I really wish he hadn't used literal zero and one arabic numerals (along with blank) as his tape symbols.
dash and pipe would have been much more evocative of 1-bit information, rather than persisting with this misleading "ones and zeroes" bullshit that has infected popular culture and tends to totally obscure the underlying concept.
Quite awesome, though I have to wonder how much computing power goes into reading the zero's and one's, controlling the servos, powering the led display, programs, etc. relative to the amount of computing power of the turing machine itself. A ratio of 10,000 to 1? A million to 1? A billion?
Isn't the whole point that they are equivalent? Both can calculate anything that can be calculated (in finite space, since both are limited by the finite length of the tape / size of RAM).
Not quite, in practice they are pretty much equivalent but they're not Universal Turing Machines. For that they would need to have infinite memory. A Universal Turing Machine is a theoretical construct, it's not something that could actually exist.
Your definition is slightly wrong. In Alan Turing's own words:
It is possible to invent a single machine which can be used
to compute any computable sequence. If this machine I is
supplied with a tape on the beginning of which is written the
S.D of some computing machine M, {242} then I will compute
the same sequence as M.
(Alan Turing, 1936, On Computable Numbers, With An Application To The Entscheidungsproblem)
So it is no merely that the Universal Machine can read and execute programs, it is that it can read and executes programs to make itself equivalent to ANY other turing machine. Thus the Universal Turing machine is Turing Complete. As the amount of tape required to solve all computable programs is unbounded the Universal Turing machine must have an infinitely long tape.
It's usually worth going straight to the source where you can in CS, but much more so with Turing as his papers are so nice and easy to read :)
the tape only has to be $steps long, for $steps steps of computation, because that's the longest distance the head can travel at all.
any program that eventually halts, has a finite number of steps, so the band can be finite.
the only thing that requires an infinite band, is a non-halting program. and non-halting programs (unlike real-world servers and OSes) don't have a result as defined by the turing machine concept.
the only thing that requires an infinite band, is a non-halting program.
Sure, that's the only type of program that would use the whole tape, but given any tape of finite length, there's a (program, input) pair that needs a longer tape.
I'd argue that our computers are no universal turing machines, but they surely are close enough for everyones needs. Just like the ocean is infinite if you try to swim through it, even though a plane crosses it in a few hours.
A very nice description of Turing machines, Abacus machines and in introduction to recursive functions can be found in the book "Computability and Logic" by Boolos, Burgess and Jeffrey.
There is the intuitive idea of an effectively computable function (of one or more natural numbers given as arguments): given the arguments, you can compute the value of the function using a "procedure" (or a set of rules) that does not require any creativity. Example: Addition of two natural numbers n and m. We know how to calculate n+m :-)
How can we formalize this notion of effective computability? On possible formalization is the notion of a Turing computable function, that means a function that can be computed by a Turing machine. One can show that addition, multiplication and many other functions can be computed by a Turing machine.
Now it is clear that every Turing computable function is effectively computable (just use pen and paper and write down what the Turing machine is doing).
The other direction is known as Church's Thesis (named after Alonzo Church) or Church-Turing-Thesis: every effectively computable function is Turing computable.
The thesis can not be proved, since "effectively computable" is an intuitive notion.
Two other formalizations of the notion of effectively computable are abacus computability (similar to Turing machines) and recursive functions. One can show that all these notions are equivalent.
dash and pipe would have been much more evocative of 1-bit information, rather than persisting with this misleading "ones and zeroes" bullshit that has infected popular culture and tends to totally obscure the underlying concept.