12. In a nuclear research submarine, a sensor measures the temperature of the reactor core. An alarm is triggered (A = true) if the sensor reading is abnormally high (S = true), indicating an overheating of the core (C = true). The alarm and/or the sensor could be defective (S_ok = false

, A_ok = false), which causes them to malfunction. The alarm system is modeled by the belief network of Figure 8.36.

(a) What are the initial factors for this network? For each factor, state what it represents and what variables it is a function of.

(b) Show how VE can be used to compute the probability that the core is overheating, given that the alarm does not go off; that is, P(c ∣ ¬a). For each variable eliminated, show which variable is eliminated, which factor(s) are removed, and which factor(s) are created, including what variables each factor is a function of. Explain how the answer is derived from the final factor.

(c) Suppose we add a second, identical sensor to the system and trigger the alarm when either of the sensors reads a high temperature. The two sensors break and fail independently. Give the corresponding extended belief network.