Exercises (Prolog)

1. A predicate length/2 describing the length of a list can be defined as follows.

     length(0,[]).
     length(N,[_|L]) :- length(M,L), N is M+1.
   

   Please define predicates member/2, fac/2, fib/2, and gcd/3 analogously.
   
   
2. Suppose predicates parent/2, female/1 i male/1 with the obvious interpretation are given.
   Please define predicates child/2, mother/2, sister/2, has_a_child/1, grandparent/2, and predecessor/2.
   
   
3. Suppose the following predicate f/2 is given.

    f(1,one).
    f(s(1),two).
    f(s(s(1)),three).
    f(s(s(s(X))),N) :- f(X,N). 

   How does Prolog answer to the following questions?
   
   
   1. f(s(1),A)?
   2. f(s(s(1)),two)?
   3. f(s(s(s(s(s(s(1)))))),C)?
   4. f(D,three)?
   
   
4. Please define a predicate latin/2, which transforms Latin numbers into arabic numbers. Latin numbers are represented by ordinary lists,
   for example [x,l,v,i,i] denotes 47. You can assume that the Latin number is correct.
   
   
5. Assume the following predicates are given.

    q1(X,Y) :- p(X,Y).                q2(X,Y) :- p(X,Z), q2(Z,Y).
    q1(X,Y) :- p(X,Z), q1(Z,Y).       q2(X,Y) :- p(X,Y).  
   
    q3(X,Y) :- p(X,Y).                q4(X,Y) :- q4(X,Z), p(Z,Y).  
    q3(X,Y) :- q3(X,Z), p(Z,Y).       q4(X,Y) :- p(X,Y). 
       
    p(tom,bob).
    p(tom,liz).
    p(bob,ann).      
    p(bob,pat).     
    p(pat,jim). 

   Using answer trees please show how Prolog answers to the following questions: ?-q_i(tom,pat). and ?-q_i(liz,jim) for i = 1, ... 4.