhibits a close link between the abstract type structures and their implementation, cast in the setting of name passing processes. Related work. First we comment on the relationship between the axiomatisation in this paper and that of [9, 10]. Both axiomatisations take the same conceptual standpoint based on the interplay between a category of partial maps and a subcategory of total maps there of. However, in order to accommodate the intensional nature of games, here we have to weaken the categorical structure for modelling product ....

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M. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science. Cambridge University Press, 1996.

....one would like these constructions to have good (i.e. meaningful) categorical properties and simple concrete definitions. One way to go about creating such a semantic universe is to concentrate on the categorical properties of the constructions. This is the route taken by axiomatic domain theory, [6, 5]. The more traditional way is to define constructions concretely and then prove the categorical properties. The latter approach frequently requires additional assumptions about the spaces employed. Let us illustrate these two alternatives with the probabilistic powerdomain construction. While it ....

M. P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994. To be published by Cambridge University Press in the Distinguished Dissertations Series.

....an isomorphism FA a A such that a is an initial F algebra and a Gamma1 is a terminal F coalgebra. C is said to be algebraically compact if every endofunctor has a free algebra, where every usually ranges over an understood class of functors, for example over suitably enriched functors [5]. Algebraically compact categories allow the construction of canonical solutions to recursive domain equations involving bifunctorial type constructors [7, 5] Our goal in this section is to show that Q P is algebraically compact relative to a good class of endofunctors including the powerdomain ....

.... endofunctor has a free algebra, where every usually ranges over an understood class of functors, for example over suitably enriched functors [5] Algebraically compact categories allow the construction of canonical solutions to recursive domain equations involving bifunctorial type constructors [7, 5]. Our goal in this section is to show that Q P is algebraically compact relative to a good class of endofunctors including the powerdomain as well as all functors derived from the type constructors considered in Section 2. More generally one wants to solve recursive domain equations for ....

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M. P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Ph.D. thesis, Department of Computer Science, University of Edinburgh. Available as ECS-LFCS-94-307, 1994.

....and recursion. This system did not provide a close integration of the two kinds of recursive type, and suffered from a reliance on the heavy machinery of fixpoint induction for reasoning about terms involving general recursion. Following recent work by Freyd and others on algebraic compactness [Fre90, Fre91, Fre92, Bar92, Sim92, Fio94], the more elegant solution presented in this paper was developed. In brief, Freyd showed how to reduce the problem of finding solutions for general recursive domain equations to that of building inductive types, provided the functors involved are algebraically bounded, i.e. the inductive and ....

Marcello P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994.

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M. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations Series. Cambridge University Press, 1996.

....of types. Those ideas are the origin of the definition of types as functors that we have given. The treatment of type constructors as functors is difficult because of contravariance. A solution is to take the category of types to be involutory (i.e. isomorphic to its dual via an involution) [8]. Given a category C there are various approaches to constructing an involutory category I(C) In the context of orderenriched categories one can exploit the duality embedding projection [21] There are also canonical choices, namely C = C op Theta C [11] or its subcategory C [1, 10] ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994. To be published by Cambridge University Press in the Distinguished Dissertations Series.

....solving recursive type equations. There he introduced algebraically compact categories and established their fundamental property: that bifunctors on them have canonical and minimal fixed points. This has been a first important step towards an axiomatic theory of recursive types (see [Sim92] and [Fio94a, Chapters 6 8] Other work on algebraic compactness can be found in [Ad a93, Bar92] Concerning fixed points of endomorphisms, it was noticed by [HP90] after studying the work of [Law64, Law69] that in the presence of cartesian closure they are inconsistent with coproducts (empty or binary) ....

....categories of partial maps [RR88] and, in particular, via the notion of partial cartesian closure [LM84] that an appropriate categorical setting emerged. With this background it was possible, for the first time, to consider categorical models for a rich type theory with recursive types. In [Fio94a, FP94] a notion of categorical model for the metalanguage FPC a type theory with sums, products, exponentials and recursive types [Plo85, Gun92, Win93] was defined. Very roughly, categorical models of FPC are algebraically compact partial cartesian closed categories with binary ....

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M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994. (To be published by Cambridge University Press in the Distinguished Dissertations Series).

....of a domain theoretic category; and hence it will guide our axiomatisation. 2 Lifting We present an axiomatisation of lifting in terms of dominances which is by now a traditional way to generate examples of Assumption 1. For the connection between this axiomatisation and partiality consult [Fio96] Convention. Subobjects obtained as pullbacks of a monomorphism m are called m subobjects. In the vein of synthetic domain theory [Ros86, Hyl91, Tay91] we consider a generic observable property , 1 , Sigma, whose role is to internalise the notion of termination and hence determine the ....

