Abstract
There is a need for a language able to reconcile the recent upsurge of interest in quantitative methods in the software sciences with logic and set theory that have been used for so many years in capturing the qualitative aspects of the same body of knowledge. Such a lingua franca should be typed, polymorphic, diagrammatic, calculational and easy to blend with traditional notation. This paper puts forward typed linear algebra (LA) as a candidate notation for such a role. Typed LA emerges from regarding matrices as morphisms of suitable categories whereby traditional linear algebra is equipped with a type system. In this paper we show typed LA at work in describing weighted (probabilistic) automata. Some attention is paid to the interface between the index-free language of matrix combinators and the corresponding index-wise notation, so as to blend with traditional set theoretic notation. © 2012 Springer-Verlag.

Abstract
In a recent article [1], David Parnas questions the traditional use of formal methods in software development, which he considers an underdeveloped body of knowledge and therefore of little hope for the software industry. He confronts the reader with the following statement, at some stage: "We must learn to use mathematics in software development, but we need to question, and be prepared to discard, most of the methods that we have been discussing and promoting for all these years." At the core of Parnas objections we find the contrast between the current ad-hoc (re)invention of mathematical concepts which are cumbersome and a burden to use and elegant (and therefore useful) concepts which are neglected, often for cultural or (lack of) background reasons. © 2012 Springer-Verlag.

Abstract
Tool interoperability as a mean to achieve integration is among the main goals of the international Grand Challenge initiative. In the context of the Verifiable file system mini-challenge put forward by Rajeev Joshi and Gerard Holzmann, this paper focuses on the integration of different formal methods and tools in modelling and verifying an abstract file system inspired by the Intel Flash File System Core. We combine high-level manual specification and proofs with current state of the art mechanical verification tools into a tool-chain which involves Alloy, VDM++ and HOL. The use of (pointfree) relation modelling provides the glue which binds these tools together. © 2009 Springer-Verlag Berlin Heidelberg.

Abstract
The pointfree transform offers to the predicate calculus what the La-place transform offers to the differential/integral calculus: the possibility of changing the underlying mathematical space so as to enable agile algebraic calculation. This paper addresses the foundations of the transform and its application to a calculational approach to extended static checking (ESC) in the context of abstract modeling. In particular, a calculus is given whose rules help in breaking the complexity of the proof obligations involved in static checking arguments. The close connection between such calculus and that of weakest pre-conditions makes it possible to use the latter in ESC proof obligation discharge, where pointfree notation is again used, this time to calculate with invariant properties to be maintained. A connection with the "everything is a relation" lemma of Alloy is established, showing how close to each other the pointfree and Alloy notations are. The main advantage of this connection is that of complementing pen-and-paper pointfree calculations with model checking support wherever validating sizable abstract models.

Abstract
Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to software systems. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as types. An outcome of such a theory is a calculus of invariants' proof obligation discharge, a fragment of which is presented in the paper. The approach has two main ingredients: one is that of adopting relations as "first class citizens" in a pointfree reasoning style; the other lies on a synergy found between a relational construct, Reynolds' relation on functions involved in the abstraction theorem on parametric polymorphism and the coalgebraic account of bisimulations and invariants. This leads to an elegant proof of the equivalence between two different definitions of bisimulation found in coalgebra literature (due to B. Jacobs and Aczel & Mendler, respectively) and to their instantiation to the classical Park-Milner definition popular in process algebra.

Abstract
Galculator is the name of the prototype of a proof assistant of a special brand: it is solely based on the algebra of Galois connections. When combined with the pointfree transform and tactics such as the indirect equality principle, Galois connections offer a very powerful, generic device to tackle the complexity of proofs in program verification. The paper describes the architecture of the current Calculator prototype, which is implemented in Haskell in order to steer types as much as possible. The prospect of integrating the Galculator with other proof assistants such as e.g. Coq is also discussed. Copyright © 2008 ACM.