Z notation: Difference between revisions

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Abrial has said that Z is so named "Because it is the ultimate language!"<ref>{{cite web|
Abrial has said that Z is so named "Because it is the ultimate language!"<ref>{{cite web|
  url=http://liacs.leidenuniv.nl/~hoogeboomhj/praatjes/formalmethods/formalmethods.pdf |
  url=https://liacs.leidenuniv.nl/~hoogeboomhj/praatjes/formalmethods/formalmethods.pdf |
  title=Formal Methods in Software Engineering | first=Hendrik Jan | last=Hoogeboom | publisher=[[University of Leiden]] | location=The Netherland | access-date=14 April 2017 }}</ref> although the name "[[Ernst Zermelo|Zermelo]]" is also associated with the Z notation through its use of [[Zermelo–Fraenkel set theory]].
  title=Formal Methods in Software Engineering | first=Hendrik Jan | last=Hoogeboom | publisher=[[University of Leiden]] | location=The Netherland | access-date=14 April 2017 }}</ref> although the name "[[Ernst Zermelo|Zermelo]]" is also associated with the Z notation through its use of [[Zermelo–Fraenkel set theory]].


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Z is based on the standard mathematical notation used in [[axiomatic set theory]], [[lambda calculus]], and [[first-order predicate logic]].<ref>{{cite book| last=Spivey | first=J. Michael | authorlink=Michael Spivey | title=The Z Notation: A Reference Manual | edition=2nd | location=Hemel Hempstead | publisher=[[Prentice Hall]] | series=International Series in Computer Science | date=1992 | isbn=978-0139785290 }}</ref> All expressions in Z notation are [[Type (model theory)|typed]], thereby avoiding some of the [[Naive set theory#Paradoxes|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. It is augmented with '''Z schema''' boxes, which can be combined using their own operators, based on standard logical operators, and also by including schemas within other schemas. This allows Z specifications to be built up into large specifications in a convenient manner.
Z is based on the standard mathematical notation used in [[axiomatic set theory]], [[lambda calculus]], and [[first-order predicate logic]].<ref>{{cite book| last=Spivey | first=J. Michael | authorlink=Michael Spivey | title=The Z Notation: A Reference Manual | edition=2nd | location=Hemel Hempstead | publisher=[[Prentice Hall]] | series=International Series in Computer Science | date=1992 | isbn=978-0139785290 }}</ref> All expressions in Z notation are [[Type (model theory)|typed]], thereby avoiding some of the [[Naive set theory#Paradoxes|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. It is augmented with '''Z schema''' boxes, which can be combined using their own operators, based on standard logical operators, and also by including schemas within other schemas. This allows Z specifications to be built up into large specifications in a convenient manner.


Because Z notation 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.<ref>{{cite web| url=https://unicode-search.net/unicode-namesearch.pl?term=Z%20NOTATION | title=Unicode Explained: Internationalize Documents, Programs, and Web Sites | first=Jukka K. | last=Korpela | website=unicode-search.net | access-date=24 March 2020 }}</ref>
Because Z notation 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.<ref>{{cite web | url=https://unicode-search.net/unicode-namesearch.pl?term=Z%20NOTATION | title=Unicode Explained: Internationalize Documents, Programs, and Web Sites | first=Jukka K. | last=Korpela | website=unicode-search.net | access-date=24 March 2020 | archive-date=24 March 2020 | archive-url=https://web.archive.org/web/20200324115418/https://unicode-search.net/unicode-namesearch.pl%3Fterm%3DZ%2520NOTATION | url-status=dead }}</ref>


