1. www.eschertech.com Fundamentals of Perfect Developer A one-day hands-on tutorial

2. www.eschertech.com Outline Session 1: Getting Started –Configuring Perfect Developer and creating a project –Expressions, types, constants, functions, properties –Classes, data, invariants, functions, schemas, constructors Session 2: Going Further –Interfacing to a Java front-end –Sequences and recursion Session 3: Refining methods and data –Statement lists, statement types, loops, nested refinement. –Internal data, retrieve functions Session 4: Inheritance –Derived and deferred classes –Defining and redefining inherited methods

3. www.eschertech.com Configuration Configure Perfect Developer to use the chosen editor –Load the Project Manager –Select Options…Editor –Browse to the editor executable file –Select the editor type Configure the editor to recognise Perfect Developer files –See instructions in the Editor Customizations folder of the Perfect Developer installation

4. www.eschertech.com Creating a project Click on the New toolbar icon Browse to a writable folder, enter a project name, click Save and then OK Create a new file called Examples Add text: property (a, b, c: int) pre a < b, b < c assert a < c; Save the file Save the project and click the Verify tool

5. www.eschertech.com Some predefined classes bool char int real set of X bag of X seq of X map of (X -> Y) Set, bag, sequence and map types are finite collections. See Quick Reference or Language Reference Manual for more details (e.g. commonly-used members)

6. www.eschertech.com Expressions All the usual arithmetic operators a < b < c means what you would expect it to a / b (integer) and a % b have precondition b > 0 a / b (integer) rounds towards -  a.. b yields the sequence {a, a+1, a+2 … b} #c yields the cardinality or length of the collection c a # c yields the number of occurrences of a in c

7. www.eschertech.com Booleans, Conditionals etc. Boolean connectives: & | ==> Conditional expression: ( [a > 0]: a, [b > 0]: b, []: 0) Let-declaration: ( let t ^= a - b; t * t ) Embedded assertion: ( assert a > 0, b > 0; a / b ) The whole shebang: ( let t ^= a - b; assert t > 0; [t > 10]: 1, []: 10 / t )

8. www.eschertech.com Constructor calls A constructor call yields a value of the specified type Without parameters: seq of int{} MyClass{} With parameters: seq of int{a, b, c} MyClass{42} Some constructors have preconditions that must be met int{s} is ok when s = "123" but not when s = "lemon" ! [precondition is: ~s.empty & (forall c::s :- c.isDigit) ]

9. www.eschertech.com Quantifiers etc. If c is of type set of T, bag of T or seq of T, and p(x) is any Boolean expression involving x: ExpressionReturn type forall x::c :- p(x)bool forall x: T :- p(x) exists x::c :- p(x)bool exists x: T :- p(x) that x::c :- p(x)T any x::c :- p(x) those x::c :- p(x)set / seq / bag of T for x::c yield v(x)set / seq / bag of type of v for those x::c :- p(x) yield v(x)

10. www.eschertech.com Declaring constants and functions Declaring a constant: const four: int ^= 2 + 2; const smallPrimes: seq of int ^= those x::2..100 :- ~(exists y::2..<x :- x % y = 0); const two ^= 2; Declaring a function: function half(x: int): int pre x > 0 ^= x / 2; Precondition (if needed) Type can be omitted in simple cases

11. www.eschertech.com Declaring properties Use property declarations to express theorems: property assert half(four) = two; property (x: int) pre x > 0, x % 2 = 0 assert half(x) < x, (let h ^= half(x); h + h = x); Implicit universal quantification over the parameters Givens to be assumed Consequences to be proved

12. www.eschertech.com Exercise 1: express the following All the integers from 0 to 100 inclusive, in ascending order. Verify that your solution contains 42 but does not contain 101. The integer j (which is known to be greater than 0) divides the integer i exactly. The squares of all the prime numbers from 2 to 100 inclusive. The highest integer in a set S of integers that is known to be non-empty

13. www.eschertech.com Declaring enumerations class Currency ^= enum unspecified, euro, pound, USdollar end; const localCurrency ^= pound@Currency; … localCurrency.toString …

