# Z notation

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.

## History

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.

## 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.

## Standards

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

• 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

## References

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. "ISO/IEC 13568:2002". Information Technology — Z Formal Specification Notation — Syntax, Type System and Semantics . ISO. 2002-07-01. 196 pp.
6. "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.
• 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.
• Toyn, Ian, Z Specification proposals, UK: York .
• WSDL 2.0, W3C , a specification containing Z notation assertions and explanation