M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as large as possible, so to be closed under various domain theoretic constructions. Gordon Plotkin defined in [Plo76] SFP domains and conjectured that the category of algebraic SFP domains is the largest cartesian closed category of domains. This conjecture was confirmed by Mike Smyth in [Smy83]. In [Jun88, Jun89] Achim Jung dropped the countability condition and showed that there are two maximal cartesian closed categories of algebraic domains, bifinite domains and L domains. Moreover, he proved that every cartesian closed category of algebraic domains is contained in one of these. ....

M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983. 3

....products, and exponents. The only problematic part for domains is the exponent, which in this setting means the space of continuous functions. Cartesian closed categories of domains are well understood and the understanding is in some sense essentially complete by the work of Jung [5] Smyth [11], and others. In this paper we consider categories of e#ective domains. Again the function space is the crucial construction in order to obtain a cartesian closed category and we therefore concentrate on that. The case for e#ective algebraic domains is satisfactory. We introduce a natural notion ....

....algebraic cpos. When restricting to the category of countably based algebraic cpos (i.e. the set of compact elements is countable) the countably based bifinite domains make up the largest cartesian closed full subcategory, i.e. it contains all other cartesian closed full subcategories (see Smyth [11]) A countably based bifinite domain is also called an sfp object, indicating that it is the limit of a sequence of finite partial orders. To motivate our definition let (P ; #) be a partial order and let us consider # as an information ordering. Suppose A # P . When is A su#ciently ....

M. B. Smyth, The largest cartesian closed category of domains, Theoretical Computer Science 27 (1983), 109 -- 119.

.... Largest Cartesian Closed Category of Domains, Considered Constructively DIETER SPREEN, Fachbereich Mathematik, Theoretische Informatik, Universitat Siegen, 57068 Siegen, Germany, E mail: spreen informatik.uni siegen.de Abstract A conjecture of Smyth [10] is discussed which says that if D and [D # D] are e#ectively algebraic directedcomplete partial orders with least element (cpo s) then D is an e#ectively strongly algebraic cpo, where it was left open what exactly is meant by an e#ectively algebraic and an e#ectively strongly algebraic cpo. ....

....at the heart of computer science, at least under the viewpoint of developing correct programs. In this respect the results of the paper are relevant to the workshop. Keywords: E#ectively given domains, SFP domains, largest Cartesian closed category of domains 1 Introduction In his seminal paper [10] Smyth showed that the category SFP introduced by Plotkin [8] is the largest Cartesian closed category of domains, thus confirming a conjecture of Plotkin. In this paper we treat Plotkin s conjecture for the case of e#ectively given domains. For various reasons one mostly uses the term domain to ....

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M. B. Smyth, The largest cartesian closed category of domains, Theoret. Comput. Sci. 27 (1983) 109--119.

....by DFG Graduiertenkolleg. 2 R. Kummetz: Posets with Projections and their Morphisms algebraic lattices with Scott continuous mappings as morphisms. Plotkin [13] introduced the cartesian closed category of SFP domains in order to obtain models for non deterministic programming languages. Smyth [15] proved that this category is the largest cartesian closed category of countably based algebraic domains with least element. Bifinite domains (FB domains in [1, 8] are a natural generalization of SFP objects. The conditions that there be a least element and only a countable basis are dropped. ....

M.B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....lattices with Scott continuous mappings as morphisms. Later it turned out that, instead of lattices, only dcpo s are needed for this construction. Plotkin [18] introduced the cartesian closed category of SFP domains, which are pointed domains arising as limits of chains of finite posets. Smyth [20] Supported by DFG Graduiertenkolleg. 2 R. Kummetz: Domains with Approximating Projections showed that SFP domains even form a maximal cartesian closed category. The conditions that there be a least element and a countable basis were dropped later by Gunter [12] and by Jung [13] Extending Smyth s ....

