2. Consider the domain of house plumbing shown in Figure 5.13 In this diagram, p1, p2, and p3 are cold water pipes; t1, t2, and t3 are taps; d1, d2, and d3 are drainage pipes.
Suppose you have the following atoms • pressurized_pi is true if pipe pi has mains pressure in it • on_ti is true if tap ti is on • off_ti is true if tap ti is off • wet_b is true if b is wet (b is either the sink, bath or floor)
• flow_pi is true if water is flowing through pi • plugged_sink is true if the sink has the plug in • plugged_bath is true if the bath has the plug in • unplugged_sink is true if the sink does not have the plug in • unplugged_bath is true if the bath does not have the plug in.

A definite-clause axiomatization for how water can flow down drain d1 if taps t1 and t2 are on and the bath is unplugged is pressurized_p1 .
pressurized_p2 ← on_t1 ∧ pressurized_p1 .
flow_shower ← on_t2 ∧ pressurized_p2 .
wet_bath ← flow_shower .
flow_d2 ← wet_bath ∧ unplugged_bath .
flow_d1 ← flow_d2 .
on_t1 .
on_t2 .
unplugged_bath .

(a) Finish the axiomatization for the sink in the same manner as the axiomatization for the bath. Test it in AILog.

(b) What information would you expect the a resident of a house to be able to provide that the plumber who installed the system, who is not at the house, cannot? Change the axiomatization so that questions about this information are asked of the user.

(c) Axiomatize how the floor is wet if the sink overflows or the bath overflows. They overflow if the plug is in and water is flowing in. You may invent new atomic propositions as long as you give their intended interpretation. (Assume that the taps and plugs have been in the same positions for one hour; you do not have to axiomatize the dynamics of turning on the taps and inserting and removing plugs.) Test it in AILog.

(d) Suppose a hot water system is installed to the left of tap t1. This has another tap in the pipe leading into it and supplies hot water to the shower and the sink (there are separate hot and cold water taps for each). Add this to your axiomatization. Give the denotation for all propositions you invent. Test it in AILog.