1) The definition of a Turing Machine (TM) works perfectly well with an unbounded tape, not an infinite tape.

2) The only TMs which need an infinite tape are non-halting TM. Every halting TM only uses a finite tape.

3) Nearly every (if not every) computational class (e.g., P, NP, PSPACE, EXP, etc) only uses a finite tape who’s length can be bounded before running; Even NP-Complete problems only take a tape proportional to a polynomial of the input size.

So if you give me a computable problem, plus it’s input, I can give you a TM with a finite tape than can solve it.

And the point of a TM was not to pass the Turing Test, it was to prove that the set of computable numbers is countable.

]]>Nor is this a problem for Turing’s original discussion point. At least one machine which we *know* would pass the Turing test – an exact simulation of a human brain – is a finite machine. We know that an infinite-capacity-tape Turing Machine is not actually necessary for the test. I concede that a finite automaton is ultimately nothing more than a nifty version of Eliza, but then… so are we. (Unless you’re claiming to possess an infinite number of neurons, in which case I will have little choice but to bow to your superior intellect.)

After all, an awful lot of maths is based on us pretending that there’s no meaningful difference between finite-but-arbitrarily-large and countably-infinite.

]]>Maybe it’s nitpicking. But the difference between a Turing Machine and a finite state automaton is like the difference between infinite and finite, or between the alphabet and the entire works of Shakespeare (times infinity) i.e. Really Big.

The whole point of a Turing Machine being infinite was that it could theoretically simulate the responses of a human being (i.e. ANY response to ANY input) and therefore pass the Turing Test. A finite automaton is little more than a version of Eliza; a finite series of responses to a finite series of inputs. It’s a nifty calculator.

Finite automata are cool in their own right. Ones with really big tapes are cooler. But seeing them presented as Turing Machines is just painful.

Yehuda

]]>A finite machine – a machine with a finite tape or finite memory – is a non-deterministic finite automata. NOT a Turing machine.

Thank you and good day,

Yehuda

Episode 4:Faith in Numbers.

http://www.youtube.com/watch?v=ORY-mXXgJg4 ]]>