How to make a circuit to detect a cut cable properly
Most alarm systems actually use a very simple system: If the input pins are connected together (or the input is connected to ground, or whatever scheme you choose) then the sensor is considered "inactive". If the sensor activates, or the wire is cut, the connection is taken open circuit and it is considered "active".

For instance:







simulate this circuit &ndash; Schematic created using CircuitLab

As long as the voltage at INPUT is pulled below the low logic input level (\$V_{IL}\$) the alarm is off. As soon as it rises above the logic high level (\$V_{IH}\$) the alarm sounds. If you have particularly lossy cable or it's in a very noisy environment you can adjust R1 to compensate.

For extra security you could make your sensor more active and have a self test mode. Add a fourth "test" wire that triggers a test, the result of which is to activate the sensor as if it were triggering the alarm - however your software knows to not sound the alarm, instead to just note that the sensor is functioning properly:







simulate this circuit