Solution

1. Give a sequence of atoms added to the consequence set for a bottom-up proof procedure. Show clearly what is the consequence set.

   Here is one such sequence:

   pressurised(p1)
   on(t1)
   pressurised(p2)
   pressurised(p3)
   on(t2)
   flow(shower)
   wet(bath)
   off(t3)
   off(t4)
   off(t5)
   on(t6)
   unplugged(bath)
   flow(d2)
   flow(d1)
   plugged(sink)
   

   This is the sequence that results from selecting the topmost applicable rule at each stage. Many other sequences are possible, as long as the body of a clause is derived before the head.
2. Give a top-down derivation for the query

   ?flow(d1).

   Here is such a derivation (always selecting the leftmost atom in the body):

   yes <- flow(d1).
   yes <- flow(d2).
   yes <- wet(bath) & unplugged(bath).
   yes <-  flow(shower) & unplugged(bath).
   yes <- on(t2) & pressurised(p2) & unplugged(bath).
   yes <- pressurised(p2) & unplugged(bath).
   yes <- on(t1) & pressurised(p1) & unplugged(bath).
   yes <- pressurised(p1) & unplugged(bath).
   yes <- unplugged(bath).
   yes <- .
   

3. Give a failing derivation for the query ?flow(d1).

   Here is a failing derivation (always selecting the leftmost atom in the body):

   yes <- flow(d1).
   yes <- flow(d3).
   yes <- wet(sink) & unplugged(sink).
   yes <- on(t3) & pressurised(p3) & unplugged(sink).