Structure of Z Formal methods Lecture
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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.
![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](https://image.slidesharecdn.com/structureformal1-230725034919-4d436a32/85/Structure-of-Z-Formal-methods-Lecture-10-320.jpg)
Structure of Z Formal methods Lecture
- 1. Formal methods in software engineering Lecture on Structure:Element of Z Prof Engr. Faiz ul haque Zeya
- 2. Topics Structure: Tuple, Records, Relations, Tables, Databases, Pairs Binary Relations, Functions, Sequences and Operators
- 3. 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)
- 5. 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
- 7. 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.
- 10. 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
- 13. Combination of domain and range restriction
- 17. Inverse operator
- 18. Composition relation Relational composition formalizes this kind of reasoning: It merges two relations into one by combining pairs that share a matching component.
- 19. 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.
- 21. 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.
- 24. 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.
- 25. Injections every element of the function's codomain is the image of at most one element of its domain.
- 26. SEQUENCES Sequences are ordered collection
- 27. Another way to declare sequence is:
- 28. Operators