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.

An example of a formal specification (in Spanish) using the 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.[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 notationEdit

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.


In 1992, “Her Majesty the Queen [was] graciously pleased to approve the Prime Minister’s recommendation that The Queen’s Award for Technological Achievement should be conferred this year upon Oxford University Computing Laboratory. Oxford University Computing Laboratory [gained] the Award jointly with IBM United Kingdom Laboratories Limited for the development of a programming method based on elementary set theory and logic known as the Z notation, and its application in the IBM Customer Information Control System (CICS) product.”[8]

See alsoEdit

  • 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.


  1. ^ Abrial, Jean-Raymond (1974), "Data Semantics", in Klimbie, J. W.; Koffeman, K. L. (eds.), 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. (eds.), 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. ^ Korpela, Jukka K. "Unicode Explained: Internationalize Documents, Programs, and Web Sites". unicode-search.net. Retrieved 24 March 2020.
  6. ^ a b c "ISO/IEC 13568:2002". Information Technology — Z Formal Specification Notation — Syntax, Type System and Semantics (Zipped PDF). ISO. 1 July 2002. 196 pp.
  7. ^ 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. 15 July 2007. 12 pp.
  8. ^ "The Queen's Award for Technological Achievement 1992". Oxford University Computing Laboratory. Archived from the original on 22 March 2012. Retrieved 17 October 2021.

Further readingEdit