14. www.eschertech.com Declaring a Class abstract internal confined interface variables, invariants, methods, constructors [never mind for now] access redeclarations, methods, constructors, properties [never mind for now] Variables declared here form the abstract data model of the class Invariants here constrain the data These methods and constructors are for use by confined and/or interface methods and constructors Access redeclarations allow abstract variables to be directly accessed from outside the class These methods and constructors can be called from outside the class Only the interface section is mandatory class Money ^= end;

15. www.eschertech.com Declaring data and invariants abstract var amt: int, ccy: Currency; invariant amt = 0 | ccy ~= unspecified@Currency; Variable amt is of type int Variable ccy is of type Currency Restriction on the values of amt and ccy

16. www.eschertech.com Functions and Operators function worthHaving: bool ^= amt > 0; function plus(m: Money): Money pre m.ccy = ccy ^= Currency{amt + m.amt, ccy} assert result.ccy = ccy; operator (n: int) * : Money ^= Currency{amt * n, ccy}; No “()” if no parameters Return type Result expression Optional postassertion Name Optional precondition Use nonmember prefix for methods & properties with no self object Operator declarations are the same as function declarations apart from the header

17. www.eschertech.com Declaring Schemas nonmember schema swap(a!, b!: Money) pre a.ccy = b.ccy post change a,b satisfy a’=b, b’=a assert a.plus(b) = b’.plus(a’); schema !inflate(howMuch: int) pre 0 < howMuch < 200 post amt! = (amt * howMuch)/100; No “()” if no parameters Parameter is modified Postcondition includes frame This one modifies instance variables Short for: change amt satisfy amt’= (amt * howMuch)/100 No self object

18. www.eschertech.com Declaring Constructors build{a: int, c: Currency} pre a > 0 post amt! = a, ccy! = c; build{!amt: int} post ccy! = euro@Currency; build{} ^= Money {0, unspecified@Currency}; Note parameter list in “{}” Initialise instance variable directly from the parameter Postcondition must initialise all instance variables * We do use “{}” if no parameters This constructor is defined in terms of another one Short for: change amt satisfy amt’= a *except for variables whose when-guards are false

19. www.eschertech.com Using access redeclarations abstract variables may be redeclared in the interface: function v1; makes v1 readable selector v2; makes v2 readable and writable Making a variable writable is generally a bad idea –Except for “convenience” classes, e.g. class pair Making a variable of a complicated type readable is generally a bad idea –Because we can’t then change its representation easily Constants may be redeclared as nonmember functions Use access redeclarations sparingly!

20. www.eschertech.com Exercise 2: Specification (followed by coffee break) You have been provided with a Perfect specification of class Money Try to verify it (there will be 3 verification errors) Fix the specification to remove the verification errors Verify that multiplying a Money object by 2 is equivalent to adding it to itself Declare a “+” operator that works like the “plus” function except that if the amount of either operand is zero, we don’t care what the corresponding currency is

21. www.eschertech.com Using Perfect with a graphical UI Java front-end Perfect back-end function f (…) schema ! s (…) Button 1 Button 2 class Application ^= interface class MyApp implements ActionListener MyApp() build{} constructor calls when pressed calls Best to avoid preconditions in methods called from Java!

22. www.eschertech.com Using Perfect with a graphical UI Declare a Perfect class Application –Declare interface functions/schemas to call from Java –Declare an interface constructor to call from Java In the graphical interface code: –Import Application.java –Instantiate a single Application object during initialisation –Call member functions/schemas when buttons are pressed –Convert parameter types as necessary We have provided you with a sample –In file TutorialExample.java

23. www.eschertech.com Exercise 3: Build the sample program Open project TutorialExample.pdp Verify the project Click the Build tool icon Check for error messages Locate and run output\App.jar Try making your own changes to Application.pd –e.g. print some of the expressions you wrote earlier [tips follow…]

24. www.eschertech.com Tips To use constants and functions from Examples.pd: –Add file Examples.pd to the project –Add to Application.pd: import "Examples.pd"; You can convert any expression to a string –By calling.toString on it To make your application robust: –Verify your version of Application.pd –Don’t add any preconditions to the methods called from Java!

