Sheaves, local homeomorphisms and local sets as Hilbert locale modules

We give a thorough account of the various equivalent notions for \sheaf" on a locale, namely the separated and complete presheaves, the local home- omorphisms, and the local sets, and to provide a new approach based on quantale modules whereby we see that sheaves can be identi¯ed with certain Hilbert modules in the sense of Paseka. This formulation provides us with an interesting category that has immediate meaningful relations to those of sheaves, local homeomorphisms and local sets. The concept of B-set (local set over the locale B) present in [3] is seen as a simetric idempotent matrix with entries on B, and a map of B-sets as de¯ned in [8] is shown to be also a matrix satisfying some conditions. This gives us useful tools that permit the algebraic manipulation of B-sets. The main result is to show that the existing notions of \sheaf" on a locale B are also equivalent to a new concept what we call a Hilbert module with an Hilbert base. These modules are the projective modules since they are the image of a free module by a idempotent automorphism On the ¯rst chapter, we recall some well known results about partially ordered sets and lattices. On chapter two we introduce the category of Sup-lattices, and the cate- gory of locales, Loc. We describe the adjunction between this category and the category Top of topological spaces whose restriction to spacial locales give us a duality between this category and the category of sober spaces. We ¯nish this chapter with the de¯nitions of module over a quantale and Hilbert Module. Chapter three concerns with various equivalent notions namely: sheaves of sets, local homeomorphisms and local sets (projection matrices with entries on a locale). We ¯nish giving a direct algebraic proof that each local set is isomorphic to a complete local set, whose rows correspond to the singletons. On chapter four we de¯ne B-locale, study open maps and local homeo- morphims. The main new result is on the ¯fth chapter where we de¯ne the Hilbert modules and Hilbert modules with an Hilbert and show this latter concept is equivalent to the previous notions of sheaf over a locale.

JML- Based formal development of a Java card application for managing medical appointments

Although formal methods can dramatically increase the quality of software systems, they have not widely been adopted in software industry. Many software companies have the perception that formal methods are not cost-effective cause they are plenty of mathematical symbols that are difficult for non-experts to assimilate. The Java Modelling Language (short for JML) Section 3.3 is an academic initiative towards the development of a common formal specification language for Java programs, and the implementation of tools to check program correctness. This master thesis work shows how JML based formal methods can be used to formally develop a privacy sensitive Java application. This is a smart card application for managing medical appointments. The application is named HealthCard. We follow the software development strategy introduced by João Pestana, presented in Section 3.4. Our work influenced the development of this strategy by providing hands-on insight on challenges related to development of a privacy sensitive application in Java. Pestana’s strategy is based on a three-step evolution strategy of software specifications, from informal ones, through semiformal ones, to JML formal specifications. We further prove that this strategy can be automated by implementing a tool that generates JML formal specifications from a welldefined subset of informal software specifications. Hence, our work proves that JML-based formal methods techniques are cost-effective, and that they can be made popular in software industry. Although formal methods are not popular in many software development companies, we endeavour to integrate formal methods to general software practices. We hope our work can contribute to a better acceptance of mathematical based formalisms and tools used by software engineers. The structure of this document is as follows. In Section 2, we describe the preliminaries of this thesis work. We make an introduction to the application for managing medical applications we have implemented. We also describe the technologies used in the development of the application. This section further illustrates the Java Card Remote Method Invocation communication model used in the medical application for the client and server applications. Section 3 introduces software correctness, including the design by contract and the concept of contract in JML. Section 4 presents the design structure of the application. Section 5 shows the implementation of the HealthCard. Section 6 describes how the HealthCard is verified and validated using JML formal methods tools. Section 7 includes some metrics of the HealthCard implementation and specification. Section 8 presents a short example of how a client-side of a smart card application can be implemented while respecting formal specifications. Section 9 describes a prototype tools to generate JML formal specifications from informal specifications automatically. Section 10 describes some challenges and main ideas came acrorss during the development of the HealthCard. The full formal specification and implementation of the HealthCard smart card application presented in this document can be reached at https://sourceforge.net/projects/healthcard/.