Inheriting Debts

My children’s real father is an alcoholic. He has some very old medical bills that he has only been paying like $10 a month - just to keep them off his back.   If he dies are the children responsible for the leftover? He doesn’t have any assets and doesn’t even have insurance to pay for his funeral. He is in very bad health and my daughter has taken over his checkbook and pays his bills out of his disability check. She asked me and I told her that I didn’t think they would be responsible but I