Problem 1:
The top number at each node is its state; the lower number is the gametree
value of the node, propagated upward.
MAX MIN MAX MIN LEAF
100 A> 97 C> 105 A> 107 C> 101
101  101  101  101 
   D> 122
  
  B> 101 C> 117
  53 
  D> 53
 
 D> 145 A> 141 C> 135
 >=135 135 
 B branch pruned D> 161

B> 98 C> 108 A> 113 C> 119
<=101 101  91 
D branch pruned  D> 91

B> 107 C> 101
101 
D> 122
The value of the starting node is 101.
X will make move A. (In fact move B is just as good  they both
lead to a final state of 101  but after doing alphabeta pruning
X does not know that B achieves 101, he just knows it can't be better
than 101.)
Problem 2
A. Converting these sentence to CNF gives
Set S0:
A. ~p V ~q V r
B. ~p V q V ~r
C. ~p V x V w
D. ~w V q.
E. ~w V ~r
F. ~x V ~p V w.
G. ~x V p V ~w
There are no singleton clauses or pure literals, so try the assignment p=TRUE
Set S1: (p=TRUE)
A. ~q V r
B. q V ~r
C. x V w
D. ~w V q.
E. ~w V ~r
F. ~x V w.
There are still no singleton clauses or pure literals, so try q=TRUE
Set S2: (p=TRUE, q=TRUE)
A. r
C. x V w
E. ~w V ~r
F. ~x V w.
A is a singleton clause, so set r=TRUE.
Set S3: (p=TRUE, q=TRUE, r=TRUE)
C. x V w
E. ~w
F. ~x V w.
E is a singleton clause, so set w=FALSE
Set S4: (p=TRUE, q=TRUE, r=TRUE, w=FALSE)
C. x
F. ~x
C is a singleton clause, so set x=TRUE
Set S5: (p=TRUE, q=TRUE, r=TRUE, w=FALSE, x=TRUE)
F. empty
F is the empty clause, so fail. Backtrack to the last choice point,
which was setting q=TRUE at S1. From S1, set q=FALSE.
Set S6: (p=TRUE, q=FALSE)
B. ~r
C. x V w
D. ~w
E. ~w V ~r
F. ~x V w.
B and D are singleton clauses, so set r=FALSE, w=FALSE.
Set S7: (p=TRUE, q=FALSE, r=FALSE, w=FALSE)
C. x
F. ~x
C is a singleton clause, so set x=TRUE
Set S8: (p=TRUE, q=FALSE, r=FALSE, w=FALSE, x=TRUE)
F. empty
F is the empty clause, so fail. Backtrack to the last choice point,
which was setting p=TRUE at S0. From S0, set p=FALSE.
Set S9: (p=FALSE)
D. ~w V q.
E. ~w V ~r
G. ~x V ~w
~w is a pure literal, so set w=FALSE.
Set S10: (p=FALSE,w=FALSE)
Empty set of clauses. Return the valuation p=FALSE, w=FALSE.
The other atoms may be set arbitrarily.