C={r(a),r(e),p(c),q(b),s(a,b),s(d,b),s(e,d)}

C ∪ {q(a)} by rule 2 with X/a,Y/b

C ∪ {p(a)} by rule 1 with X/a

C ∪ {q(d)} by rule 2 with X/d,Y/b

C ∪ {q(e)} by rule 2 with X/e,Y/d

C ∪ {p(e)} by rule 1 with X/e

yes(R) ⇐ two_doors_east(R,r107)
   resolve with two_doors_east(E1,W1) ←
        imm_east(E1,M1) ∧imm_east(M1,W1).
   substitution: {E1/R,W1/r107}
yes(R) ⇐ imm_east(R,M1) ∧imm_east(M1,r107)
   select leftmost conjunct
   resolve with imm_east(E2,W2)←imm_west(W2,E2)
   substitution: {E2/R,W2/M1}
yes(R) ⇐ imm_west(M1,R) ∧imm_east(M1,r107) (*)
   select leftmost conjunct
   resolve with imm_west(r107,r109)
   substitution: {M1/r107,R/r109}
yes(r111) ⇐ imm_east(r107,r107)
   resolve with imm_east(E3,W3)←imm_west(W3,E3)
   substitution: {E3/r107,W3/r107}
yes(r111) ⇐ imm_west(r107,r107)
   no substitution available
   BACKTRACK to (*) with another resolvent, 
      until imm_west(r107,r109) is chosen 

By selecting the rightmost conjunct, there would be only one choice:

yes(R) ⇐ two_doors_east(R,r107)
   resolve with two_doors_east(E1,W1) ←
        imm_east(E1,M1) ∧imm_east(M1,W1).
   substitution: {E1/R,W1/r107}
yes(R) ⇐ imm_east(R,M1) ∧imm_east(M1,r107)
   select rightmost conjunct
   resolve with imm_east(E2,W2)←imm_west(W2,E2)
   substitution: {E2/M1,W2/r107}
yes(R) ⇐ imm_east(R,M1) ∧imm_west(r107,M1)
   select rightmost conjunct
   resolve with imm_west(r107,r109)
   substitution: {M1/r109}
yes(R) ⇐ imm_east(R,r109)
   resolve with imm_east(E3,W3)←imm_west(W3,E3)
   substitution: {E3/R,W3/r109}
yes(R) ⇐ imm_west(r109,R)
   resolve with imm_west(r109,r111)
   substitution: {R/r111}
yes(r111) ⇐ 

It is optimal because the query had a constant in its second argument, therefore instantiating the variable W of the two_doors_east predicate. But it would not be optimal for queries like:

ask two_doors_east(r107,R).

(a)

yes(R) ⇐ two_doors_east(r107,R)
   resolve with two_doors_east(E1,W1) ←
        imm_east(E1,M1) ∧imm_east(M1,W1).
   substitution: {E1/r107,W1/R}
yes(R) ⇐ imm_east(r017,M1) ∧imm_east(M1,R)
   select leftmost conjunct
   resolve with imm_east(E2,W2)←imm_west(W2,E2)
   substitution: {E2/r107,W2/M1}
yes(R) ⇐ imm_west(M1,r107) ∧imm_east(M1,R)
   select leftmost conjunct
   resolve with imm_west(r105,r107)
   substitution: {M1/r105}
yes(R) ⇐ imm_east(r105,R)
   resolve with imm_east(E3,W3)←imm_west(W3,E3)
   substitution: {E3/r105,W3/R}
yes(R) ⇐ imm_west(R,r105)
   resolve with imm_west(r103,r105)
   substitution: {R/r103}
yes(r103) ⇐ 

(b)

yes(R) ⇐ next_door(R,r107)  (*)
   resolve with next_door(E1,W1) ⇐ imm_east(E1,W1).
   substitution: {E1/R,W1/r107}
yes(R) ⇐ imm_east(R,107)
   resolve with imm_east(E2,W2)←imm_west(W2,E2)
   substitution: {E2/r105,W2/R}
yes(R) ⇐ imm_west(r107,R)
   resolve with imm_west(r107,r109)
   substitution: {R/r109}
yes(r109) ⇐ 

Another derivation by backtracking to (*) and selecting another clause to resolve against:

yes(R) ⇐ next_door(R,r107)
   resolve with next_door(W1,E1) ⇐ imm_west(W1,E1).
   substitution: {W1/R,E1/r107}
yes(R) ⇐ imm_west(R,r107)
   resolve with imm_west(r105,r107)
   substitution: {R/r105}
yes(r105) ⇐ 

(c)

…similar to (d)…

(d)

yes(R) ⇐ west(r107,R)  (*)
   resolve with west(W1,E1) ⇐ imm_west(W1,E1).
   substitution: {W1/r107,E1/R}
yes(R) ⇐ imm_west(r107,R)
   resolve with imm_west(r107,r109)
   substitution: {R/r109}
yes(r109) ⇐ 
 

Another derivation by backtracking to (*) and selecting another clause to resolve against:

yes(R) ⇐ west(r107,R)  (*)
   resolve with west(W1,E1) ⇐
        imm_west(W1,M1) ∧
        west(M1,E1).
   substitution: {W1/r107,E1/R}
yes(R) ⇐ imm_west(r107,M1) ∧ west(M1,R).
   select leftmost conjunct
   resolve with imm_west(r107,r109)
   substitution: {M/r109}
yes(R) ⇐ west(r109,R).
  resolve with west(W2,E2) ⇐ imm_west(W2,E2).
   substitution: {W2/r109,E2/R}
yes(R) ⇐ imm_west(r109,R)
   resolve with imm_west(r109,r111)
   substitution: {R/r111}
yes(r111) ⇐ 

(a)

f(X,X,c,X,c)

(b)

yes(c(l,X1),L) ⇐ append(c(l,X1),c(L,nil),c(l,c(i,c(s,c(t,nil)))))

(c)

append(c(l,X1),c(L,nil),c(l,c(i,c(s,c(t,nil))))) ⇐ 
   append(X1,c(L,nil), c(l,c(i,c(s,c(t,nil)))))

(a)

{Z/f(X),Y/g(b)}

(b)

{W/f(t),X/t,Q/t}

(c)

{P/val(X,bb),Z/val(X,bb)}

(a) Here is a top-down derivation:

yes(Y) ⇐ adj(b,Y,c(a,c(b,c(b,c(a,emp))))).
  choose clause 3, with { A/b,B/Y,L/c(a,c(b,c(b,c(a,emp)))),F/F1,E/E1 }
  
yes(Y) ⇐ ap(F1,c(b,c(Y,E1)),c(a,c(b,c(b,c(a,emp)))))
  choose clause 2, under
  { F1/c(a,T2),L/c(b,c(Y,E1)),H/a,R/c(b,c(b,c(a,emp))),T/T2 } 
  
yes(Y) ⇐ ap(T2,c(b,c(Y,E1)),c(b,c(b,c(a,emp))))   (*)
  choose clause 1 under { T2/emp,Y/b,E1/c(b,c(a,emp)) } 
  
yes(b) ⇐

(b) Yes, there is one more answer .

at (*) 
  choose clause 2 under 
  { T2/c(b,T3),L/c(b,c(Y,E1)),H/b,R/c(b,c(a,emp)),T/T3 } 
  
yes(Y) <- ap(T3,c(b,c(Y,E1)),c(b,c(a,emp))) 
  choose clause 1 under { T3/emp,L/c(b,c(a,emp)),Y/a,E1/emp 
  
yes(a) <-