15.1
a. none

b. p(Roy, Tim), [Roy/y, Tim/x]

c. p(1,1) [1/x]

d. none

e. p(1,1) [1/x, 1/y]

f. p(f(x,y)) [f(x,y)/z]

g. none

h. none (an occurs check)

15. 2 p(1,1,1) [1/r, 1/y, 1/w, 1/q, 1/x]

16.1
a. x is the parent of y if x is the father of y or x is the mother of y.

x is the uncle of y if:
x is the brother of  z and z is a parent of y
or x is the husband of  z and z is the sister of w and w is a parent of y

b. Vx1 x2 x3 x4 x5 y1 y2 y3 y4 y5 z3 z4 z5 w5|
(is_father(x1,y1) => is_parent(x1,y1)) and
(is_mother(x2,y2) => is_parent(x2,y2)) and
(is_brother(x3,z3) and is_parent(z3,y3) => is_uncle(x3, y3)) and
 (is_husband(x5,z5) and is_sister(z5, w5) and is_parent(w5,y5) =>
is_uncle(x5, y5))

c. Vx1 x2 x3 x4 x5 y1 y2 y3 y4 y5 z3 z4 z5 w5|
(~is_father(x1,y1) V is_parent(x1,y1)) ^
(~is_mother(x2,y2) V is_parent(x2,y2)) ^
(~is_brother(x3,z3) V ~is_parent(z3,y3) V is_uncle(x3, y3)) ^
 (~is_husband(x5,z5) V ~is_sister(z5, w5) V ~is_parent(w5,y5) V
is_uncle(x5, y5))

d. is_parent(x,y):-is_father(x,y).
is_parent(x,y):- is_mother(x,y).
is_uncle(x,y) :- is_brother(x,z), is_parent(z,y).
is_uncle(x,y) :- is_husband(x,z), is_sister(z,w), is_parent(w,y).

Extra Credit.

15.6  with
Clause 1: A(f(x,y),x,y)
Clause 2: ~A(z, g(z,1),3)
Clause 3: ~A(r, w, u) v A(w, v, 3) v ~A(r, 3, t) v A (u, v, t)
Clause 4: ~A(h(m), m, h(m)).

C2, C3 |- C5: [z/w, g(z,1)/v] ~A(r, z, u) v ~A(r, 3, t) v A (u, g(z,1), t)
C5, C2 |- C6: [z/u, 3/t] ~A(r, z, z) v ~A(r, 3, 3)
C6, C1 |- C7: [f(x,y)/r, 3/x, 3/y] ~A(f(3,3), 3, 3)
C7: C1 |- c8: [3/x, 3/y] empty clause. done.

16-16
1: adjective(the).
2: adjective(a).
3: adjective(an).
4: verb(solved).
5: verb(ate).
6: noun(student).
7: noun(problem).
8: noun(test).
9: noun(apple).
10: verbphrase((x.nil)):-verb(x).
11: verbphase((x.y)):-verb(x), nounphrase(y).
12: nounphrase((x.(y.nil)):-adjective(x), noun(y)).
13: sentence(x):-concat(n,v,x), nounphrase(n), verbphase(v).
I assume concat defined elsewhere

14: ?-sentence((the student solved the problem)).
14, 13 |- 15: ?- concat((the student), (solved the problem)),
nounphrase((the student)), verbphrase((solved the problem)).
15, 12 |- 16: ?-adjective(the), noun(student), verbphrase((solved the
problem)).
16, 1 |- 17: ?-noun(student), verbphrase((solved the problem)).
17, 6 |- 18: ?-verbphrase((solved the problem)).
18, 11 |- 19: ?-verb(solved), nounphrase((the problem)).
19, 4 |- 20: ?-nounphrase((the problem)).
20, 12 |- 21: ?-adjective(the), noun(problem).
21, 1 |- ?-noun(problem).
22, 7 |- 23: empty. done