Semantic Domains and Denotational Semantics

The theory of domains was established in order to have appropriate spaces on which to define semantic functions for the denotational approach to programming-language semantics. There were two needs: first, there had to be spaces of several different types available to mirror both the type distinctions in the languages and also to allow for different kinds of semantical constructs - especially in dealing with languages with side effects; and second, the theory had to account for computability properties of functions - if the theory was going to be realistic. The first need is complicated by the fact that types can be both compound (or made up from other types) and recursive (or self-referential), and that a high-level language of types and a suitable semantics of types is required to explain what is going on. The second need is complicated by these complications of the semantical definitions and the fact that it has to be checked that the level of abstraction reached still allows a precise definition of computability

The Mixed Powerdomain

This paper characterizes the powerdomain constructions which have been used in the semantics of programming languages in terms of formulas of first order logic under a preordering of provable implication. The goal is to reveal the basic logical significance of the powerdomain elements by casting them in the right setting. Such a treatment may contribute to a better understanding of their potential uses in areas which deal with concepts of sets and partial information such as databases and computational linguistics. This way of viewing powerdomain elements suggests a new form of powerdomain - called the mixed powerdomain - which expresses data in a different way from the well-known constructions from programming semantics. It is shown that the mixed powerdomain has many of the properties associated with the convex powerdomain such as the possiblity of solving recursive equations and a simple algebraic characterization

Retraction map categories and their applications to the construction of lambda calculus models

This paper deals with categorical models of the λ-calculus. We generalize the inverse limit method Scott used for his construction of D∞, and introduce order-enriched ccc's, retraction map categories and ɛ-categories. An order-enriched ccc is a cartesian closed category C equipped with a partial order relation ⩽ on the set of the arrows. A retraction map category of C is R=(R, ⩽, i, j), where ⩽ is a partial order relation on the set |C| of all the objects of C, R is the category of the poset (|C|, ⩽), and i and j are functors from R to C and from Rop to C that satisfy the conditions: (1) j a, b ∘ i a, b ⩾ ida and (2) i a, b ∘ j a, b ⩽ idb for every arrow a, b: a → b in R (i.e., a⩽b). The ɛ-category E=E(C, R) of C w.r.t. R is the category whose objects are ideals of (|C|, ⩽) and whose arrows are ideals of (C, ⊑), where ⩽ is the partial order relation in R and ⊑ is the partial order relation defined by f ⊑ g iff dom(f)⩽dom(g), cod(f)⩽cod(g) in R and f⩽j a, b ∘ g ∘ i(a, b in C. We show that every ɛ-category E=E(C, R) is also an order-enriched ccc. Moreover when E and R satisfy a particular condition, E(C, R) has a reflexive object. For example, if there is an ideal U of (|C|, ⩽) satisfying the following conditions, then U is isomorphic to UU in E and a λ-algebra is constructed from E and U: (1) for every pair of a, b ∈ U, U contains ba, and (2) for every c ∈ U, there are a, b ∈ U such that c ∈ ba. We reconstruct Pω and D∞ using ɛ-categories

The Mixed Powerdomain

This paper introduces an operator M called the mixed powerdomain which generalizes the convex (Plotkin) powerdomain. The construction is based on the idea of representing partial information about a set of data items using a pair of sets, one representing partial information in the manner of the upper (Smyth) powerdomain and the other in the manner of the lower (Hoare) powerdomain where the components of such pairs are required to satisfy a consistency condition. This provides a richer family of meaningful partial descriptions than are available in the convex powerdomain and also makes it possible to include the empty set in a satisfactory way. The new construct is given a rigorous mathematical treatment like that which has been applied to the known powerdomains. It is proved that M is a continuous functor on bifinite domains which is left adjoint to the forgetful functor from a category of continuous structures called mix algebras. For a domain D with a coherent Scott topology, elements of M D can be represented as pairs (U, V) where U ⊆ D is a compact upper set, V ⊆ D is a closed set and the downward closure of U ∩ V is equal to V. A Stone dual characterization of M is also provided

Developing new communities: the progress of three private sector new settlements 1960-1993

In the light of increasing interest in new settlements through the 1980s and early 1990s, this thesis examines the development of three existing private sector new settlements. The study examines the development of three case study sites, East Goscote in Leicestershire, Bar Hill in Cambridgeshire and Martlesham Heath in Suffolk. It assesses what new light the development of such sites can shed upon existing understandings of the development of place and community, and the changing nature of urban form in the context of the modem to post-modem transition noted by geographers over the past thirty years. It also aims to place new settlements in a wider historical context, examining the theoretical assumptions they inherit from earlier attempts to create new places, notably the garden cities movement, and the state new towns programme. The case studies examine the origins of each of the three sites, and follow their physical development to the present day, analysing what events, personalities and assumptions shaped their final form. They address the extent to which the development of each site was a response to local contingencies, or to wider forces, and draw out both the similarities and the differences between each site. The study also looks at the way in which community organisations have developed in each of the three sites, and to what extent each community has developed a sense of its own identity and cohesion. In doing so, it determines the extent to which concepts of place and community are relevant in the context of new settlements, and in the context of contemporary urban forms. The thesis illustrates that these three new settlements were primarily local, contingent responses, but also indicates that there are common patterns to their growth. The study also shows the considerable similarities new settlements share with early garden cities, and the extent to which they were also affected by the state new towns programme. In addition, it illustrates that notions of place identity and community are shaped by a small number of individuals, and concludes that such concepts remain valid, though subject to constant change and renegotiation