Lecture 11 Formal Modeling: Z, OCL homework Alloy background

1. Lecture 11Formal Modeling: Z, OCL homeworkAlloy background

2. Review Lecture 10 Z Roadmap for Today • Z language: elements of Z - review with small examples • Z modeling: walk through 3 case studies – put it together • break • Alloy background

3. Z Basic Constructs • Declarations introduce variables. • Expressions describe values that variables can assume. • Predicatesplace constraints on the values that variables do assume.

4. Set and Type Set Name: Z, DICE Set Expression: {i: Z|1 i  6} Every object belongs to a set called its type. Free Type Basic Type Z, [NAME] - include indefinite number of elements

5. Exercise: Set Expression in Z SetName == {x: T | pred(x)• expr(x)} Natural numbers: N == {i: Z|i 1} Odd numbers: ODD == {i: Z• 2*i+1} // the first part introduces local variables; their scope includes the expression after • Prime numbers:

6. Introduce Variables in Z Declaration: x: Z Axiomatic definitions: define global, optional constraints size: N size > 640 Normalized Declaration: s, b: Z s  SMALL b  BIG

7. Expressions and Predicates • Expressions: computing values • Constants: 1, 2, red • Operators on constants and variables: arithmetic, set • Predicate: constraining values – return true/false • Equality • Arithmetic relations • Set membership • Set relations • Logic: building complex predicate

8. Exercise: Reading Z logic operations and predicate

9. Structures • Tuples: instances of Cartesian product types [NAME] ID==N DEPT::= admin|manufacturing|research EMPLOYEE == ID× NAME ×DEPT • Relations: a set of tuples (table/database)

10. Binary Relation and Relation Calculus

11. Binary Relations and Functions Function: binary relations where each element in the domain appears just once phonef(doug) = 4107

12. Lambda Expressions Define functions ( declaration | predicate expression) Use functions without writing declarations Compare to set definitions! •

13. Sequence: Model Array and Lists Sequence: <>

14. let/if-then-else let In the let-expression (let x1 == E1; … ; xn == En •E), the variables x1 … xnare local; their scope includes the expression E (let r == iroot(a) • r*r < a < (r+1)*(r+1)) if then else - conditional expression: if P then E1 else E2 – E1 and E2 type should be the same | x | = if x > 0 then x else -x

15. Exercise: Formal Reasoning Philip works on the adhesives team in the materials group, which is part of the research division Is Philip in the research division?

16. Exercise: Formal Reasoning A train moves at a constant velocity of sixty miles per hour for four hours Z spec: Query the spec: How far does the train travel?

17. Z Schemas • Schema: math in a box, with a name attached • Schema calculus builds large schemas from smaller ones. • Model states and operations. • Can be used as declarations, predicates, expressions…

18. Schema calculus: Conjunction

19. Example - Conjunction  Quotient n, d, q, r: N d  0 n = q * d + r Remainder r, d: N r < d Division = Quotient  Remainder Division n, d, q, r: N d  0 n = q * d + r r < d

20. Schema calculus : Disjunction

21. Example - Disjunction  Division n, d, q, r: N d  0 n = q * d + r r < d DivideByZero d, q, r: N report!: TEXT d = 0  q = 0  r = 0 report! = Error T_Division = Division  DivideByZero T_Division n, d, q, r: N report!: TEXT (d  0  r < d  n = q * d + r )  (d = 0  q = 0  r = 0  report! = Error)

22. Z Specification Yes Yes No Yes Yes Yes Yes NL: newline Ident: variable/type names, e.g, ch?

23. Z Specification Process

24. Modeling Large Systems Large systems are composed of components collected into subsystems. Complex components can be built up from simpler ones using schema inclusion.

