The way that's worded, does that mean only cooking equipment with a fire extinguishing equipment needs to be shut down automatically?
Cooking equipment without fire extinguishing equipment does not require an automatic shut down?
Is the only cooking equipment requiring fire extinguishing equipment hoods?
Assuming the hood has fire extinguishing system (Sprinkers, ANSUL system ...) anything under under the hood is to be shut down.
I am familiar with ANSUL systems which have a switch that indicates operation of the extinguisher.
Here that would trigger several things:
1) send an alarm to the fire alarm system of the building, initiate evacuation; low voltage supervised circuit like alarm pull station
2) force the exhaust hood fan to RUN even if turned off manually; vent smoke
3) force any incoming air fan to STOP even if turned on manually; don't add combustion air.
4) trip the gas valve for appliances under the hood.
5) trip electrical power under the hood, typically shunt trip a main breaker feeding a subpanel for equipment under the hood.
6) the overload relays of the exhaust fan may be overridden, causing the fan to run to failure, if necessary (like a fire pump).
7) not turn off all lighting in kitchen; may rely on emergency lighting.
This often takes careful design so that things "fail-safe" on broken control wires.