Perhaps the TV was not a good analogy.
Consider this: I am thinking about my reply to your post. I write my reply to your post. Throughout all this action, my heart continues to beat - and yet I expend none of my thought process to this task. It is an involuntary action. Yes, you could argue, that your very intelligence is connected to the fact of my heartbeating but, if I were to suffer some kind of trauma which affected my self-awareness, my heart would still keep on beating.
This is how I see the levels within HAL's AI.