....is a bifinite domain. ii) D; is a P domain and K(D) has property M. Proof: We need only prove (ii) i) Let A K(D) be finite. We show that U 1 (A) is finite. We do this by contradiction and assume that U 1 (A) is infinite. Then we may follow the arguments of Smyth s proof of Lemma 4 in [20], where K onig s Lemma is applied to show that there is a strictly increasing sequence in U 1 (A) in contradiction to Corollary 5.19. Alternatively, to prove the existence of such a sequence we can also use selection functions as done in Jung [13] proof of Lemma 2.2. 2 Plotkin s so called ....

Smyth, M.B.: The largest cartesian closed category of domains. Theor. Comput. Sci. 27 (1983), 109-119.

....is said to have property MW. Property M is familiar from ideas in topology (where it is an ordertheoretic formulation of an important property of compact subsets called coherence [15] and in domain theory (where it is a necessary condition for the bases of domains with good closure properties [14]) Mellish identified these conditions in [8] he too noted the need for finite quasimeets and quasi joins of pairs of elements of P , and by fiat introduced a top and bottom element for P so that the first part of the M and W properties hold. Given an MW poset P , we can now express in greater ....

M. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....of D. Define AlgCpo and ContCpo to be the full categories of Cpo with respectively algebraic cpos and continuous cpos for objects. Neither of these categories is cartesian closed. However, their cartesian closed full subcategories have been extensively studied, and the largest ones identified [9, 4, 5]. Let D and E be cpos. D is a retract of E if there exist continuous f : D E and g : E D such that g ffi f = 1D . If, in addition, f ffi g 1E then f is called an embedding and g its associated projection (each of f and g being determined by the other) If K is a full subcategory of Cpo ....

....then: 1. The cartesian closed structure of K is inherited from Cpo. 2. K has a least fixed point operator, lfp, which is a fix dinatural. 3. RK is cartesian closed. 4. If K is a full subcategory of AlgCpo then RK is a full subcategory of ContCpo. Proof. Statement 1 is (essentially) Lemma 5 of [9]. For 2, we know (from 1) that, for any object D of K, D D is (isomorphic to) D D] so the components of the least fixed point operator are given by lfp D : D D] D. It also follows from 1 that K is a well pointed, Poset enriched cartesian closed category, so the fix dinaturality of lfp ....

M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....can be categorized as the first kind. The second kind of results has a different emphasis. Rather than establishing the suitability of a certain class of domains for semantic modeling, these results show that categories with certain properties just do not exist beyond a limit. The work of Smyth [17], Jung [13] and Huth [12] falls into the second kind. These results have been established by showing that a certain category is maximal, or the largest. Clearly, this paper belongs to the second list. It should be noted that, beyond Scott domains, the category of L domains with stable maps has ....

M.B. Smyth. The largest cartesian closed category of domains, Theoretical Computer Science 27 (1983) 109--120.

....of their categorical definition. They were first defined by Plotkin [Plo76] where they are called SFP objects ) and the term bifinite is due to Paul Taylor. Bifinite domains (and various closely related classes of cpo s) have also been discussed under other names such as strongly algebraic [Smy83a, Gun86] and profinite [Gun87] domains. 6.1 Plotkin orders. As we suggested earlier, the image of a finitary projection p : D D on a domain D can be viewed as an approximation to D. A bifinite domain is one which is a directed limit of its finite approximations. But what is this really saying ....

....about bifinite domains which makes them special. They are the largest class of domains which are closed under the operators listed in the Lemma. In fact, there is the following: Theorem 18 If D and D D are domains, then D is bifinite. The theorem is due to Smyth and its proof may be found in [Smy83a] It is carried out by analyzing each of the cases pictured in Figure 3 and showing that if D D is not a domain, then D cannot be bifinite. A similar result for the bounded complete domains can be found in [Gun86] Semantic Domains 33 7 Recursive definitions of domains. Many of the data ....

M. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....cartesian product is quite pleasing; it is a categorical product in each case. But we can say even more. Lemma 3.2.5. Let C be a full subcategory of DCPO or DCPO which has finite products. Then these are isomorphic to the cartesian product. In a restricted setting this was first observed in [Smyth, 1983a] The general proof may be found in [Jung, 1989] A useful property which does not follow from general categorical or topological considerations, is the following. Lemma 3.2.6. A function f : D Theta E F is continuous if and only if it is continuous in each variable separately. Proof. Assume ....

....there are no other ways of constructing a cartesian closed full subcategory of CONT or ALG than those exhibited in the previous two sections. The idea that such a result could hold originated with Gordon Plotkin, Plotkin, 1981] For the particular class ALG it was verified by Mike Smyth in [Smyth, 1983a] for the other classes by Achim Jung in [Jung, 1988, Jung, 1989, Jung, 1990] All these classification results depend on the Axiom of Choice. 40 Samson Abramsky and Achim Jung 4.3.1 Domains with least element Let us start right away with the crucial bifurcation lemma on which everything else ....

M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

.... Domain Theory [AJ94] It will be shown that the full subcategory CL of continuous lattices is not closed and one of our main results characterizes the largest closed full subcategory of CL (under one extra condition) The result is reminiscent of similar theorems for cartesian closed categories [Smy83, Jun90] it would be very interesting to find a deeper reason for this similarity. 1 In fact, Barr works with infima rather than suprema but this difference is immaterial. From a different perspective, this paper introduces a new model for Classical Linear Logic [Gir87] One the surface of ....

M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....is quite pleasing; it is a categorical product in each case. But we can say even more. Domain Theory 41 Lemma 3.2.5. Let C be a full subcategory of DCPO or DCPO which has finite products. Then these are isomorphic to the cartesian product. In a restricted setting this was first observed in [Smyth, 1983a] The general proof may be found in [Jung, 1989] A useful property which does not follow from general categorical or topological considerations, is the following. Lemma 3.2.6. A function f : D 2E F is continuous if and only if it is continuous in each variable separately. Proof. Assume f : D ....

....there are no other ways of constructing a cartesian closed full subcategory of CONT or ALG than those exhibited in the previous two sections. The idea that such a result could hold originated with Gordon Plotkin, Plotkin, 1981] For the particular class ALG it was verified by Mike Smyth in [Smyth, 1983a] for the other classes by Achim Jung in [Jung, 1988, Jung, 1989, Jung, 1990] All these classification results depend on the Axiom of Choice. 4.3.1 Domains with least element Let us start right away with the crucial bifurcation lemma on which everything else in this section is based. Lemma ....

M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....bounded complete, the latter have property SFP, and SFP implies M. Membership in these domain classes is preserved by sum and product. Exponentiation, i.e. forming the space of continuous functions, only preserves (bounded) completeness and SFP, but neither M nor the property to be a plain domain [11]. The Lawson topology In this section, we introduce the Lawson topology [7, 8] for a fixed underlying M domain D. The closed sets of this topology have many pleasing mathematical properties. They may be used to define the classical power domains. Following Plotkin [8] we define the Lawson ....

Smyth, M.B.: The Largest Cartesian Closed Category of Domains, Theoretical Computer Science 27, (1983), 109-119

....work with when it comes to concrete calculations. Within approximated domains, that is, within the categories CONT and ALG we may look for additional structure to model computational phenomena. Higher types, for example, require cartesian closure. Neither CONT nor ALG is cartesian closed but from [Smy83a] and [Jun90] we know essentially all cartesian closed full subcategories. Among the maximal ones are FS domains (in the continuous case) and bifinite domains (in the algebraic case) see [AJ94b, Chapter 4] Recursive types pose no problem for ALG, CONT, nor for any of its cartesian closed ....

M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....8 Carl A. Gunter 3 Plotkin Orders. In this section we introduce the category of Plotkin orders 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 and simplifies the definitions of some of the operators (such as the powerdomains) which we ....

