Intro to Z Language

1. Intro to Z Language • Z is a “specification” language based on typed sets, relations, and functions to express: • What are the functionalities of the system • And what the desired results are • But without stating the “how” part. • Thus Z is a declarativelanguage - - meaning that it is non-procedural (not like Java or C) • Z “notation” developed at the Programming Research Group at Oxford University for recording, in precise way, some of the many decisions that are made in the course of software development.

2. Z Alphabet and Lexical elements • Z has both forms: • textand • graphical • Z identifiers are constructed from letters, digits, and the “_” character. • Upper and lower case are distinct • No length restriction (cool but nuts!) • First character must be a letter; it may also be a special symbol (state change) or (no state change). • May end with special characters of ?, !, ' .

3. Z-Schema • A schema is a structuring mechanism, a diagrammatic presentation, that allows the specification of: a) data b) operations that access that data • The Z- schema may be defined in two ways: • graphical form • text form

4. Z- Schema Forms • The Graphical form: schema name declarative or “signature” part predicate part • The Text form : schema name = [ declarative part I predicate part ]

5. Z-Schema “Parts” • The declarative or the “signature” part: • Introduces the identifiers (or variables) • And assigns them the set type • Each line statement is “assumed” to be terminated with ; • The predicate part refers to the identifiers in the declaration part or some global identifier in other schemas • The predicates, when there is more than one, are logical “conjunctions”, /\, of the predicates.

6. Examples of Schemas Graphical form: - Schema is named B_Team B_Team Member : [students] Team : IP students • Member is of type “students” • (which is defined elsewhere) • Team is a subset (type) of the • power set of students # (Team) < 11 - Cardinality of Team is < 11 Text form: B_Team = [ Member: students; Team: IP students I # (Team) < 11 ]

7. Z types and declarations • Z is “strongly” typed --- that is , every identifier must be declared. • Z has 3 “types” : • built-in : Z (integers), N (natural numbers), N1 ( positive integers) • basic: application dependent set (assumed to be pre-defined) • free type : enumerations • ( note that real numbers and characters are not part of built-in) • Basic type may be written in bracket, [ ] : • [ students ] , [ employees ] , [ books ] • Or [ students, employees, books ] • but we often just use these and not clearly (or forget to) define them ! • Freetype may be written as enumerations: • Students ::= { William I Shilpa I Harish I Carolyn I Amed I Joel } • and then identifiers x and y may be declared: • x , y : Students • Z allows one to “build up” the data types for declarations and usage.

8. Global constant and variable • Global constant is declared with the “==“ symbol: minimum == 0 • Global “variable” that has some constant associated with it may be declared via an “axiomatic description” : age : N age < 150 The global “variable”, age, is really some “constant” that has a value less than 150 and age may be used globally. Axiomatic description is like the schema without the top and bottom lines.

9. More Sophisticated Types • The variables in the signature may introduce any set theoretic type, including any higher order functions. The symbols used to depict relations and different types of functions are : • a relation • a partial function • a total function • a partial injection • a total injection • a partial surjection • a total surjection • a bijection

10. A Z simple schema example • The schema is titled Library Books and describes the book status Library_Books Library_Books -books, in_lib, and checked_out are of type, power set of Books; -lent_to is of the type, partial function from Books to Person; -on_shelf is a partial function from Books to Shelves books , in_lib, checked_out : IP Books lent_to : Books Person on_shelf : Books Shelves In_lib = dom (on_shelf) Checked_out = dom (lent_to) books = in_lib U checked_out • -in_lib is the domain of on_shelf; • -checked_out is the domain of • of lent_to; • books is the sum of in_lib • and checked_out

11. Z- Schema inclusion • A feature of Z-schema is that it allows the incremental and independent development of schema(s) • There is a mechanism for one schema to refer to another schema via schema inclusion. • Schema inclusion is accomplished by stating the name of the referred to schema in the signature part of the referring schema. • The result of schema inclusion is that : • the refered to schema’s signature portion is included into the signature portion of the referring schema, and • the predicate portion of the referred to schema is logically conjoined with the predicates of the referring schema The schema inclusion is a bit like the “inheritance” or “include” functions

12. Schema Inclusion Example Temp_Users SPSU_Users ::= Students I Faculty I Administrators temp_u, users : IP SPSU_Users n_of_days : N1 exp_days : users n_of_days Note : exp_days is a total function temp_u users temp_u = { u: users I ran(exp_days) < 30 u } Users Note :Temp_Users is included in the Users schema. So we can use all the terms from Temp_Users Temp_Users reg_u : IP SPSU_Users reg_u = { u : users I ran(exp_days) >= 30 u } users = temp_u U reg_u

13. GeneralGuidelinefor Z-Schema Specification • Static portion: • The states (or variables and constants) the system can have. • Invariant characteristics • “Dynamic” portion: • The functions (operations) that the system performs • The relationships among the inputs, outputs and other variables • The change of states in terms of “pre” and “post” conditions