The Z Notation: Whence the Cause and Whither the Course?

Abstract

Abstract. The Z notation for the formal specification of computer-based systems has been in existence since the early 1980s. Since then, an international Z community has emerged, academic and industrial courses have been developed, an ISO standard has been adopted, and Z has been used on a number of significant software development projects, especially where safety and security have been important. This chapter traces the history of the Z notation and presents issues in teaching Z, with examples. A specific example of an industrial course is presented. Although subsequent notations have been developed, with better tool support, Z is still an excellent choice for general purpose specification and is especially useful in directing software testing to ensure good coverage. Whence the Cause?"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity." -Alan M. Turing The Purpose of Ordinal Logics (1938) HistoryThe computing pioneer Alan Turing [13,22] and the physicist Albert Einstein (1879Einstein ( -1955, who died only a year after Turing although after a considerably longer lifespan, are two of the western scientists who are celebrated with busts on the campus of Southwestern University in Chongqing, China (see Figure 1). Einstein is of course extremely well-known internationally, but Turing has been less well-known until more recently when his achievements have become increasingly visible to the public. He is considered by many to be the founding father of modern computer science with his 1936/7 paper on the Entscheidungsproblem ("decision problem" in German), establishing what is computable through the theoretical computing device that has become known as a Turing Machine [42]. This is an abstract model of a computing machine that is useful for demonstrating what is and what is not computable. This astoundingly novel approach has had a profound effect on the theory of computation subsequently, effectively founding the field of theoretical computer science. Turing produced what is considered by many as the first paper on proving a program correct [43]. However, sadly this short paper had little impact on the development of formality in computing until it was rediscovered and evaluated in its historical context much later [34]. Instead, Tony Hoare's much later 1969 paper on an axiomatic basis for computer programming, introducing Hoare logic and assertions, was a turning point in providing a formal approach to program proving [29].Formal methods [18] emerged during the 1970s as a mathematically-based approach to software development. Jean-Raymond Abrial's 1974 paper on data semantics [1] was in hindsight a seminar paper leading to the development of the Z notation for the formal 2 specification of computer-based systems. In the early 1980s, Abrial visited the Programming Research Group (PRG) in the Oxford University Computing Laboratory (OUCL) and sowed the seeds for the development of the Z notation there. A Z course...

The Impact of Alan Turing: Formal Methods and Beyond

“…The following convention is used: (i) a dashed letter indicates the value at the end of the process represented by the box; (ii) an undashed letter represents the initial value of a quantity. This could be describing an operation schema box in the modern-day Z notation [28]. Further work on Turing's 1949 paper and its impact has been undertaken more recently by Cliff Jones [104,105,106].…”

“…The paper Laws of Programming presents some general algebraic laws for imperative programming languages [94] on Programming from Specifications [121] explicitly depends on [68,75,93], as stated in the book's preface. Jean-Raymond Abrial (born 1938), progenitor of the Z notation [28], later produced The B-Book: Assigning Programs to Meanings [1] on the B-Method to derive a program from a formal specification in a rigorous manner with tool support.…”

Alan Turing: Founder of Computer Science

“…Turing worked on software for the Mark 1 and even produced what may have been the first proof of correctness for a program [51]. Although this had little influence at the time, its importance in the field of formal methods [6,15] was realised much later [11,41].…”