25. www.eschertech.com Sequences and Strings The standard collection types are: set of X, bag of X, seq of X (where X is any type you like) string  seq of char Members of class seq of X include: head tail front back append(x) prepend(x) ++(s) # (x)in take(n) drop(n) slice(offset, length) isndec isninc permndec permninc isOrdered(cp) !sort(cp) findFirst(x) findLast(x) Useful global functions include: flatten(ss: seq of seq of X) interleave(ss, s) See the Library Reference for details

26. www.eschertech.com Recursive and templated functions function reverse(s: seq of class T): seq of T decrease #s ^= ( [#s <= 1]: s, []: reverse(s.front).prepend(s.last) ); Indicates that T can be any type Recursion variant Recursive call

27. www.eschertech.com Recursion variants General form is: decrease e1, e2, e3 … e1, e2 … are of int, bool, char or an enumeration type The variant must decrease on each recursive call –Either e1 must decrease –Or e1 stays the same and e2 decreases –Or e1 and e2 stay the same and e3 decreases… Integer components must not go negative

28. www.eschertech.com Exercise 4: Sequences Specify the following functions: numLeadingSpaces(s: string): nat –returns the index of the first character in s that is not a space, or the length of s if it is all spaces removeLeadingSpaces(s: string): string –returns s with any leading spaces removed firstWord(s: string): string –returns the leading characters of s up to but not including the first space character in s splitIntoWords(s: string): seq of string –splits the sentence s into individual words (hint: use recursion!)

29. www.eschertech.com Lunch break!

30. www.eschertech.com Refinement There are three types of refinement in Perfect: –Refining result expressions to statement lists –Refining postconditions to statement lists –Refining abstract data to implementation data When you refine the abstract data of a class, you normally need to refine the method specifications as well So we will start with refining specifications

31. www.eschertech.com Refining specifications Specification refinement serves these purposes: –To implement a specification where Perfect Developer fails to –To implement a specification more efficiently –To take account of data refinement in affected methods You can refine these sorts of specification: –Expressions that follow the “^=” symbol –Postconditions To refine a specification, append to it: via statement-list end

32. www.eschertech.com function square(x: int): int ^= x^2 via value x*x end; schema swap(a!, b!: class X) post a!= b, b!= a via let temp ^= a; a! = b; b! = temp end; Some simple refinements value statement returns a value from the via..end Semicolon separates and sequences the statements A postcondition can be used as a statement

33. www.eschertech.com Nested Refinements You can refine not just method specifications but also: –Postcondition statements –Let-declarations in statement lists function fourth(x: int): int ^= x^4 via let x2 ^= x^2 via value x*x end; value x2*x2 end; value yielded by the inner via..end value yielded by the outer via..end

34. www.eschertech.com Types of Statement Let-declaration Assertion Variable declaration Postcondition pass statement If-statement value and done statements Loop statement Block statement Exactly the same as in bracketed expressions Omit the “post” keyword! Does nothing Same as in postconditions

35. www.eschertech.com If-statement if [c in `a`..`z`]: isAletter! = true; valid! = true; [c in `0`..`9`]: isAletter! = false; valid! = true; []: valid! = false fi Guard Statement list Optional “else” part [] fi means the same as []: pass fi

36. www.eschertech.com “value” statement function max(a,b,c: class X): X satisfy result >= a & result >= b & result >= c & (result=a | result=b | result=c) via if [a > b]: value max(a, c); []: value max(b, c) fi end; Every path in an expression refinement must end at a value statement

37. www.eschertech.com “done” statement schema max(a!,b,c: class X) post change a satisfy a’ >= a & a’ >= b & a’ >= c & (a’= a | a’= b | a’= c) via if [a > b]: a!= max(a, c); done; [] fi; a!= max(b, c) end; A postcondition refinement may contain one or more done statements Implicit done statement here

38. www.eschertech.com Loop statement // Calculate a^b var rslt: int! = 1; loop var j: nat! = 0; change rslt keep rslt’ = a^j’ until j’= b decrease b - j’; rslt! * b, j! + 1 end; Loop variant List of what the loop can change Start of loop statement Loop variable declaration Termination condition Loop invariant list Loop body End of loop statement

39. www.eschertech.com Loop statement loop local variable declarations (optional) change list (optional) invariant termination condition (optional) variant body statements end If no change list is given, only the local variables can be changed If no termination condition is given, the loop terminates when the variant can decrease no more