....appears in a pullback A B j B LB for a unique characteristic map A LB) and ffl the composite 1 j1 , L1 jL1 , L 1 and the unique map 0 1 are j 1 subobjects. Theorem 2.5 Lifting structures and dominances that classify 0 1 are in bijective correspondence. Proof: See [Fio96, Chapter 3] Lifting structures are easily seen to provide the structure of a commutative monad: Definition 2.6 (Lifting) Let K be a cartesian category with an initial object. Every lifting structure (L; j) on K induces a lifting monad L = L; j; t ) on K with multiplication and tensorial ....

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M.P. Fiore. Axiomatic domain theory in categories of partial maps. To be published by Cambridge University Press in the Distinguished Dissertations Series. (Ph.D. thesis, Department of Computer Science, University of Edinburgh, 1994.), 1996.

....classifier) monad becomes a KZ doctrine. 1 Introduction One of the concerns of axiomatic domain theory , and a particular concern of this paper, is the analysis of notions of approximation aiming at explaining and justifying (order theoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and order enrichment we considered contextual approximation which, in the framework we were working in, coincided with the specialisation preorder . But in the applications carried out in [FP94, Fio94a] we had to work with an axiomatised notion ....

....of categories of domains. For example, in [Fio94c, Fio94a] while studying the interaction between partiality and order enrichment we considered contextual approximation which, in the framework we were working in, coincided with the specialisation preorder . But in the applications carried out in [FP94, Fio94a] we had to work with an axiomatised notion of approximation, instead of the aforementioned one, for the following two reasons: first, the specialisation preorder is not appropriate in categories of domains and stable functions (see [Fio94c] and, second, we do not know of non order theoretic ....

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M. P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994. (Available as technical report ECS-LFCS-94-307 or from http://www.dcs.ed.ac.uk/home/mf/thesis.dvi.Z).

....In particular, Paul Taylor [30] investigated the limit colimit coincidence for categories with filtered colimits. In some respects his work is a precursor to ours; however, we take a step further and develop an axiomatic theory in accordance with the approach to Axiomatic Domain Theory adopted in [6, 24, 9, 10]. Conceptually, the categorical theory of domains that we put forward may be seen as the traditional theory of Smyth and Plotkin [28] where cpos ( complete partial orders) are replaced with their categorical analogue (viz. small categories with colimits of chains) Technically, this 1 is not ....

....Rel (the category of sets and relations, with hom sets ordered by inclusion) pCpo (the category of cpos and partial continuous functions, with hom sets ordered pointwise) Prof , and Prof M . From Theorem 2.1, Corollary 2.2, and Lemma 3. 1, we can deduce pseudoalgebraic compactness (see [8, 6]) Corollary 3.2 (Pseudo algebraic compactness) Kcats are pseudo algebraically compact with respect to pseudo Cat functors. Thus, every pseudo Cat functor T : K Theta K K on a Kcat K has a free pseudo dialgebra T (D; D) D characterised by the following universal property: for every ....

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M. P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. CUP,

.... the endofunctor BS = 1 # f (S) V ) C # f (V S) C # f (S) which we will consider below in the following uniform form BS # = C# # # f (S) V ) C# # # f (V S) 1# # # f (S) 23) where # # : pSet op pSet ## Set is the partialexponential functor (see e.g. [11]) In this setting, a coalgebra h : S ## BS induces the early transition relation s c #v# ## s # iff s # # # 1 (hx) c) v) c # C, v # V, s, s # # S) s c #v# ## s # iff (v, s # ) # # 2 (hx) c) c # C, v # V, s, s # # S) s # ## s # iff s # # # 3 (hx) s, s # # S) ....

.... For a mono preserving presheaf P : I ## Set we define P# # : Set I ## Set I as the functor mapping a presheaf Q to the presheaf P # #Q with action given by (P # #Q)n = Pn # #Qn and (P # #Q) #) P (#)# #Q(#) u # ## Q(#) # u # P (#) R where P (#) R (q) p iff P (#) p) q (see [11]) This construction extends that of products in that we have an injection P Q ## ## P# #Q given by: Pn Qn ## ## Pn # #Qn p, q # ## p R # #q (25) where (p R # #q) x) if x = p then q) In the vein of the treatment of early bisimulation for value passing CCS given in (23) we ....

M. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations Series. Cambridge University Press, 1996.

.... been customary to interpret recursive types as initial algebras (the notion dual to that of final coalgebra) We remark that in certain categories of domains this traditional view and the viewpoint adopted here coincide for initial algebras and final coalgebras are canonically isomorphic (see e.g. [Fre90, Smy91, Fre92, FP92, Fio94]) The mysterious definition of bisimulation in our example is an instance of an abstract notion of bisimulation on a coalgebra (taken from [AM89] motivated by concurrency theory. This has two important methodological consequences. 1. We are able to provide bisimulations for recursive data ....

....result for the canonical model of the untyped call by value calculus. This section is technically involved; in particular the technique used to solve recursive type equations of mixed variance functors, namely via algebraic compactness, is merely sketched (for a thorough treatment consult [Fio94, Chapters 6 and 7]) Finally, in Section 9 we indicate plans for future work. There are some relationships between this and the work of [RT93] and [Pit94, Pit93] In [RT93] Rutten and Turi study final semantics and strong extensionality for the categories of non well founded sets, complete metric spaces and ....

