Z notation

An example of a formal specification using the Z notation.

The Z notation ( /ˈzɛd/), named after Zermelo–Fraenkel set theory, is a formal specification language used for describing and modelling computing systems. It is targeted at the clear specification of computer programs and computer-based systems in general.

History

In 1974, Jean-Raymond Abrial published "Data Semantics". 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.^{[citation needed]} The Z notation is used in the 1980 book Méthodes de programmation.

Z was originally proposed by Abrial in 1977 with the help of Steve Schuman and Bertrand Meyer. 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 answers the question "Why Z?" with "Because it is the ultimate language!"^{[citation needed]}

Usage and notation

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 catalog (called the mathematical toolkit) of commonly used mathematical functions and predicates.

Although 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.

Standards

The ISO completed a Z standardization effort in 2002. This standard can be obtained directly from ISO. The standard and a technical corrigendum are available from the ISO site.

Tools

• ERZ: Tool for transform ER model to Z Notation equivalent, by Luis Espino
• Community Z Tools (CZT) project
• Tools for developing and checking Z specifications in Microsoft Word
• Mike Spivey's Fuzz Type-Checker for Z
• Z/Eves — A proof checker for the Z notation (German site but all manuals in English)
• Z/EVES Documentation, papers, and manuals on Z/EVES
• ZETA open-source system for development software specifications in Z
• HOL-Z open-source proof environment for Z in Isabelle/HOL
• CADiZ, a set of free software tools that assist use of Z notation
• ProofPower, a suite of open-source tools supporting specification and proof in HOL and in the Z notation

See also

• Z User Group (ZUG)
• Community Z Tools (CZT) project
• Other formal methods (and languages using formal specifications)  :
  • Z++ and Object-Z : object extensions for the Z notation
  • Abstract Machine Notation (AMN), used in B-Method
  • Alloy, a specification language inspired by Z notation and implementing the principles of Object Constraint Language (OCL).
• Fastest is a model-based testing tool for the Z notation.

Further reading

• J. Michael Spivey (1992). The Z Notation: A reference manual (2nd edition ed.). Prentice Hall International Series in Computer Science. http://spivey.oriel.ox.ac.uk/mike/zrm/. 
• Jim Davies and Jim Woodcock (1996). Using Z: Specification, Refinement and Proof. Prentice Hall International Series in Computer Science. ISBN 0-13-948472-8. http://www.usingz.com/text/online/. 
• Jonathan Bowen (1996). Formal Specification and Documentation using Z: A Case Study Approach. International Thomson Computer Press. ISBN 1-85032-230-9. http://www.zuser.org/zbook. 
• Jonathan Jacky (1997). The Way of Z: Practical Programming with Formal Methods. Cambridge University Press. ISBN 0-521-55976-6. http://staff.washington.edu/jon/z-book/. 

External links

General

• The World Wide Web Virtual Library: The Z notation, by Jonathan Bowen
• Specification proposals by Ian Toyn
• W3C WSDL 2.0 a specification containing Z notation assertions and explanation