40. www.eschertech.com Designing loop invariants Variables in loop invariants may be primed or unprimed –Primed = current values at the start of an iteration –Unprimed = value before the loop started The invariant is the only source of information about current values of changing variables The state when the loop completes is given by: invariant & until-part The invariant should comprise: –A generalisation of the state that the loop is meant to achieve; –Additional constraints needed to make the invariant, until-part, variant and body well-formed

41. www.eschertech.com Example of invariant design Given s: seq of int we wish to achieve total’= + over s Generalise this to tot’= + over s.take(j’) for some loop counter j –When j = #s then the generalisation becomes the required state because s.take(#s) = s This generalisation forms part of the invariant –But s.take(j’) has precondition 0 <= j ’<= #s –So we must either add this precondition as an earlier invariant… –Or as a type constraint in the declaration of j

42. www.eschertech.com Loop example (incorrect) var totl: int! = 0; loop var j: int! = 0; change totl keep totl’= + over s.take(j’) until j’= #s decrease #s - j’; totl! + s[j], j! + 1 end; Problem! These expressions are not universally well- formed

43. www.eschertech.com Loop example (version 1) var totl: int! = 0; loop var j: int! = 0; change totl keep 0 <= j’<= #s, totl’= + over s.take(j’) until j’ = #s decrease #s - j’; totl! + s[j], j! + 1 end; Added this extra invariant at the start This is now well-formed This is also well-formed (provided the until- condition is false)

44. www.eschertech.com Loop example (version 2) var totl: int! = 0; loop var j: (int in 0..#s)! = 0; change totl keep totl’= over s.take(j’) until j’= #s decrease #s - j’; totl! + s[j], j! + 1 end; Added this type constraint This is now well-formed This is also well-formed (provided the until- condition is false)

45. www.eschertech.com function rev(s: seq of int): seq of int decrease #s ^= ([s.empty]: s, []: rev(s.tail).append(s.head)) via var rslt: seq of int! = seq of int{}; loop var j: (nat in 0..#s)! = 0; change rslt keep rslt’= rev(s.take(j’)) until j’= #s decrease #s - j’; rslt! = rslt.prepend(s[j]), j! + 1 end; value rslt end; Refining recursion to loops

46. www.eschertech.com Refining recursion to loops Is the preceding example correct? –Probably! –But Perfect Developer cannot verify it! The definition builds the result from front to back –Using append The implementation builds the result from back to front –Using prepend They are equivalent only because of associativity (a ++ b) ++ c = a ++ (b ++ c) reverse(x.tail).append(x.head) = reverse(x.front).prepend(x.last) To prove this we need an inductive prover!

47. www.eschertech.com function rev(s: seq of int): seq of int decrease #s ^= ([s.empty]: s, []: rev(s.tail).append(s.head)) via var rslt: seq of int! = seq of int{}; loop var j: (nat in 0..#s)! = #s; change rslt keep rslt’= rev(s.drop(j’)) until j’= 0 decrease j’; j! - 1, rslt! = rslt.append(s[j’]) end; value rslt end; Refining recursion to loops

48. www.eschertech.com Loops: a summary Getting the invariant correct is critical –It must describe the relationships between all variables changed by the loop (including the local loop variables) Its main part is a generalisation of the desired state after the loop –When the until condition is satisfied, the generalisation must reduce to the desired state You may also need to include constraints on variables –To make expressions in the loop well-formed If refining a recursive definition, make sure that the loop builds the result in the same order as the definition

49. www.eschertech.com Exercise 5: Method refinement Refine the following specifications function min2(x, y: int): int satisfy result <= x, result <= y, result = x | result = y; function findFirst(s: seq of int, x: int): int satisfy 0 <= result <= #s, result = #s | s[result] = x, forall j::0..<result :- s[j] ~= x; –Function numLeadingSpaces from exercise 4 –Function splitIntoWords from exercise 4

50. www.eschertech.com Data refinement When designing a class, we should always use the simplest possible abstract data model –Avoid redundancy! –Don’t be concerned with efficiency at this stage! The methods are specified in terms of this model –This keeps the specifications simple! The data should not be directly accessible from outside –So we can change the implementation of the data without changing the class interface –[Except for very simple classes like pair]