25. Case Study A: Using Z to define an abstract data type state space (a set of variables and their constrains), initial state, operations State space: Initial State: Operation:

26. Case Study B: Birthday book Model a system which records people's birthdays, and is able to issue a reminder when the day comes round. Step 1 Define given sets and types [NAME] [DATE] Step 2 Define state space • knownis the set of names with birthdays recorded • birthdayis a function which, when applied to certain names, gives the birthdays associated with them

27. Case Study B: Birthday book Step 3 initial state Step 4 operations Add, find, reminder

28. Case Study B: Birthday book Add Birthday

29. Case Study B: Birthday book Find Birthday

30. Case Study B: Birthday book Remind

31. Case Study C: Editor Example Natural language (English) description: • User can type in text • User can move cursor forward and backward through the text • User can delete the character in front of the cursor

32. State schemas Editor left, right: TEXT # ( left  right ) maxsize left is before the cursor, right is after

33. Initialization schemas InitEditor Editor left = right = <> conventional name begins with Init Editor is included left and right come from Editor

34. Operation schemas printing: P Char Insert  Editor ch?: CHAR ch?  printing left’ = left  <ch?> right’ = right global declaration of printing characters “” means that the state will change “?” means input (“!” means output) precondition left’ is the value after the operation

35. Schema Inclusion InitEditor Editor left = right = <> InitEditor left, right: TEXT # ( left  right )  maxsize left = right = <> Editor left, right: TEXT # ( left  right )  maxsize

36. Homework 3: OCL If an order item is linked to a StockItemDescobject then the associated order is linked to the warehouse linked to the StockItemDescobject. contextOrderItem inv: self.stock->notEmpty() implies self.itemord.itemSources.Stocks→ includesAll(self.stock) includesAll: collection->includesAll(collection) Includes: collection->include(object) self.itemord.itemSources→includesAll(self.stock.Stocks) collection? the same type

37. Reading OCL Context Booking inv: self.bookedseats.hasSeats→forAll(f |f = self.bookingFor) forAll: collection→ forall (condition must hold for all elements) Step 1: direct translation: the flights where the booked seats belong to should be the flights we aim to book Step 2: meaning: All the booked seats should belong to the booked flights Context FlightPlane inv: self.numberOfSeats = self.hasSeats→size() All the seats on the FightPlane are available for booking (ambiguity on the UML)

38. Implies Context FlightSection inv: (seatClass = Section::first or seatClass = Section::business or seatClass= Section::premier) impliesisLocatedIn collect (f.booking.passenger.status) forAll(s | s = PassengerClass:: executive or s = PassengerClass:: premier) reading implies: if a is true then b is true Truth table for implies

39. implies and if-then-else DVD orders: Give an OCL invariant that specifies that the sum attribute will be zero if no DVDs are ordered context Order inv: if self.OrderedVideos→size()=0 thenself.sum = 0 incorrect! self.OrderedVideos→size()=0 impliesself.sum= 0 if <boolexpr> then < bool expr1> // the result of the if-then-else expression is either bool expr1 else <bool expr2> // or boolexpr2, thus else cannot be omitted endif

40. Operations on Collections Specifies that the sum attribute really describes the price of the DVDs ordered context order inv: self.sum = self.OrderedVideos→ collect(price)→ sum() Specifies that different instances of DVD have different Id context DVD inv: DVD.allInstances()  isUnique(Id) context DVD inv: DVD.allInstances() forAll(b1, b2 | b1 <> b2 implies b1.Id <> b2.Id) context DVD inv: DVD.allInstances() forAll(b | b <> self implies b.Id <> self.Id)

41. forAll, exists Syntax: forAll, exists collection->forAll(elem : T | expr) collection->forAll(elem | expr) collection->forAll(expr) • The result is a Boolean value • forAll(expr) --- expr should be about attributes or operations • forAll(elem| expr) --- elem is a local variable you specifies, if no type is specified for elem, elem has the same type as the one specified under context

42. To Do • Homework 4: Z - due 10/18 (Tues) 11:00am • Reading assignment - in class • Alloy (Wed 10/13) – Corey • An Approach to Combine Model-based and Scenario based Software Architecture Analysis (Mon 10/17) – Fernado We are moving to Software Architecture next time!

43. Wild & Crazy Ideas - How can we learn better in the next half quarter? Thank you for the nice words  Discuss some of your comments !