Infinite Time Turing Machines (or Hamkins-Kidder machines) have been introduced in [HaLe00] and their computability theory has been investigated in comparison to the usual computability theory in a sequence of papers by Hamkins, Lewis, Welch and Seabold: [HaLe00], [We00a], [We00b], [HaSe01], [Hale02], [We04], [We05] (cf. also the survey papers [Ha02], [Ha04] and [Ha05]). Infinite Time Turing Machines have the same hardware as ordinary Turing Machines, and almost the same software. However, an Infinite Time Turing Machine can continue its computation if it still hasn't reached the Halt state after infinitely many steps (for details, see §, 1).
['Benedikt Löwe']
Space bounds for infinitary computation