51. www.eschertech.com Data Refinement (cont’d) Perfect supports two sorts of data refinement: Replacing abstract variables by internal variables –Use a retrieve function to indicate that a variable is replaced –Examples: see Dictionary.pd and Queue.pd in C:\Program Files\Escher Technologies \Perfect Developer\Examples\Refinement Supplementing abstract variables by internal variables –The new internal data adds no new information –Declare internal invariants to specify the relationship –Example: add an index to a data structure –The internal data may be changed even within a function

52. www.eschertech.com Data Refinement Example We have a class that maintains a list of numbers Its constructor creates an empty list We provide a method to append a number to the list We provide a method to return the sum of all the numbers in the list

53. www.eschertech.com List of integers class function sum(s: seq of int): int decrease #s ^= ([s.empty]: 0, []: sum(s.front) + s.last); class ListOfNumbers ^= abstract var list: seq of int; interface function list; build{} post list! = seq of int{}; schema !addNumber(n: int) post list! = list.append(n); function getSum: int ^= sum(list); end;

54. www.eschertech.com Data Refinement Example Suppose that method “sum” is called frequently Save time by caching the sum of the list

55. www.eschertech.com Refined list of integers class class ListOfNumbers ^= abstract var list: seq of int; internal var totl: int; invariant totl = sum(list); interface function list; build{} post list! = seq of int{} via list! = seq of int{}, totl! = 0; end; …

56. www.eschertech.com Refined list of integers class … schema !addNumber(n: int) post list! = list.append(n) via list! = list.append(n), totl! + n end; function getSum: int ^= sum(list) via value totl end; end;

57. www.eschertech.com Exercise 6: Data Refinement Write a recursive function “longest” which, given a list of strings, returns the longest string in the list (or the empty string if the list is empty, or the latest one of several equal-length longest strings) Write a class that maintains a list of strings. You should provide: –A constructor, which sets the list to an empty list –A member schema to append a new string to the list –A member function to return the “longest” string in the list Refine the class to make the implementation of the “longest” member function more efficient

58. www.eschertech.com Inheritance When declaring a class you can inherit another class –Declare class UniversityMember … –Then class Student ^= inherits UniversityMember … –And class StaffMember ^= inherits UniversityMember … –And class Professor ^= inherits StaffMember … A derived class inherits the variables of its parent –But they are not [normally] visible in the derived class A derived class inherits the methods of its parent –But only confined and interface members of the parent are visible

59. www.eschertech.com The confined section The confined section behaves like the interface section –You can put the same types of declaration in it –i.e. Methods, operators, constructors, access redeclarations –Not constants, variables or invariants But confined declarations are only visible within the current class and its descendents –Not to the public at large! –cf. protected in Java and C++

60. www.eschertech.com Redefining methods Functions, selectors, schemas and operators that are inherited from a parent class may be redefined This must be indicated using the redefine keyword The parameter and result types in the redefinition must be identical to those in the overridden function

61. www.eschertech.com Example of overriding class UniversityMember ^= abstract var firstNames: seq of string, lastName: string; interface function getSalary: Money ^= 0; … end; class StaffMember ^= inherits UniversityMember abstract var salary: Money; interface redefine function getSalary: Money ^= salary; … end;

62. www.eschertech.com from types and Dynamic Binding You may declare a variable (or parameter, or result) to be of type “from C” where C is a class –e.g. var member: from UniversityMember The variable may be assigned a value of type C or any of its descendants –So member may be assigned a value of type Student, Professor … –“from C” actually means “the union of all non-deferred classes in the set comprising C and its direct and indirect descendents When calling a member function on such a variable, the [re]definition appropriate to the actual type is called –e.g. the relevant version of getSalary

63. www.eschertech.com Deferred methods You can also declare a method in a class deferred The method is left undefined in that class This avoids the risk of classes inheriting what may be an unsuitable definition The class itself must also be flagged deferred and may not be instantiated Descendent classes may define the method using the define keyword Any descendent class that does not define it is also a deferred class

