Consider using abductive diagnosis on the problem in the previous question, with the following elaborations.

$$ Valves can be

$$

$$

$$

$$

$$

$$

$$

or

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$.$ Some valves may be specified as open or closed.

$$ A valve can be

$$

$$

$$

$,$ in which case the gas will flow if the valve is open and not if it is closed;

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$,$ in which case gas never flows;

$$

$$

$$

$$

$$

$$

$$

$$

$$

$,$ in which case gas flows independently of whether the valve is open or closed; or

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$,$ in which case gas flowing into the valve leaks out instead of flowing through.

$$ There are three gas sensors that can detect gas leaking $($but not which gas$)$; the first gas sensor detects gas from the rightmost valves $($

$$

$1$

$,$

$...$

$,$

$$

$4$

$),$ the second gas sensor detects gas from the center valves $($

$$

$5$

$,$

$...$

$,$

$$

$12$

$),$ and the third gas sensor detects gas from the leftmost valves $($

$$

$13$

$,$

$...$

$,$

$$

$16$

$).$

$($a$)$ Axiomatize the domain so the system can explain thrust or no thrust in engine

$$

$1$

and the presence of gas in one of the sensors$.$ For example, it should be able to explain why

$$

$1$

is thrusting. It should be able to explain why

$$

$1$

is not thrusting and there is a gas detected by the third sensor$.$

$($b$)$ Test your axiomatization on some non$-$trivial examples.

$($c$)$ Some of the queries have many explanations. Suggest how the number of explanations could be reduced or managed so that the abductive diagnoses are more useful.