Bemærkninger til vintereksamen 2000

Opgave 1

Spørgsmål a:

    map2 :: [a -> b] -> [[a] -> [b]]
    mapd :: (a -> b) -> [[a]] -> [[b]]

Spørgsmål b:

(i)   X = 1
      X = 2
(ii)  X = 1  Y = 1
      X = 1  Y = 2
      X = 2  Y = 1
      X = 2  Y = 2
(iii) X = 1  Y = 1
      X = 1  Y = 2

Opgave 2

Opgave 3

Spørgsmål a:

    levelsWith _ [] = []
    levelsWith j l = p : levelsWith (2*j) r
                     where (p,r) = splitAt j l

Spørgsmål b:

    addLayer [] _ = []
    addLayer (x:xs) [] = Fork x Null Null : addLayer xs []
    addLayer (x:xs) (t:[]) = Fork x t Null : addLayer xs []
    addLayer (x:xs) (t0:t1:ts) = Fork x t0 t1 : addLayer xs ts

Spørgsmål c:

    mkHtree = head . mkHtrees . levels

Opgave 4

Spørgsmål a:

    maxtree(OldTree,NewTree) :-
      maxnode(OldTree,M),forcevalue(OldTree,M,NewTree).

    maxnode([],0).
    maxnode(n(V,L,R),M) :-
      maxnode(L,ML),maxnode(R,MR),max_list([V,ML,MR],M).

    forcevalue([],_,[]).
    forcevalue(n(_,OL,OR),M,n(M,NL,NR)) :-
      forcevalue(OL,M,NL),forcevalue(OR,M,NR).

Spørgsmål b:

    maxtree2(OldTree,NewTree) :- mt(OldTree,NewTree,M,0,M).

    mt(n(V,A,B),n(H,A1,B1),H,AC,N) :-
        V < AC,
        mt(A,A1,H,AC,ACA),
        mt(B,B1,H,ACA,N).
    mt(n(V,A,B),n(H,A1,B1),H,AC,N) :-
        V>= AC,
        mt(A,A1,H,V,ACA),
        mt(B,B1,H,ACA,N).
    mt([],[],_,A,A).

Opgave 5

Spørgsmål a:

    p(x)
    p(y) # ~q(x,Z) # p(Z)
    ~p(f(x,y)) # ~q(x,Z) # p(Z)

Spørgsmål b:

    ~r & (~p # ~q # r)  |==  ~p # ~q
    !! & (~t # p)  |==  ~q # ~t
    !! & (~u # q)  |==  ~u # ~t
    !! & (~s # u)  |==  ~s # ~t
    !! & (~s # t)  |==  ~s
    !! & s  |==