Z notation

In 1974, Jean-Raymond Abrial published "Data Semantics".[1] He used a notation that would later be taught in the University of Grenoble until the end of the 1980s. While at EDF (Électricité de France), Abrial wrote internal notes on Z. The Z notation is used in the 1980 book Méthodes de programmation.[2]

Z was originally proposed by Abrial in 1977 with the help of Steve Schuman and Bertrand Meyer.[3] It was developed further at the Programming Research Group at Oxford University, where Abrial worked in the early 1980s, having arrived at Oxford in September 1979.

Abrial has said that Z is so named "Because it is the ultimate language!"[4] although the name "Zermelo" is also associated with the Z notation through its use of Zermelo–Fraenkel set theory.

Z is based on the standard mathematical notation used in axiomatic set theory, lambda calculus, and first-order predicate logic. All expressions in Z notation are typed, thereby avoiding some of the paradoxes of naive set theory. Z contains a standardized catalogue (called the mathematical toolkit) of commonly used mathematical functions and predicates, defined using Z itself.

Because Z notation (just like the APL language, long before it) uses many non-ASCII symbols, the specification includes suggestions for rendering the Z notation symbols in ASCII and in LaTeX. There are also Unicode encodings for all standard Z symbols.[5]

ISO completed a Z standardization effort in 2002. This standard[6] and a technical corrigendum[7] are available from ISO free:

• the standard is publicly available[6] from the ISO ITTF site free of charge and, separately, available for purchase[6] from the ISO site;
• the technical corrigendum is available[7] from the ISO site free of charge.

• Z User Group (ZUG)
• Community Z Tools (CZT) project
• Other formal methods (and languages using formal specifications):
  • FDM (Formal Development Methodology), revolving around the Ina Jo and Ina Flo specification sub-languages, quite popular in the 1980s and 1990s
  • VDM-SL, the main alternative to Z
  • B-Method, developed by Jean-Raymond Abrial (creator of Z notation)
  • Z++ and Object-Z : object extensions for the Z notation
  • Alloy, a specification language inspired by Z notation and implementing the principles of Object Constraint Language (OCL).
  • Verus, a proprietary tool built by Compion, Champaign, Illinois (later purchased by Motorola), for use in the multi-level secure UNIX project pioneered by its Addamax division.
• Fastest is a model-based testing tool for the Z notation.

• Spivey, John Michael (1992). The Z Notation: A reference manual. International Series in Computer Science (2nd ed.). Prentice Hall.
• Davies, Jim; Woodcock, Jim (1996). Using Z: Specification, Refinement and Proof. International Series in Computer Science. Prentice Hall. ISBN 0-13-948472-8.
• Bowen, Jonathan (1996). Formal Specification and Documentation using Z: A Case Study Approach. International Thomson Computer Press, International Thomson Publishing. ISBN 1-85032-230-9.
• Jacky, Jonathan (1997). The Way of Z: Practical Programming with Formal Methods. Cambridge University Press. ISBN 0-521-55976-6.