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

Wetting Phase Transition at the Surface of Liquid Ga-Bi alloys: An X-ray Reflectivity Study

X-ray reflectivity measurements of the binary liquid Ga-Bi alloy reveal a dramatically different surface structure above and below the monotectic temperature $T_{mono}=222^{\circ}$ C. A Gibbs-adsorbed Bi monolayer resides at the surface at both regimes. However, a 30 {\AA} thick, Bi-rich wetting film intrudes between the Bi monolayer and the Ga-rich bulk for $T > T_{mono}$. The internal structure of the wetting film is determined with {\AA} resolution, showing a theoretically unexpected concentration gradient and a highly diffuse interface with the bulk phase.Comment: 5 RevTex pages, 3 figures, Phys. Rev. Let

X-ray study of the liquid potassium surface: structure and capillary wave excitations

We present x-ray reflectivity and diffuse scattering measurements from the liquid surface of pure potassium. They strongly suggest the existence of atomic layering at the free surface of a pure liquid metal with low surface tension. Prior to this study, layering was observed only for metals like Ga, In and Hg, the surface tensions of which are 5-7 fold higher than that of potassium, and hence closer to inducing an ideal "hard wall" boundary condition. The experimental result requires quantitative analysis of the contribution to the surface scattering from thermally excited capillary waves. Our measurements confirm the predicted form for the differential cross section for diffuse scattering, $d\sigma /d\Omega \sim 1/q_{xy}^{2-\eta}$ where $\eta = k_BT q_z^2/2\pi \gamma$, over a range of $\eta$ and $q_{xy}$ that is larger than any previous measurement. The partial measure of the surface structure factor that we obtained agrees with computer simulations and theoretical predictions.Comment: 7 pages, 7 figures; published in Phys. Rev.

Surface Crystallization in a Liquid AuSi Alloy

X-ray measurements reveal a crystalline monolayer at the surface of the eutectic liquid Au_{82}Si_{18}, at temperatures above the alloy's melting point. Surface-induced atomic layering, the hallmark of liquid metals, is also found below the crystalline monolayer. The layering depth, however, is threefold greater than that of all liquid metals studied to date. The crystallinity of the surface monolayer is notable, considering that AuSi does not form stable bulk crystalline phases at any concentration and temperature and that no crystalline surface phase has been detected thus far in any pure liquid metal or nondilute alloy. These results are discussed in relation to recently suggested models of amorphous alloys.Comment: 12 pages, 3 figures, published in Science (2006

Pairing Interactions and Gibbs Adsorption at the Liquid Bi-In Surface: A Resonant X-Ray Reflectivity Study

Resonant x-ray reflectivity measurements from the surface of liquid Bi22In78 find only a modest surface Bi enhancement, with 35 atomic % Bi in the first atomic layer. This is in contrast to the Gibbs adsorption in all liquid alloys studied to date, which show surface segregation of a complete monolayer of the low surface tension component. This suggests that surface adsorption in Bi-In is dominated by attractive interactions that increase the number of Bi-In neighbors at the surface. These are the first measurements in which resonant x-ray scattering has been used to quantify compositional changes induced at a liquid alloy surface.Comment: 11 pages, 2 figures, 2 tables, published in Phys. Rev. Let