# 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.

## Contents

## History[edit]

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[edit]

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.

## Standards[edit]

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.

## Tools[edit]

*Community Z Tools (CZT)*(project), Source forge*Z Word tools*(project), Source forge for developing and checking Z specifications in Microsoft Word- Fuzz, a 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
- z-vimes Z-Vimes: type checker and (eventually) theorem prover for the Z specification language
- ProB is an animator and model checker originally written for the B-Method that provides also support for Z specifications ("ProZ") that conform to the Fuzz type checker

## See also[edit]

- Z User Group (ZUG)
- Community Z Tools (CZT) project
- Other formal methods (and languages using formal specifications):
- VDM-SL, the main alternative to Z (compare)
- 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.

## References[edit]

**^**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**^**Meyer, Bertrand; Baudoin, Claude (1980),*Méthodes de programmation*(in French), Eyrolles**^**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).**^**Hoogeboom, Hendrik Jan. "Formal Methods in Software Engineering" (PDF). The Netherland: University of Leiden. Retrieved 14 April 2017.- ^
^{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. - ^
^{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[edit]

- 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. Archived from the original on 2009-06-27. - Bowen, Jonathan (1996).
*Formal Specification and Documentation using Z: A Case Study Approach*. International Thomson Computer Press. 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.

## External links[edit]

- Toyn, Ian,
*Z Specification proposals*, UK: York. *WSDL 2.0*, W3C, a specification containing Z notation assertions and explanation