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M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994. To be published by Cambridge University Press in the Distinguished Dissertations Series.

.... due to Peter Freyd [5,3,4] In particular, after his insight of considering canonical maps from initial algebras to final coalgebras, the main infinitary axiom of SDT, see [8] reads as follows: L1) c is an iso, where 1 is a terminal object, L is (the underlying endofunctor of) a lifting monad [8,2] internalizing partial 1 Research supported by EC project Programming Language Semantics and Program Logics grant SC1000 795. 2 Research supported by EC project no.6811, CLICS II, and from MURST 40 . Elsevier Preprint computations, and c is the canonical map from the initial L algebra to the ....

....a representation. We shall denote by L the action of the representation functor on H which underlies the so called lifting monad. The above abstract construction extends a well known behaviour on the category Cpo. Indeed, the notion of partial map is that expected from partial computations, see [2]. In Cpo one can consider partial maps defined on a Scott open subset. A Scott open subset is a monomorphism in Cpo which reflects the order and whose image is upward closed and inaccessible by sups of chains. The representing functor adds an element below the given order. The lifting monad ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge University Press Distinguished Dissertations in Computer Science, 1996.

....and of the well complete ones. 1 Basic concepts Lifting monad. A monad (L; j; internalizing possibly non terminating computations is usually called lifting and it is axiomatized by requiring that the unit j classifies (certain) partial maps 1 that are closed under composition; see e.g. [21, 4]. Precisely, in a category with terminal object 1, the conditions are that the naturality diagrams for the unit j are pullbacks (i.e. the natural transformation j is cartesian) and that, in the situation 1 pb fflffl j 1 fflffl D fflffl fflffl oo B L1 A , there exists a unique classifying ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge University Press Distinguished Dissertations in Computer Science, 1996.

....a first step, we concentrate on the enrichment of models of ADT. The intention is that the enriched Yoneda Grothendieck Dedekind Cayley embedding [27] will provide the desired representation (c.f. 15, 11] Axiomatic versions of various traditional results in domain theory can be found in e.g. [39, 16, 17, 38, 13, 9, 11, 32]. For instance, in [39] the crucial role Research supported by EPSRC grant GR J84205, Frameworks for Programming Language Semantics and Logic. y Research supported by an EPSRC Senior Fellowship. of Cpo enrichment in the solution of recursive domain equations was recognised and made the central ....

....model of linear type theory such that its induced commutative monad is domain theoretic. A domain theoretic model of recursive types with respect to a domain theoretic enrichment base F : C Gamma Gamma D : U is a D category M such that the C category U M is C algebraically compact (see [17, 9, 32] for this notion) The role of algebraic compactness is to provide a universal approach for solving recursive type equations. What is domaintheoretic about these models is that fixed points of endofunctors (viz. free algebras) are obtained by traditional domain theoretic methods; viz. the basic ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge University Press Distinguished Dissertations in Computer Science, 1996.

....is proposed. It would be interesting to find natural axioms on partiality which would yield an (absolute) domain theoretic model. Such axioms would provide a computational justification of Scott s original consideration of ordered structures. For an initial investigation in this direction see [Fio94]. A related possibility is to generalise from order enrichment to general enrichment. The first problem there is to provide a good notion of V domain structure; any V category with such a structure should yield a V category of partial maps. As to adequacy, we believe that if Lemma 7.11 ....

.... consult [Kel82] In the rest of the paper we let V stand for either Poset (the category of small posets and monotone functions) or Cpo (the category of small cpos and continuous functions) But we stress that all our V notions with their associated results generalise to arbitrary cartesian V (see [Fio94]) A Poset category (Cpo category) is a locally small category whose hom sets come equipped with a partial order (complete partial order) with respect to which composition of morphisms is a monotone (continuous) operation. Both Poset and Cpo with each hom set ordered pointwise are examples of ....

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M.P. Fiore. Axiomatic domain theory in categories of partial maps. Forthcoming Ph.D. thesis, 1994.

....of types. Those ideas are the origin of the definition of types as functors that we have given. The treatment of type constructors as functors is difficult because of contravariance. A solution is to take the category of types to be involutory (i.e. isomorphic to its dual via an involution) [8]. Given a category C there are various approaches to constructing an involutory category I(C) In the context of orderenriched categories one can exploit the duality embedding projection [21] There are also canonical choices, namely C = C op Theta C [11] or its subcategory C D [1, 10] ....

M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994. To be published by Cambridge University Press in the Distinguished Dissertations Series.

No context found.

M. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertation Series, Cambridge University Press, 1996.

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Marcelo P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science. Cambridge University Press, 1996.

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Marcelo P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Distinguished Dissertations in Computer Science. Cambridge University Press, 1996.

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M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge Univ. Press, 1996.

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M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge Univ. Press, 1996.

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M.P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. Cambridge University Press Distinguished Dissertations in Computer Science, 1996.

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