==Standards==
==Standards==
[[International Organization for Standardization|ISO]] completed a Z standardization effort in 2002. This standard<ref name = "Standard">{{cite book | title = Information Technology — Z Formal Specification Notation — Syntax, Type System and Semantics | chapter-url = http://www.iso.ch/iso/en/CatalogueDetailPage.CatalogueDetail?CSNUMBER=21573 | chapter = ISO/IEC 13568:2002 | date = 2002-07-01 | publisher = ISO | url = http://standards.iso.org/ittf/PubliclyAvailableStandards/c021573_ISO_IEC_13568_2002(E).zip | format = [[Zip (file format)|Zip]]ped [[PDF]]}}  196 pp.</ref> and a technical corrigendum<ref name = "Corrigendum">{{cite book | title = Information Technology — Z Formal Specification Notation — Syntax, Type System and Semantics — Technical corrigendum 1 | url = http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=46112 | chapter = ISO/IEC 13568:2002/Cor.1:2007 | publisher = ISO | date = 2007-07-15 | format = PDF}}  12 pp.</ref> are available from ISO free:
[[International Organization for Standardization|ISO]] completed a Z standardization effort in 2002. This standard<ref name = "Standard">{{cite book | title = Information Technology — Z Formal Specification Notation — Syntax, Type System and Semantics | chapter-url = http://www.iso.ch/iso/en/CatalogueDetailPage.CatalogueDetail?CSNUMBER=21573 | chapter = ISO/IEC 13568:2002 | date = 2002-07-01 | publisher = ISO | url = https://standards.iso.org/ittf/PubliclyAvailableStandards/c021573_ISO_IEC_13568_2002(E).zip | format = [[Zip (file format)|Zip]]ped [[PDF]]}}  196 pp.</ref> and a technical corrigendum<ref name = "Corrigendum">{{cite book | title = Information Technology — Z Formal Specification Notation — Syntax, Type System and Semantics — Technical corrigendum 1 | url = http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=46112 | chapter = ISO/IEC 13568:2002/Cor.1:2007 | publisher = ISO | date = 2007-07-15 | format = PDF}}  12 pp.</ref> are available from ISO free:
* the standard is publicly available<ref name = "Standard" /> from the ISO ITTF site free of charge and, separately, available for purchase<ref name = "Standard" /> from the ISO site;
* the standard is publicly available<ref name = "Standard" /> from the ISO ITTF site free of charge and, separately, available for purchase<ref name = "Standard" /> from the ISO site;
* the technical corrigendum is available<ref name = "Corrigendum" /> from the ISO site free of charge.
* the technical corrigendum is available<ref name = "Corrigendum" /> from the ISO site free of charge.
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  | title = The Z Notation: A reference manual | edition = 2nd | year = 1992
  | title = The Z Notation: A reference manual | edition = 2nd | year = 1992
  | publisher = [[Prentice Hall]] | series = International Series in Computer Science | url=https://spivey.oriel.ox.ac.uk/corner/Z_Reference_Manual }}
  | publisher = [[Prentice Hall]] | series = International Series in Computer Science | url=https://spivey.oriel.ox.ac.uk/corner/Z_Reference_Manual }}
*{{cite book|title=Using Z: Specification, Refinement and Proof|last1=Davies|first1=Jim|author-link1=Jim Davies (computer scientist)|
*{{cite book|title=Using Z: Specification, Refinement and Proof|last1=Davies|first1=Jim|author-link1=Jim Davies (computer scientist)|last2=Woodcock|first2=Jim|author-link2=Jim Woodcock|publisher=Prentice Hall|year=1996|isbn=0-13-948472-8|series=International Series in Computer Science|url=http://www.usingz.com/|archive-date=5 April 2007|access-date=22 March 2006|archive-url=https://web.archive.org/web/20070405133311/http://www.usingz.com/|url-status=dead}}
last2=Woodcock|first2=Jim|author-link2=Jim Woodcock| publisher=Prentice Hall|year=1996|isbn=0-13-948472-8|series=International Series in Computer Science | url=http://www.usingz.com/ }}
*{{cite book|title=Formal Specification and Documentation using Z: A Case Study Approach|last=Bowen|first=Jonathan| publisher=International Thomson Computer Press, [[International Thomson Publishing]] |year=1996|isbn=1-85032-230-9|author-link=Jonathan Bowen | url=https://www.researchgate.net/publication/2480325 }}
*{{cite book|title=Formal Specification and Documentation using Z: A Case Study Approach|last=Bowen|first=Jonathan| publisher=International Thomson Computer Press, [[International Thomson Publishing]] |year=1996|isbn=1-85032-230-9|author-link=Jonathan Bowen | url=https://www.researchgate.net/publication/2480325 }}
*{{cite book|title=The Way of Z: Practical Programming with Formal Methods|last=Jacky|first=Jonathan| publisher=[[Cambridge University Press]] |year=1997|isbn=0-521-55976-6| url=https://staff.washington.edu/jon/z-book/ }}
*{{cite book|title=The Way of Z: Practical Programming with Formal Methods|last=Jacky|first=Jonathan| publisher=[[Cambridge University Press]] |year=1997|isbn=0-521-55976-6| url=https://staff.washington.edu/jon/z-book/ }}

Latest revision as of 17:25, 4 September 2025

Template:Short description Template:Use dmy dates

File:Agendacumple en Z.svg
An example of a formal specification (in Spanish) using the Z notation, with named schema boxes, including declarations and predicates

The Z notation Template:IPAc-en is a formal specification language used for describing and modelling computing systems.[1] 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".[2] 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), working with Bertrand Meyer, Abrial also worked on developing Z.[3] The Z notation is used in the 1980 book Méthodes de programmation.[4]

Z was originally proposed by Abrial in 1977 with the help of Steve Schuman and Bertrand Meyer.[5] 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!"[6] although the name "Zermelo" is also associated with the Z notation through its use of Zermelo–Fraenkel set theory.

In 1992, the Z User Group (ZUG) was established to oversee activities concerning the Z notation, especially meetings and conferences.[7]

Usage and notation

Z is based on the standard mathematical notation used in axiomatic set theory, lambda calculus, and first-order predicate logic.[8] 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. It is augmented with Z schema boxes, which can be combined using their own operators, based on standard logical operators, and also by including schemas within other schemas. This allows Z specifications to be built up into large specifications in a convenient manner.

Because Z notation 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.[9]

Standards

ISO completed a Z standardization effort in 2002. This standard[10] and a technical corrigendum[11] are available from ISO free:

  • the standard is publicly available[10] from the ISO ITTF site free of charge and, separately, available for purchase[10] from the ISO site;
  • the technical corrigendum is available[11] from the ISO site free of charge.

Award

In 1992, Oxford University Computing Laboratory and IBM were jointly awarded The Queen's Award for Technological Achievement "for the development of ... the Z notation, and its application in the IBM Customer Information Control System (CICS) product."[12]

Further reading

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  • Script error: No such module "citation/CS1".
  • Script error: No such module "citation/CS1".
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See also

References

Template:Reflist

Template:ISO standards Template:List of IEC standards Template:Authority control

  1. ↑ Script error: No such module "citation/CS1".
  2. ↑ Script error: No such module "citation/CS1".
  3. ↑ Script error: No such module "citation/CS1".
  4. ↑ Script error: No such module "citation/CS1".
  5. ↑ Script error: No such module "citation/CS1". (describes early version of the language).
  6. ↑ Script error: No such module "citation/CS1".
  7. ↑ Script error: No such module "citation/CS1".
  8. ↑ Script error: No such module "citation/CS1".
  9. ↑ Script error: No such module "citation/CS1".
  10. ↑ a b c Script error: No such module "citation/CS1". 196 pp.
  11. ↑ a b Script error: No such module "citation/CS1". 12 pp.
  12. ↑ Script error: No such module "citation/CS1".
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