Question

Formal Methods Introduction

1. What are some possible disadvantages of formal methods?
2. Given the sets

             A = {1,2,5,6},    B = {1,3,5,7,9},     C= {x|x Element_of A and x < 4}
   

   What are the following:
   • A union B
   • B intersection C
   • A - B (also written A \ B)
   • P A (ie Powerset of A)
   • A \ A (also written A - A)
3. Given the function f = { (1,1), (2,3), (3,7), (4,0), (5,10) }, write down the following:
   • dom(f)
   • ran(f)
   • f override_op { (1,0) }
   • f domain_subtraction_op {3,4}
   • f range_subtraction_op {0,3}
4. Given the declarations

           x,y,z : Z;   (where Z is the set of integers)
           a : Author;
           b : Book;
           on_shelves : P Book;
           novelists : P Author;
   

   What is the type of each of the following:
   1. x
   2. x + y
   3. {x,y}
   4. {{x}, {y}}
   5. {on_shelves}
   6. on_shelves union {b}
5. Develop expressions using logic and set operators for the following statements:Use the function expressions M(x,y) and G(x,y) to represent the "mother of" and the "grandmother of" functions respectively.
   Use A(x) for the function that given a persons name returns the age of the person in years.
   Write down the type of each of the above functions.
   • "For all x and y, if x is mother of y and y is the mother of z, then x is the grandmother of z."
   • "Everyone older than 30 is a mother."
6. Consider the sets

                Lecturer = {Ray, Lindsay, Kim, Ken, Wendy}
                Subject  = {SE, DB, HC, CP}
   
                teaches = { (Ray,SE), (Lindsay,DB), (Kim,HC), (Ken,CP), (Wendy,HC) }
   

   1. Is "teaches:Lecturers->Subjects" a legitimate function? Why or why not?
   2. Is it possible to write the inverse function that given a Subject returns a Lecturer?
7. Develop expressions using logic and set operators for the following statements:
   • "There exists some integer less than 100"
   • "For every integer x there is an integer y such that x-y = 1"
   • "Every person x has a friend y that has a friend z that is an enemy of x"
   • "All the girls love a sailor"

Answer 1

1. Disadvantages of formal methods: Here I assume that it is with respect to software engineering approach.
   • System customers are unlikely to be familiar with formal specification techniques.
   • Some classes of problems are hard to describe using formal methods
   • Some software engineers are not trained to use and develop the formal models
   • Extensive training for personnel is required
   • It may become time consuming in some cases
2. Given
   4-11, 2, 5,6
   ,
   B= {1.3.5.7.9}
   and $C = \{x|x\in A \land x<4\} = \{1,2\}$
   • 
     AUB-1,2,3,5, 6,7,9^
     
   • $B\cap C = \{1\}$
   • 
   • $\mathcal{P}(A) = \{\emptyset, \{1\},\{2\}, \{5\}, \{6\}, \{1,2\},\{1,5\}, \{1,6\}, \{2,5\}, \{2,6\}, \{5,6\}, \{1,2,5\}, \{1,2,6\}, \{1,5,6\}, \{2,5,6\}, \{1,2,5,6\} \}$
   • $A-A = \emptyset$
3. Given
   f = {(1, 1). (2.3). (3.7). (4:0). (5. 10)}
   
   • 
     don( f" ) { 1. 2. 3. 4. 5}
     
   • 
     range(f)-(0. 1. 3. 7. 10}
     
   • $f~ \texttt{override\_op} ~\{(1,0)\} = \{ (1,0), (2,3), (3,7), (4,0), (5,10) \}$
   • $f ~\texttt{domain\_subtraction\_op} ~\{3,4\} = \{ (1,1), (2,3), (5,10) \}$
   • $f ~\texttt{range\_subtraction\_op} ~\{0,3\} = \{ (1,1), (3,7), (5,10) \}$​​​​​​​