1. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 1 Chapter 9 Formal Specifications

2. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 2 Objectives l To explain why formal specification helps discover problems in system requirements. l To describe the use of:  Algebraic specification techniques, and  Model-based specification techniques (including simple pre- and post-conditions).

3. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 3 Formal methods l Formal specification is part of a more general collection of techniques known as “formal methods.” l All are based on the mathematical rep- resentations and analysis of requirements and software.

4. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 4 Formal methods l Formal methods include:  Formal specification  Specification analysis and property proofs  Transformational development  Program verification (program correctness proofs) l Specifications are expressed with precisely defined vocabulary, syntax, and semantics.

5. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 5 Acceptance and use l Formal methods have not become main-stream as was once predicted, especially in the US. Some reasons why: 1.Other, less costly software engineering techniques (e.g., inspections / reviews) have been successful at increasing system quality. (Hence, the need for formal methods has been reduced.)

6. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 6 Acceptance and use 2.Market changes have made time-to- market rather than quality the key issue for many systems. (Formal methods do not reduce time-to- market.) 3.The scope of formal methods is limited. They are not well-suited to specifying user interfaces. (Many interactive applications are “GUI- heavy” today.)

7. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 7 Acceptance and use 4.Formal methods are hard to scale up to very large systems. (Although this is rarely necessary.) 5.The costs of introducing formal methods are high. 6.Thus, the risks of adopting formal methods on most projects outweigh the benefits.

8. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 8 Acceptance and use l However, formal specification is an excellent way to find requirements errors and to express requirements unambig- uously. l Projects which use formal methods invariably report fewer errors in the delivered software.

9. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 9 Acceptance and use l In systems where failure must be avoided, the use of formal methods is justified and likely to be cost-effective. l Thus, the use of formal methods is increasing in critical system development where safety, reliability, and security are important.

10. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 10 Formal specification in the software process elicitation

11. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 11 Formal specification techniques l Algebraic approach – the system is specified in terms of its operations and their relationships via axioms. l Model-based approach (including simple pre- and post-conditions) – the system is specified in terms of a state model and operations are defined in terms of changes to system state.

12. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 12 Formal specification languages

13. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 13 Use of formal specification l Formal specification is a rigorous process and requires more effort in the early phases of software development. l This reduces requirements errors as ambiguities, incompleteness, and inconsistencies are discovered and resolved. l Hence, rework due to requirements problems is greatly reduced.

14. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 14 Development costs with formal specification

15. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 15 Algebraic Specification of sub-system interfaces l Large systems are normally comprised of sub-systems with well-defined interfaces. l Specification of these interfaces allows for their independent development. l Interfaces are often defined as abstract data types or objects. l The algebraic approach is particularly well-suited to the specification of such interfaces.

16. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 16 Sub-system interfaces

17. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 17 The structure of an algebraic specification sort imports < LIST OF SPECIFICATION NAMES > Informal description of the sort and its operations Operation signatures setting out the names and the types of the parameters to the operations defined over the sort Axioms defining the operations over the sort < SPECIFICATION NAME > (Generic Parameter)

18. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 18 Specification components l Introduction – defines the sort (type name) and declares other specifications that are used l Description – informally describes the operations on the type --------------------------------------------------------------------- l Signature – defines the syntax of the operations in the interface and their parameters l Axioms – defines the operation semantics by defining axioms which characterize behavior

19. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 19 Types of operations l Constructor operations: operations which create / modify entities of the type l Inspection operations: operations which evaluate entities of the type being specified l Rule of thumb for defining axioms: define an axiom which sets out what is always true for each inspection operation over each constructor.

20. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 20 Operations on a “List” ADT l Constructor operations which create or modify sort List: Create, Cons and Tail l Inspection operations which discover attributes of sort List: Head and Length ( LISP fans: Tail = CDR, Head = CAR) l Tail is defined using Create and Cons, so Head and Length need not be defined for Tail.

21. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 21 List specification Head (Create) = Undefined exception (empty list) Head (Cons (L, v)) = if L =Create then v else Head (L) Length (Create) = 0 Length (Cons (L, v)) = Length (L) + 1 Tail (Create ) = Create Tail (Cons (L, v)) = if L =Createthen Createelse Cons (Tail (L), v) sort List imports INTEGER Create → List Cons (List, Elem) → List Head (List) → Elem Length (List) → Integer Tail (List) → List LIST ( Elem ) Defines a list where elements are added at the end and removed from the front. The operations are Create, which brings an empty list into existence; Cons, which creates a new list with an added member; Length, which evaluates the list size; Head, which evaluates the front element of the list; and Tail, which creates a list by removing the head from its input list. Undefined represents an undefined value of type Elem. A FIFO linear list Operator names + type info for argument(s) & result 123456123456

22. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 22 Recursion in specifications l Tail (Cons (L, v)) = if L=Create then Create else Cons (Tail (L), v) Tail (Cons ( [5, 7], 9)) = ? = Cons (Tail ( [5, 7] ), 9) (axiom 6) = Cons (Tail (Cons ([5], 7) ), 9) = Cons (Cons (Tail ([5]), 7), 9) (axiom 6) = Cons (Cons (Tail (Cons ([], 5) ), 7), 9) = Cons (Cons (Create, 7), 9) (axiom 6) = Cons ( [7], 9) = [7, 9]

23. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 23 Exercise What does Head (Tail (L)) do? L Head (Tail (L)) [ ] [a] [a, b] [a, b, c] undefined b b

24. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 24 Exercise Are axioms 1-6 sufficient to prove ANY true assertion of the form Head (Tail (L) ) = v ? Consider ONE EXAMPLE: Head (Tail ([a, b]) = b

25. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 25 Proof that Head (Tail ([a, b]) = b Head (Tail ([a, b])) = Head (Tail (Cons ([a], b))) = Head (Cons (Tail ([a]), b)) (axiom 6) = Head (Cons (Tail (Cons ([ ], a)), b)) = Head (Cons ([ ], b)) (axiom 6) = b (axiom 2)

26. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 26 Exercise So, we can prove Head (Tail ([a, b]) = b using the given axioms, but how could one show that the axioms are sufficient to prove ANY true assertion of the form Head (Tail (L) ) = v ? Moral: Showing correctness and completeness of algebraic specifications can be very tricky!

27. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 27 Model-based specification l Algebraic specification can be cumber- some when object operations are not independent of object state (i.e., previous operations). l (System State) Model-based specification exposes the system state and defines operations in terms of changes to that state.

28. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 28 Model-based specification l Z is a mature notation for model-based specification. It combines formal and informal descriptions and incorporates graphical highlighting. l The basic building blocks of Z-based specifications are schemas. l Schemas identify state variables and define constraints and operations in terms of those variables.

29. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 29 The structure of a Z schema Natural numbers (0, 1, 2, …) “invariants”, pre- & post- conditions ≤ Defines schema state

30. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 30 An insulin pump insulin delivery glucose level text messagesdose delivered

31. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 31 Modelling the insulin pump l The schema models the insulin pump as a number of state variables  reading? from sensor  dose, cumulative_dose  r0, r1, r2 last 3 readings  capacity pump reservoir  alarm! signals exceptional conditions  pump! output for pump device  display1!, display2! text messages & dose l Names followed by a ? are inputs, names followed by a ! are outputs.

32. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 32 Schema invariant l Each Z schema has an invariant part which defines conditions that are always true (schema predicates) l For the insulin pump schema, it is always true that:  The dose must be less than or equal to the capacity of the insulin reservoir.  No single dose may be more than 5 units of insulin and the total dose delivered in a time period must not exceed 50 units of insulin. This is a safety constraint.  display1! shows the status of the insulin reservoir.

33. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 33 Insulin pump schema ≤ dose ≤ capacity ^ dose ≤ 5 ^ cumulative_dose ≤ 50 capacity ≥ 40 => display1! = “ “ (capacity ≤ 39 ^ capacity ≥ 10) => display1! = “Insulin low” capacity ≤ 9 => (alarm! = on ^ display1! = “Insulin very low”) r2 = reading? every 10 minutes…

34. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 34 The dosage computation l The insulin pump computes the amount of insulin required by comparing the current reading with two previous readings. l If these suggest that blood glucose is rising then insulin is delivered. l Information about the total dose delivered is maintained to allow the safety check invariant to be applied. l Note that this invariant always applies – there is no need to repeat it in the dosage computation.

35. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 35 DOSAGE schema operations change state imports state & predicates (( r1 ≥ r0) ^ ( r2 = r1)) V (( r1 > r0) ^ ( r2 < r1)) V (( r1 (r0-r1))) (( r1 ≤ r0) ^ (r2 = r1)) V (( r1 < r0) ^ ((r1 – r2) ≤ (r0 – r1))) (( r1 ≤ r0) ^ (r2 > r1)) V (( r1 > r0) ^ ((r2 – r1) ≥ (r1 – r0))) x ′ - value of x after operation

36. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 36 Output schemas l The output schemas model the system displays and the alarm that indicates some potentially dangerous condition. l The output displays show the dose computed and a warning message. l The alarm is activated if blood sugar is very low – this indicates that the user should eat something to increase his blood sugar level.

37. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 37 Output schemas conversion Note conflict with “insulin low” msg. ^ ≥ 3≤ 30 ≥≤ ^ ^

38. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 38 Schema consistency l It is important that schemas be consistent. Inconsistency suggests a problem with system requirements. l The INSULIN_PUMP schema and the DISPLAY are inconsistent:  display1! shows a warning message about the insulin reservoir (INSULIN_PUMP)  display1! Shows the state of the blood sugar (DISPLAY) l This must be resolved before implemen- tation of the system.

39. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 39 Specification via Pre- and Post- Conditions l Predicates that (when considered together) reflect a program’s intended functional behavior are defined over its state variables. l Pre-condition: expresses constraints on program variables before program execution. An implementer may assume these will hold BEFORE program execution.

40. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 40 Specification via Pre- and Post- Conditions l Post-condition: expresses conditions / relationships among program variables after execution. These capture any obligatory conditions AFTER program execution. l Language: (first order) predicate calculus  Predicates (X>4)  Connectives ( &, V, →, , NOT)  Universal and existential quantifiers (for every…, there exists…)  Rules of inference (if A & (A → B) then B)

41. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 41 Example 1 Sort a non-empty array LIST[1..N] into increasing order. Pre-cond: N ≥ 1 Post-cond: For_Every i, 1 ≤ i ≤ N-1, LIST[i] ≤ LIST[i+1] & PERM(LIST,LIST’)

42. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 42 Example 2 Search a non-empty, ordered array LIST[1..N] for the value stored in KEY. If present, set FOUND to true and J to an index of LIST which corresponds to KEY. Otherwise, set FOUND to false.

43. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 43 Example 2 Pre-cond: N ≥ 1 & [(“LIST is in increasing order”) V (“LIST is in decreasing order”)] (Exercise: express the “ordered” predicate above FORMALLY.) Post-cond: [(FOUND & There_Exists i, 1 ≤ i ≤ N | J=i & LIST[J]=Key) V (NOT FOUND & For_Every i, 1 ≤ i ≤ N, LIST[i]≠KEY)] & UNCH(LIST,KEY)

44. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 44 Key points l Formal system specification complements informal specification techniques. l Formal specifications are precise and unambiguous. They remove areas of doubt in a specification. l Formal specification forces an analysis of the system requirements at an early stage. Correcting errors at this stage is cheaper than modifying a delivered system.

45. ©Ian Sommerville 2000Software Engineering, 6th edition. Chapter 9 Slide 45 Key points l Formal specification techniques are most applicable in the development of critical systems. l Algebraic techniques are particularly suited to specifying interfaces of objects and abstract data types. l In model-based specification, operations are defined in terms of changes to system state.