Your child was moving back/forth and mining in a VERY methodical pattern. It very much looked like a bot, to everyone who watched. After multiple players were not responded to, I was called in and I attempted to use /tells, /says, and /warns. (/warns stick the text in big red letters in the middle of the screen) I got no response, and after a while I chose to ban.
PS is an MMO and communication is required. I have nothing against letting a child play while supervised, as I already told you. I already unbanned you under the condition that you get no more GM warnings and play with your child. (as per our IRC discussion)