Functions

1. Functions Chapter 10 Formal Specification using Z

2. A function is a relation • In a programming language a function is a way of specifying some processing which produces a value as a result. • In Z a function is a data structure. • The two views are not incompatable; the programming language view is just a restrictive form of the Z view and in both cases a function provides a result value given an input value or values.

3. A function in C int addTwo(int arg) Example of usage { y = addTwo(7); return (arg+2); } addTwo() arg1 arg2 return1 return2 Return value Argument

4. A function is a relation • A function is a special case of a relation in which there is at most one value in the range for each value in the domain. A function with a finite domain is also known as a mapping. Note in diagram no diverging lines from left to right. f x5 x1 x2 x4 x6 y1 y2 X x3 Y dom f ran f

5. A function is a relation • In Z a function f from type X to type Y is declared as • f: X Y ‘the function f, from X to Y’ • Is equilanent to relation f: X  Ywith the restriction that for each x in the domain of f, f relates x to at most one y; f x5 x1 x2 x4 x6 y1 y2 X x3 Y dom f ran f

6. Is equilanent to relation f: X  Ywith the restriction that for each x in the domain of f, f relates x to at most one y; • Which can be stated formally as: • x dom f  1y:Y  xfy Target Source x5 f x1 x2 x4 x6 y1 y2 X x3 Y dom f ran f

7. Examples of functions • The relation between persons and their identity numbers • identityNo: PERSON  • is a function if there is a rule that a person may only have one identity number . In that case it could be declared as: • identityNo: PERSON  • There would probably also be a rule that only one identity number may be associated with any one person, but that is not indicated here.

8. Examples of functions • A functions source and target can be the same: • hasMother: PERSON PERSON • any person can have only one mother and several people could have the same mother.

9. Function Application • All the concepts which pertain to relations apply to functions. • In addition a function may be applied. • Since there will be at most only one value in the range for a given x it is possible to designate that value directly. • The value of fapplied to x is the value in the range of function of the function f corresponding to the value of x in its domain.

10. Function Application • The application is undefined if the value of x is not in the domain of f. • It is important to check that the value of x is in the domain of the function f before attempting to apply f to x. • The application of the function to the value x (the argument) is written: fx or f(x)

11. Function Application • We can check the applicability of f by writing: • x  domf • fx = y • These predicates must be true if f(x) = y is a function.

12. Partial and Total Functions • The identyNo function is a partial function there may be values in the sources that are not in the domain (some people may have no identity number). • A total function is one where there is a is a value for every possible value of x, so fx is always defined. It is written • f: X Y • Because: • domf = X (or source) • function application can always be used with a total function. Note no ‘bar’ on total function arrow.

13. Examples of Total Functions • The function age, from PERSON to natural number, is total since every person has an age: • age: PERSON   • The function hasMother, is total since every person has one mother (including a mother!): • hasMother : PERSON  PERSON

14. Formal Partial and Total functions • The set of Partial Functions from X to Y: X  Y =={f:X Y | ( x:X | xdom f  (1 y:Y xfy)) f} • The set of Total Functions from X to Y: X  Y =={f:X  Y | (dom f =X f}

15. Other Classes of Functions: injective functions (one-to-one) • An injection or (injective function) is a function which maps different values of the source on to different values of the target, e.g. f: X Y • The inverse relation of an injective function f from X to Y , f~, is itself an injective function from, from Y to X. • f~Y  X • An injective function may be partial f: X Y or total f: X Y . The functions identityNo and MonogamousMarriage areinjective.

