Good point. Turing machines can do any polynomial-time computation in polynomial time, but they might take a higher order polynomial length of time than some other type of machine. I was trying to compose a snappy reply, and got sloppy.

Still, when we talk about "polynomial time", what we really mean is "polynomial in the length of the problem using a reasonable encoding scheme." Not "polynomial in the number of entities in the input set," though we sometimes ignore the distinction. If the point and the line in your geometrical computation are specified with 512 bit numbers, that's a larger problem than if they were only 16 bits. You can't reduce that - it's information theory.

Any physical machine you build is going to have a finite number of bits of processing power available to it. We can't make computers out of infinitesimally thin Euclidian line segments. A quantum computer might be able to consider a virtually unlimited number of possibilities at the same time, but it has a limited number of qubits and they have to collapse down into one answer before it can be printed out on the screen.

If we start talking about machines that are impossible to build, are we still doing computer science? I don't say this to be facetious, but just to ask what computer science really is.

In reply to Re: Re: about Coolness, Impatience and Complexity by no_slogan
in thread about Coolness, Impatience and Complexity by Blop

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