One of your best friends forever Joe Logik works as a product manager in a high$-$tech startup AngelShield and is designing a new smart home security system product HomeShield $100.$ The product is required to monitor various sensors and trigger appropriate actions based on the state of the system. The system includes motion sensors$,$ door$/$window sensors$,$ and a smoke detector. Hearing you're taking the $2201$ AI class, he asks you for help with the design of the system. You have accepted his invitation and decided to use propositional logic to represent and reason about the security conditions.

Problem: Represent the following security rules using propositional logic and identify the conditions under which the alarm should be triggered:

Motion Detected:

If motion is detected in the living room $($L$)$ or the hallway $($H$),$ set the variable Motion to true.

Doors$/$Windows Open:

If any of the doors $($D$1,$ D$2)$ or windows $($W$1,$ W$2)$ are open, set the variable Open to true.

Smoke Detected:

If smoke is detected in the kitchen $($K$),$ set the variable Smoke to true.

Alarm Conditions:

The alarm $($Alarm$)$ should be triggered if there is motion in the hallway or living room $($Motion is true$)$ and either a door or window is open $($Open is true$).$

Additionally, the alarm should be triggered if smoke is detected in the kitchen $($Smoke is true$).$

Propositional Logic Representation:

Let:

M represent motion detected,

O represent doors$/$windows open,

S represent smoke detected,

A represent the alarm being triggered.

The logical representation can be expressed as follows:

A$=($M$$O$)$S

How does the propositional logic representation capture the conditions for triggering the alarm in the smart home security system?

Can you identify situations where the alarm will be triggered based on the logical conditions?

Discuss potential enhancements or modifications to the propositional logic representation to make it more robust for real$-$world smart home scenarios.