64. www.eschertech.com Example of deferred method deferred class UniversityMember ^= abstract var firstNames: seq of string, lastName: string; interface deferred function getSalary: Money; … end; class StaffMember ^= inherits UniversityMember abstract var salary: Money ; interface define function getSalary: Money ^= salary; … end;

65. www.eschertech.com Final classes and methods A method can be declared final to prevent it from being redefined in a descendent class final function getSalary: Money ^= … You can also declare a method final when defining or redefining it define final function getSalary: Money ^= … redefine final function getSalary: Money ^= … You can declare a class final to mean that no other class may inherit it final class Professor ^= …

66. www.eschertech.com Some consequences If D is a deferred class: var x: D is illegal –But you can use var x: from D If F is a final class: var x: from F is illegal –But you can use: var x: F If C is a non-final class, f is a final method and g is a non-final method, and given var x: from C : In … x.f … the prover can assume the full postcondition of f In … x.g … the prover can assume only the postassertion of g

67. www.eschertech.com Preconditions and inheritance When a method is inherited, by default the precondition is inherited too You may override the inherited precondition by giving a new one in the method definition or redefinition The new precondition must be satisfied whenever the old one would have been satisfied –i.e. you may only weaken the precondition To get round this, have the precondition call a method that you can redefine

68. www.eschertech.com Inherited precondition example Suppose we declare a deferred method isPaid in class UniversityMember define this to return false for class Student, true for class StaffMember Add precondition pre isPaid to method getSalary

69. www.eschertech.com Inherited precondition example deferred class UniversityMember ^= abstract var firstNames: seq of string, lastName: string; interface deferred function isPaid: bool; deferred function getSalary: Money pre isPaid; … end; class StaffMember ^= inherits UniversityMember abstract var salary: int; interface define function isPaid: bool ^= true; define function getSalary: Money ^= salary; …

70. www.eschertech.com Inherited precondition example That worked OK for class StaffMember, but what about class Student? Method getSalary can never be called on class Student because its precondition is always false How should we declare it?

71. www.eschertech.com Absurd method example deferred class UniversityMember ^= abstract var firstNames: seq of string, lastName: string; interface deferred function isPaid: bool; deferred function getSalary: Money pre isPaid; … end; class Student ^= inherits UniversityMember interface define function isPaid: bool ^= false; absurd function getSalary; …

72. www.eschertech.com Absurd methods Declaring a method absurd means that its precondition is always false Repeat the parameter list of the method but not its return type An absurd method declaration has these consequences: –The method is defined or redefined such that calling it will raise an exception –A verification condition is generated (i.e. that the precondition is always false) –It avoids the “Given false, so proof is trivial” warnings you will otherwise see

73. www.eschertech.com Inheritance and postassertions When defining or redefining an inherited method, by default the postassertion is inherited You may override the postassertion by giving a new one The old postassertion must be satisfied whenever the new one is –i.e. you may only strengthen the postassertion You can also use: assert …, q –This means assert the inherited postassertion and then q

74. www.eschertech.com Inheritance tips When using inheritance, declare methods final where possible –This is not necessary in leaf classes which are declared final For non-final methods of non-final classes, postassertions are very important –Because the prover needs them when the methods are called on from types –Does not apply to “defining” methods like isPaid Only use from types where you really need to allow different types at run-time

75. www.eschertech.com Inheritance exercises Design a UniversityMember or Employee class hierarchy that reflects the properties and privileges of members of your organisation Specify a family of shopping scanners, as outlined at the end of the Shopping Scanner worked example at: http://www.eschertech.com/teaching/scanner_example.pdf

76. www.eschertech.com What we didn’t cover Lots! –after expressions –over expressions –Guarded variable declarations –Selectors –Members of classes set and bag –Declaring templated classes –Other library classes, e.g. map of (X -> Y) –How to solve verification problems –Serialization –Declaring axioms –…

77. www.eschertech.com Further Reading Perfect Developer 3.0 Language Reference Manual –Start -> Programs -> Perfect Developer -> Documentation –Or click on the book tool in the Project Manager –Also available at www.eschertech.com Online tutorial –via Support -> Self-help section of the web site Teaching materials –via Support -> Self-help -> Teaching materials

78. www.eschertech.com Thank you for participating!