1. 12 April 2009Instructor: Tasneem Darwish1 University of Palestine Faculty of Applied Engineering and Urban Planning Software Engineering Department Formal Methods Functions

2. 12 April 2009Instructor: Tasneem Darwish2 Outlines Partial functions Lambda notation Functions on relations

3. 12 April 2009Instructor: Tasneem Darwish3 Partial Functions A partial function from X to Y is a relation that maps element of X to at most one element of Y ; we write to denote the set of all such relations Example 8.1 An organisation has a system for keeping track of its employees while they are on the premises. Each employee is issued with an active badge which reports their current position to a central database. If the set of all people is Person, and the set of all locations is Location, then the information provided by the system may be described by a relation where_is of type Person↔Location. It is impossible for an employee to be in two places at once, so this relation will be a partial function:

4. 12 April 2009Instructor: Tasneem Darwish4 Partial Functions These relations are called partial functions because there may be elements of X that are not related to any element of Y. If each element of X is related to some element of Y, then the function is said to be total; we write X →Y to denote the set of all total functions from X to Y Example 8.2 We may define a relation double on the set of natural numbers N as follows: This relation is a total function: for every natural number m there is a unique number n such that

5. 12 April 2009Instructor: Tasneem Darwish5 Partial Functions If a lies within the domain of a function f, we write f (a) to denote the unique object that is related to x We say that f (a) is the result of applying function f to argument a There are two inference rules associated with function application:

6. 12 April 2009Instructor: Tasneem Darwish6 Partial Functions Example 8.3 If Rachel is an employee, then we may write where_is rachel to denote her current location. Where_is = =

7. 12 April 2009Instructor: Tasneem Darwish7 Lambda notation Suppose that f is a function whose domain is precisely those elements of X that satisfy a constraint p. If the result of applying f to an arbitrary element x can be written as the expression e, then f can be described as The lambda notation offers a more concise alternative; Example 8.4 The function double could also be defined by

8. 12 April 2009Instructor: Tasneem Darwish8 Lambda notation Example 8.5 The function min maps a set of natural numbers to the least number present in that set: the minimum value If the declaration part of a lambda expression introduces more than one variable, then the source type of the function is given by the resulting characteristic tuple The source is the Cartesian product set

9. 12 April 2009Instructor: Tasneem Darwish9 Lambda notation Example 8.6 The function pair takes two functions f and g as arguments, each of which must be a homogeneous relation on N. The result is a function that takes a natural number n and returns the pair formed by applying each of f and g to n: Example 8.7 Let triple be the function

10. 12 April 2009Instructor: Tasneem Darwish10 Functions on Relations Example 8.8 The domain and range operators may be defined by

11. 12 April 2009Instructor: Tasneem Darwish11 Functions on Relations Example 8.9 The restriction operators may be defined by

12. 12 April 2009Instructor: Tasneem Darwish12 Functions on Relations Example 8.10 The relational composition operator may be defined by

13. 12 April 2009Instructor: Tasneem Darwish13 Functions on Relations Example 8.11 The relational inverse operator may be defined by

14. 12 April 2009Instructor: Tasneem Darwish14 Functions on Relations Example 8.12 The transitive closure operator may be defined by