Revising Z: part II - logical development

This is the second of two related papers. In "Revising Z: Part I - logic and semantics" (this journal) we introduced a simple specification logic ZC comprising a logic and a semantics (in ZF set theory). We then provided an interpretation for (a rational reconstruction of) the specification language Z within ZC. As a result we obtained a sound logic for Z, including the basic schema calculus. In this paper we extend the basic framework with more sophisticated features (including schema operations) and we mount a critique of a number of concepts used in Z. We further demonstrate that the complications and confusions which these concepts introduce can be avoided without compromising expressibility

Revising Z: part I - logic and semantics

This is the first of two related papers. We introduce a simple specification logic ZC comprising a logic and a semantics (in ZF set theory) within which the logic is sound. We then provide an interpretation for (a rational reconstruction of) the specification language Z within ZC. As a result we obtain a sound logic for Z, including a basic schema calculus

Asymptotically Hilbertian Modular Banach Spaces: Examples of Uncountable Categoricity

We give a criterion ensuring that the elementary class of a modular Banach space E (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of E) consists of all direct sums E\oplus_m H, where H is an arbitrary Hilbert space and \oplus_m denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures.Comment: 20 page

Generic orbits and type isolation in the Gurarij space

We study the question of when the space of embeddings of a separable Banach space $E$ into the separable Gurarij space $\mathbf G$ admits a generic orbit under the action of the linear isometry group of $\mathbf G$. The question is recast in model-theoretic terms, namely type isolation and the existence of prime models. We characterise isolated types over $E$ using tools from convex analysis. We show that if the set of isolated types over $E$ is dense, then a dense $G\_\delta$ orbit exists, and otherwise all orbits are meagre. We then study some (families of) examples with respect to this dichotomy. We also point out that the class of Gurarij spaces is the class of models of an $\aleph\_0$-categorical theory with quantifier elimination, and calculate the density character of the space of types over $E$, answering a question of Avil{\'e}s et al

Results on formal stepwise design in Z

Stepwise design involves the process of deriving a concrete model of a software system from a given abstract one. This process is sometimes known as refinement. There are numerous refinement theories proposed in the literature, each of which stipulates the nature of the relationship between an abstract specification and its concrete counterpart. This paper considers six refinement theories in Z that have been proposed by various people over the years. However, no systematic investigation of these theories, or results on the relationships between them, have been presented or published before. This paper shows that these theories fall into two important categories and proves that the theories in each category are equivalent

An analysis of total correctness refinement models for partial relation semantics I

This is the first of a series of papers devoted to the thorough investigation of (total correctness) refinement based on an underlying partial relational model. In this paper we restrict attention to operation refinement. We explore four theories of refinement based on an underlying partial relation model for specifications, and we show that they are all equivalent. This, in particular, sheds some light on the relational completion operator (lifted-totalisation) due to Wookcock which underlines data refinement in, for example, the specification language Z. It further leads to two simple alternative models which are also equivalent to the others