Throughout this homework, whereever you present a substitution, you may either use the format of the instructor, {... term/var ...} or of the text, {... var/term ...}, but you must use the same format throughout the homework.

1. (6) Consider the following predicates on intervals of time (see page 239).
   • Meets(x,y): interval x precedes interval y, with no intervening interval.
   • Before(x,y): interval x precedes interval y, with a gap between them.
   • Starts(x,y): interval x is an initial proper subinterval of y
   • During(x,y): interval x is a proper subinterval of interval y
   Assuming all intervals have non-zero lengths, define Before, Starts, and During using only Meets. In particular, do not use any functions, and do not assume there are any time points.

2. (8) For each of the following pairs of atomic sentences, give a most general unifier, if one exists, or state that they are not unifiable and why.
   1. P(A, B, x), P(y, z, y).
   2. Q(A, F(x), x), Q(y, F(y), B).
   3. Greater(Successor(x), x), Greater(Successor(Predecessor(5)), y).
   4. Greater(Successor(x), x), Greater(Successor(Predecessor(y)), y).

3. (10) Translate the following formula into clause form, showing each of the 10 steps.
   AxAy(Cousin(x, y) <=> EuEv(Parent(u, x) & Parent(v, y) & Sibling(u, v)))

4. (5) Rewrite the tree of Figure 9.6 (page 280) to show a proof that S(A) follows from the KB in (9.41) using refutation resolution with the set-of-support strategy (making the negation of the query the initial set of support).