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Z notation

Updated: 2017-06-26T05:49Z
An example of a formal specification (in Spanish) using the Z notation.

The Z notation /ˈzɛd/ 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.


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.[citation needed] 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.

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 catalogue (called the mathematical toolkit) of commonly used mathematical functions and predicates, defined using Z itself.

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. There are also Unicode encodings for all standard Z symbols.


ISO completed a Z standardization effort in 2002. This standard[5] and a technical corrigendum[6] are available from ISO for free:

  • the standard is publicly available[5] from the ISO ITTF site free of charge and, separately, available for purchase[5] from the ISO site;
  • the technical corrigendum is available[6] from the ISO site free of charge.


See also


  1. ^ Abrial, Jean-Raymond (1974), "Data Semantics", in Klimbie, J. W.; Koffeman, K. L., Proceedings of the IFIP Working Conference on Data Base Management, North-Holland, pp. 1–59 
  2. ^ Meyer, Bertrand; Baudoin, Claude (1980), Méthodes de programmation (in French), Eyrolles 
  3. ^ Abrial, Jean-Raymond; Schuman, Stephen A; Meyer, Bertrand (1980), "A Specification Language", in Macnaghten, A. M.; McKeag, R. M., On the Construction of Programs, Cambridge University Press, ISBN 0-521-23090-X  (describes early version of the language).
  4. ^ Hoogeboom, Hendrik Jan. "Formal Methods in Software Engineering" (PDF). The Netherland: University of Leiden. Retrieved 14 April 2017. 
  5. ^ a b c "ISO/IEC 13568:2002". Information Technology — Z Formal Specification Notation — Syntax, Type System and Semantics (Zipped PDF). ISO. 2002-07-01.  196 pp.
  6. ^ a b "ISO/IEC 13568:2002/Cor.1:2007". Information Technology — Z Formal Specification Notation — Syntax, Type System and Semantics — Technical corrigendum 1 (PDF). ISO. 2007-07-15.  12 pp.

Further reading

External links

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