In more civilized countries, you file your current address and all change of address forms with the government, you register any separation agreements with the government, etc.
The government absolutely should know enough to apply this reasonably accurately, and things that they can't correctly derive (like 'paid more than half the costs of keeping up the home') should simply not be part of tax law, or should be a checkbox when registering your address of "I am head of household" so the government knows.
The US government can't derive all this, but they should update it so they can.
Heck, with the NSA's hooks into banks and cell phone companies, there's a non-zero chance they actually could derive this all now if they wanted.
> a checkbox when registering your address of "I am head of household" so the government knows.
The American mind cannot handle such invasions of privacy.
>and things that they can't correctly derive (like 'paid more than half the costs of keeping up the home')
In most situations where you paid significantly more than half deriving this information should be easy.
Obviously edge cases where someone paid 55% would not be easily derived, but probably most people wouldn't care too much about that.