.... H H H H H H H H H H H H Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Gamma Figure 1: Posets that are not Plotkin orders. property M and the acc is a Plotkin order. A proof of this uses Konig s lemma and can be found in [27]. On the other hand, Proposition 7 Any Plotkin order has property M. Proof. Let A be a poset and suppose u A is finite. If a complete set u 0 of upper bounds of u is finite, then it contains a complete set of minimal upper bounds. If A is a Plotkin order, then there is a finite B A with u ....

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Smyth, M. B., The largest cartesian closed category of domains. Theoretical Computer Science, vol. 27, 1983, pp. 109--119.

....distributive Scott domains; and finally, without any restriction, disjunctive systems give rise to L domains. 1 Introduction Discovered by Coquand [Co90] and Jung [Ju90] independently, L domains form one of the maximal cartesian closed categories of algebraic cpos. Together with the work of Smyth [Sm83] which shows that the SFP domains of Plotkin [Pl76] is the largest cartesian closed category inside the algebraic cpos, we have a better picture of good categories of domains that may be used in denotational semantics of programming languages. The primary contribution of this paper is to give ....

Smyth, M.B., The largest cartesian closed category of domains, Theoretical Computer Science 27, (1983).

....embedding projection pairs respectively. It is possible to show that B is a cartesian closed category and B ep has bilimits of directed families [Gun85, Gun87] Bifinite domains with a countable basis and least element are the SFP objects of Plotkin [Plo76] We will follow Smyth s terminology [Smy83] and refer to them as bifinite domains. We write B for the category with continuous functions and B ep for the category with embedding projection pairs. It is not hard to see that B is a cartesian closed category and B ep has bilimits for countable directed families. 6 Carl A. Gunter ....

....basis. So why not restrict oneself to these The problem is that the L domains with countable basis are not closed under the exponential Consider the poset K pictured in Figure 2. This is an L domain with a countable basis but K K has a basis with continuum many members. Since M. Smyth [Smy83] has proved that any domain which has an algebraic function space is in fact bifinite, it is reasonable to investigate the category BL of bifinite L domains which have countable bases and least elements, i.e. the bifinite L domains. The poset in Figure 2 is a typical example of an L domain ....

M. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....at all those different definitions of domains the novice in the field will naturally ask for some orientation. And indeed, it is possible to give a rather complete overview once the basic assumption is shared that a collection of domains should form a cartesian closed category. Michael Smyth [9] showed 1983 that there is a largest cartesian closed full subcategory in the class of all countably based algebraic dcpo s with least element. In his doctoral thesis [6] the present author completely described all categories of algebraic domains with respect to that criterion of cartesian ....

....cartesian closed if one starts with a cartesian closed category. This immediately gives us a class of continuous domains for each class of algebraic domains. It is then an obvious question whether this will give us the whole variety on the continuous side. As Smyth notes at the end of his paper [9] (this result) does not come out by manipulating retractions . It turned out to be a very hard problem, indeed. The solution, which was partly provided in [6] and is completed here, involves the definition of two new classes of domains: Ldomains and FS domains. We will show below that each ....

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M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

....can be similarly treated. 6 SFP information systems The category of SFP domains with continuous mappings, denoted by SFP, is the largest cartesian closed full subcategory of the category of algebraic cpo s with continuous mappings, and is closed under the three main power functors. See [Plo81, Smy83, Jun88]. Recall that an algebraic cpo, A, is 2 = 3 SFP if for every nite S KA , the set of minimal upperbounds of S, denoted by Mub(S) is nite and complete in the sense that for every upperbound x of S, there exists some y 2 Mub(S) with y v x. Furthermore, A is SFP if, in addition, for every ....

M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109-119, 1983.

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M. B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109--119, 1983.

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M.B. Smyth. The largest cartesian closed category of domains. Theoretical Computer Science, 27:109119, 1983.

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