16. An Injective function. The function f: X  Yis injective (or one-to-one) because each element in Y has one arrow pointing to it. A injective can be partial, f: X  Y, or total f: X Y ; Target Source f x1 x2 x3 y1 y2 y3 X Y dom f ran f

17. An Injective function can be: partial, f: X  Y =={f:X  Y | f~ Y  X f} (has inverse) or total f: X  Y=={f:X  Y | dom f = X  f} Target Source f x1 x2 x3 y1 y2 y3 X Y dom f ran f

18. A Surjective function. The function f: X  Yis surjective (or onto) because range of is all of Y. The range is the whole of the target. A surjective can be partial, f: X  Y, or total f: X  Y ; Target Source f x1 x2 x3 x4 x4 y1 y2 y3 X Y dom f ran f

19. A Surjective function can be: partial f: X  Y == {f:X  Y | ran f = Y  f} or total f: X  Y == {f:X  Y | dom f = X } Target Source f x1 x2 x3 x4 y1 y2 y3 X Y dom f ran f

20. A Bijective function. The function f: X  Yis a total function that is both injective (one-to-one) and surjective (or onto) is called a bijective function, ; Target Source f x1 x2 x3 y1 y2 y3 X Y dom f ran f

21. A Bijective function. The function f: X  Yis a total function that is both injective (one-to-one) and surjective (or onto) is called a bijective function, f: X  Y == (X  Y)  (X  Y); Target Source f x1 x2 x3 y1 y2 y3 X Y dom f ran f

22. Some examples, range on the left domain on the right A function is called a bijection , if it is onto and one-to-one. A bijective function has an inverse Function Application A function f from X to Y is a relation from X to Y having the properties: 1. The domain of f is in X. 2. If (x,y), (x,y’) f, then y=y’ (no fan out from domain). A total function from X to Y is denoted f: X Y.

23. Example Function Given SIDE ::= left | right the function driveON COUNTRY  SIDE side of the road vehicles drive on is a surjection, because the range includes all the values of the type SIDE. It would usually be considered total because driveOn should be defined in all countries. driveON COUNTRY SIDE

24. What is a Hash Function? • A hash function takes a data item to be stored or retrieved (say to/from disk) and computes the first choice for a storage location for the item. For example assume StudentNumber is the key field of a student record, and one record is stored on one disk block. To store or retrieve the record for student number n in a choice of 11 disk locations numbered 0 to 10 we might use the hash function. • h(n) = n mod 11

25. Describe the steps needed to store a record for student number 257 using in locations disk 0-10 using h(n) = n mod 11 558 32 132 102 15 5 0 1 2 3 4 5 6 7 8 9 10 Steps to store 257 using h(n)=n mod 11 into locations 0-10 1. Calculate h(257) giving 4. 2. Note location is already occupied, a collision has occurred. 3. Search for next free space , with 0 assumed to follow 10. 4. If no free space then array is full, otherwise store 257 in next free space 558 32 132 102 15 5 257 0 1 2 3 4 5 6 7 8 9 10

26. Hash Function Collision, Search Criterion • A collision occurs if h(x) = h(y), for x  y • A collision resolution policy is required. One simply approach is to find the next unoccupied cell. • The search criterion must be an equality condition on a single field. Not suited to range searches (e.g. Find StudentNumber < 123) • Range: StudentNumber 1…. 99,999,999.

27. Hash Function Collision • h(x) does not guarantee that distinct key values will hash to distinct disk address . • The domain may be much larger than the range If that is the case some keys will map onto the same disk address.

28. Hash Function Collision • The hash function maps the hash field space to the disk address space. It is usually a partial surjective function • HashFieldSpace  DiskAddressSpace. • What are the implications of the hash function being a total function?

29. Hash Function Collision • Problem if the the hash function is total i.e. defined for all elements in the domain. Many collisions • h(n) = n mod 3 Disk Block space =3 Key Space = 6 x1 x2 x3 x4 x6 y0 y1 y2 N A

30. Hash Function Collision • Also important is the selection of keys from the key space. • Will they be random or clustered?

31. What properties should the function possess? • If the domain is greater than the range it is should a partial surjective function. • surjective: The range is the whole of the target, all the disk addresses could be used. • partial there may be values in the sources that are not in the domain, not all keys will be used . HashFieldSpace  DiskAddressSpace

32. Is hash efficient? • If collisions occur infrequently and if when one does occur it is resolved quickly, then hashing provides a very fast method for storing and retrieving data. As an example, personnel data are frequently stored and retrieved by hashing on employee identification number. • The hash function h should distribute the records uniformly over the storage locations; otherwise, search time will be increased because many collisions occur.

