I’m not going to give a formal definition because I think the following informal description will suffice.

An enumeration machine is a finite automaton equipped with two tapes. The work tape is read/write and the output tape is write-only. The machine’s output tape head moves only after writing a letter to the tape, and then only by one cell to the right each time. It may only write letters from its input alphabet to the output tape, but it may write any letter from the tape alphabet to the work tape. There are no accept or reject states, but a distinguished enumeration state, which is used to signal that it has finished writing a word to the output tape.

The machine starts with both tapes blank and proceeds to transition from one configuration to the next according to its transition function. Each step is determined by its state and the letter underneath the head of the work tape. The actions it can perform are to overwrite the letter under the work-tape head and, optionally, to write a letter to the blank cell under the output-tape head. It can move the work-tape head left or right by one cell.

When the machine enters the enumeration state, the word on the output tape is considered to be enumerated, the contents of the output tape are erased and the output-tape head returned to the leftmost position. The work-tape and work-tape head are unaffected.

The machine runs forever, and the language recognised by an enumeration machine \(E\) is just the set of words that are ever enumerated by \(E\).