1. Formal Methods in SE Lecture 16 Formal Methods in SE Qaisar Javaid Assistant Professor

2. Formal Methods in SE Lecture 16 Specification team. Responsible for developing and maintaining the system specification. Development team. Responsible for developing and verifying the software. The software is NOT executed or even compiled during this process. Certification team. Responsible for developing a set of statistical tests to exercise the software after development. Reliability growth models used to determine when reliability is acceptable. Cleanroom process teams

3. Formal Methods in SE Lecture 16 The results of using the Cleanroom process have been very impressive with few discovered faults in delivered systems. Independent assessment shows that the process is no more expensive than other approaches. There were fewer errors than in a 'traditional' development process. However, the process is not widely used. It is not clear how this approach can be transferred to an environment with less skilled or less motivated software engineers. Cleanroom process evaluation

4. Formal Methods in SE Lecture 16 Free and Bound Variables 4 In the expression Q x : A/B p we say that variable x is bound by the quantifier. The scope of x extends from the vertical bar (or the spot if there is no constraint) to the next closing bracket. If the variable x appears in a predicate q but is not bound by any quantifier, we say that x is free in q.

5. Formal Methods in SE Lecture 16 Substitution 5 We write p[t/x ] to denote the predicate that results from substituting t for each free occurrence of x in predicate p. The substitute expression t need not be another variable; it can be any expression whose possible values match those of x.

6. Formal Methods in SE Lecture 16 Universal Quantification 6 The universal quantifier ∀ is a generalized form of ∧.

7. Formal Methods in SE Lecture 16 Existential Quantification 7 The existential quantifier ∃ is a generalized form of ∨.

8. Formal Methods in SE Lecture 16 Universal Quantification 8

9. Formal Methods in SE Lecture 16 Existential Quantification 9

10. Formal Methods in SE Lecture 16 Equality 10 The = (equality) symbol is used to show that expressions are identical Equalities are one form of atomic propositions in our logical language (the other form is set membership).

11. Formal Methods in SE Lecture 16 Equality 11

12. Formal Methods in SE Lecture 16 If the identity of a bound variable is revealed within the quantified expression, then we may replace all instances of that variable, and remove the existential quantier. Consider following predicate. This states that there is a value of x in a for which p ^ x = t is true. If t is in the set a and p holds then we can replace x by t. One Point Rule 12

13. Formal Methods in SE Lecture 16 Data Structures Objects from discrete mathematics can model data structures. –Tuples (records) –Relations (tables, linked data structures) –Functions (lookup tables, trees and lists) –Sequences (lists, arrays) 13

14. Formal Methods in SE Lecture 16 TUPLES Tuples can resemble C structures or Pascal records. Tuples are instances of Cartesian product types. First declare types for each component. [NAME] ID == N DEPT ::= admin | manufacturing | research Define the Cartesian product type EMPLOYEE. EMPLOYEE == ID X NAME X DEPT Declare tuples which are instances of the type. 14

15. Formal Methods in SE Lecture 16 RELATIONS Relations are sets of tuples. They can resemble tables or databases. In Z this can be expressed 15

16. Formal Methods in SE Lecture 16 PAIRS The projection operators first and second extract the components of a pair. first(aki,4117) = aki second(aki, 4117) = 4117 Pairs are tuples with just two components. (aki, 4117) The maplet arrow provides alternate syntax without parentheses. 16

17. Formal Methods in SE Lecture 16 BINARY RELATIONS Binary relations are sets of pairs. P (NAME X PHONE) OR NAME  PHONE Binary relations can model lookup tables Binary relations are many-to- many relations NAMEPHONE Aki4019 Philip4107 Doug4107 Doug4136 Philip0113 Frank0110 Frank6190 …… 17

18. Formal Methods in SE Lecture 16 BINARY RELATIONS 18

19. Formal Methods in SE Lecture 16 RELATIONAL CALCULUS Restriction operators can model database queries. Domain restriction selects pairs based on their first component. 19

20. Formal Methods in SE Lecture 16 RELATIONAL CALCULUS Overriding can model database updates. 20

21. Formal Methods in SE Lecture 16 RELATIONAL CALCULUS Inverse reverses domain and range by exchanging the components of each pair. 21

22. Formal Methods in SE Lecture 16 RELATIONAL CALCULUS contd.. Composition merges two relations by combining pairs that share a matching component. 22

23. Formal Methods in SE Lecture 16 OPERATOR SYMBOLS Domain & Range restriction operators can also be combined 23

24. Formal Methods in SE Lecture 16 FUNCTIONS Functions are binary relations where each element in the domain appears just once. Each domain element is a unique key. A function cannot be a many-to-many or even one-to-many relation Function application is a special case of relational image. It associates a domain element with its unique range element. 24

25. Formal Methods in SE Lecture 16 Types of Functions Surjective Function A function is surjective (onto) if every element of the domain is mapped to some element of the range. some domain elements may be mapped to more than one range elements. (Total Injections) Injective Function Injections are the functions that associate each element in their domain with a different element in their range. It’s a one-to-one relation. (Partial injections) Bijective Function A function is bijective (one-to-one and onto) iff it is both injective and surjective. (Equivalently, every element of the domain is mapped to exactly one element of the range.) A bijective function is a bijection (one-to-one correspondence), and is reversible. 25

26. Formal Methods in SE Lecture 16 Examples 26

27. Formal Methods in SE Lecture 16 BINARY RELATIONS AND FUNCTIONS 27