
If Real computers have finite number of states, why Turing machines are needed? - pinchn
http://cstheory.stackexchange.com/q/34398/1037
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tgflynn
This is a great question. What do we lose by thinking of real computers as
Turing machines when they are in fact finite ? For one thing I believe the
halting-problem doesn't exist in reality because it only holds for infinite
systems (ie. Turing machines), not finite ones (ie. physical computers).

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dalke
We lose very little. The Busy Beaver problem shows that even a small number of
states can take time that's described with Knuth's up arrow notation.

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tgflynn
I never heard of the BB problem before but according to Wikipedia it requires
an N-state _Turing Machine_ so I don't see how that's relevant to the question
I posed. Could you elaborate ?

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dalke
Sure. By definition, BB-N halts, which means it uses a finite number of cells
because it halts in finite time. Thus, it doesn't need a infinite tape, just a
very large one.

Consider this variation. Given 4 GB of tape cells, and 2 symbols, what is the
longest number of sequential '1's that can be written by a 10 state machine,
where the machine halts and where the machine does not reach the end of the
tape?

If you can solve that, use 1 TB of cells. Then 1 gogoolplex of cells. If you
can keep on doing this, then you're able to compute S(n) and Σ(n), and solve
the Busy Beaver problem.

