Turing Machines with a Schedule to Keep

It is shown that if a sequence of digits α = α 1 α 2 α 3 ··· is generated by a multitape Turing machine so that the n th digit α n is produced no later than time T ( n ), where T is a real-time-countable function, then there exists a multitape Turing machine which generates α and produces α n at exactly time T ( n ). This result makes it possible to strengthen some theorems of Hartmanis and Stearns and to answer a question of Yamada. Other applications yield a more general approach to countability and a new technique for dealing with time-limited computational complexity.