33. Functions and Hash Functions • When a hash function is used, the following conditions could hold: #source > #target and #domain < #source and #range=#target . • Where source is all possible record keys and target is actual disk blocks available. The domain is actual records keys and the range is the actual disk blocks available. • What are the implications of a hash function being a total function? • What are the implications of a hash function being an invective function? • What are the implications of a hash function being a bijective function? • In general what are the desirable properties that a hash function should possess?

34. Functions and Hash Functions • When a hash function is used, the following conditions could hold: #source > #target and #domain < #source and #range=#target . • Where source is all possible record keys and target is actual disk blocks available. The domain is actual records keys and the range is the actual disk blocks available. • What are the implications of the hash function being a total function? • It is not stated wheather the #domain>#range if it is the problem is many collisions occur when the hash function is total i.e. defined for all elements in the domain.

35. Functions and Hash Functions • When a hash function is used, the following conditions could hold: #source > #target and #domain < #source and #range=#target . • Where source is all possible record keys and target is actual disk blocks available. The domain is actual records keys and the range is the actual disk blocks available. • What are the implications of a hash function being a invective function? • One to one, an element in the domain will have a corresponding unique element in the range:- no collisions. But some source elements may not be mapped into the the range.

36. Functions and Hash Functions • When a hash function is used, the following conditions could hold: #source > #target and #domain < #source and #range=#target . • Where source is all possible record keys and target is actual disk blocks available. The domain is actual records keys and the range is the actual disk blocks available. • What are the implications of a hash function being a bijective function? • The property of having an inverse is not normally of interest as we are usually working from domain (HashFieldSpace) to the range(DiskAddressSpace).

37. Exercise 1 • A system records the booking of hotel rooms on one night. Given base types: • [ROOM] set of all rooms in hotel • [PERSON] set of all possible persons • the state of the hotel’s bookings can be represented by the following schema:

38. Exercise 1 Hotel___ bookedTo: ROOM  PERSON _______ (a) Explain why bookedTo is a function. (b) Explain why bookedTo is partial.

39. Exercise 1 Answer Hotel___ bookedTo: ROOM  PERSON _______ (a) Explain why bookedTo is a function. • bookedTo is a function since it maps rooms to persons and for any given room(domain) at most one person (range) can book it. A person can book any number of rooms.

40. Exercise 1 Answer Hotel___ bookedTo: ROOM  PERSON _______ (b) Explain why bookedTo is partial. The function is partial since not all rooms have been booked.

41. Exercise 2 Given: Init___ Hotel’ ______ bookedTo =  _______

42. Exercise 2 Given, first version of accept booking operation: AcceptBooking0___ Hotel p?:PERSON r?: ROOM _____________ r? dom bookedTo bookedTo’ = bookedTo  {r?  p?} _________________ Explain the meaning of each line

43. Exercise 2 Answer  AcceptBooking0_ (schema name) Hotel (incorporates Hotel and its variables) p?:PERSON ( person input variable) r?: ROOM ( room input variable) _____________ r? dom bookedTo ( room not booked) bookedTo’ = bookedTo  {r?  p?} _________________ ( the maplet relating the room to the person is included in the new value of the function bookedTo)

44. Exercise 3 Write a schema cancelBooking0 which cancels a booking for a given person and a given room. It should deal with error conditions in a similar manner as AcceptBooking0

45. Exercise 3 Answer (a) CancelBooking0_ Hotel p?:PERSON r?: ROOM _____________ {r?  p?} dom bookedTo bookedTo’ = bookedTo \ {r?  p?} _________________ Explain the meaning of each line

46. Exercise 3 Answer (b) CancelBooking0_ (schema name) Hotel (incorporates Hotel and its variables) p?:PERSON ( person input variable) r?: ROOM ( room input variable) _____________ {r?  p?} dom bookedTo ( room is booked) bookedTo’ = bookedTo \ {r?  p?} _________________ ( the maplet relating the room to the person is removed from the new value of the function bookedTo)