Structure of Z Formal methods Lecture

Jul 25, 2023Download as PPT, PDF0 likes11 views

This document provides an overview of different structures in formal methods and software engineering, including tuples, records, relations, tables, databases, pairs, binary relations, functions, and sequences. It discusses topics such as tuples associating elements in a fixed order, relations modeling tables and databases, pairs associating two components, functions associating a single item to each element in a set, and sequences being ordered collections. It also covers operators for these structures like projection, relation image, domain restriction, range restriction, composition, and inverse.

Formal methods in
software engineering
Lecture on Structure:Element of Z
Prof Engr. Faiz ul haque Zeya

Topics
 Structure:
 Tuple,
 Records,
 Relations,
 Tables,
 Databases,
 Pairs
 Binary Relations,
 Functions,
 Sequences and
 Operators

Tuple
 Tuples associate particular elements of any type, in a fixed order
 If we use set (2,11,11) and (11,2,11) won’t be different.
 Tuples are instances of Cartesian product types, sometimes called cross
product types. We define product types using the cross product symbol x. This
abbreviation definition introduces a set of tuples named DATE
 DATE=DAY X MONTH X YEAR
 DATE has product type P(Z X Z X Z)

Relation, Table and Database
 A set of tuples is called a relation. Relations can model tables and databases.
You have probably heard of the relational database, which is just a database
where the data are stored in one or more relations

Pair and binary relation
 It has two components.
 . We can use a pair to associate a name with a telephone extension number,
as in (aki, 4117). Z provides an alternate syntax for pairs that uses the maplet
arrow ---> to emphasize the asymmetry between the two components. The
pair (aki, 4117) can also be written aki -->4117
 Z also provides the first and second operators for extracting each component
from a pair: first(aki,41) = aki second(aki,41) = 41 operators like these that
extract components from structures are called projection operators.

Operators for relations
The relation image operator
 The relational image operator can model table lookup. Its first argument is a
relation, its second argument is a set of elements from the domain, and its
value is the set of corresponding elements from the range. It is notated in an
unusual mixfix syntax: Thick brackets (]...[) surround the second argument.
To look up the numbers for Doug and Philip in the phone relation, we use the
relational image: phone

Combination of domain and range
restriction

Composition relation
 Relational composition formalizes this kind of reasoning: It merges two
relations into one by combining pairs that share a matching component.

Binary relation and linked data
structure.
 Relations are not just for modelling tables and flat databases. They can model
linked data structures as well. Linked data structures are often pictured as
graphs: networks of nodes connected by arcs. Data flow diagrams, state
transition diagrams, and syntax trees are all examples of graphs.

Function
 Sometimes we need to associate a single item with each element in a set. For
this we use a special kind of relation, called & function. A function is a binary
relation where an element can appear only once as the first element in a pair.

Partial and total function
 The domain of a partial function might not include every element of the
source set.
 Total function apply to every element of the source set
 Square of x.

Injections
 every element of the function's codomain is the image of at most one element
of its domain.

SEQUENCES
 Sequences are ordered collection

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