Could we design a Turing machine that output every bit of $\pi$ one after another correctly with limited tape length?

Question

Could we design a Turing machine that output every bit of $\pi$ one after another correctly?

I think the answer is yes because we have computer programs that calculate the value of $\pi$ day after day forever, and thus it's viable.

However, could we do that if the Turing machine has a limited tape length?

score 7 · Answer 1 · answered Jun 09 '17 at 19:22

7

A Turing machine with a bounded-length tape is basically just a finite-state automaton, since there are only finitely many possible configurations of data on the tape and only finitely many states in the machine itself. What this means is that any sequence of numbers output by such a machine is eventually going to be periodic - in particular, no irrational number can be produced this way.

answered Jun 09 '17 at 19:22

Noah Schweber

245,398

Interestingly, this argument would also apply to any computer we can build - although it would take quite a while to exhaust all the possible states of a 1 TB hard drive. – Jair Taylor Jun 09 '17 at 20:02
@JairTaylor Yup. Anything with a finite amount of memory. – Noah Schweber Jun 09 '17 at 20:03

Could we design a Turing machine that output every bit of $\pi$ one after another correctly with limited tape length?

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