[Replying to multiple messages here.] ["William R. Buckley" <wrb(a)wrbuckley.com>] This has nothing to do with my point, which is that defining "computer" in a way that makes all computers equal to Turing machines is not a useful definition, since there are then no "computer"s. The location of code is not immaterial when it is in an inherently read-only medium, since you are specifying the possibility of self-modification of code as part of your definition. Perhaps you are right; perhaps not - but any such claim of equivalence either springs from ignorance or from linguistic shorthand; anyone, professor or not, who persists in claiming such equivalence after having had it pointed out that real computers have finite storage is simply _wrong_. And in any case, I don't see how what you wrote has anything to do with my point, which is that what looks like a Harvard architecture when considered from one point of view may look like a von Neumann architecture when considered from another, and that restricting yourself to points of view which include the possibility of code-which-writes-code (I hesitate to call it *self*-modifying code) is unnecessarily restrictive, not to mention contrary to the usual uses of the words to the point of being confusing. That's not what I was trying to do at all. I read your statement about APL both (a) running on both von Neumann and Harvard architectures and (b) including the execute operator as implying that (c) both vN and H architectures can do run-time code generation. My point was that this implication is false: execute does not require run-time code generation, so (c) does not necessarily follow from (a) and (b). There is when you are drawing distinctions based on whether the hardware can modify its own code or not. For example, it is likely that the machine I'm typing this on is microprogrammed with mask-programmed microcode ROM, in which case the real hardware is a strict Harvard architecture, with no possibility whatever of code-that-writes-code. Based on what you've written, it seems to me that this implies you do not think it is reasonable to call it a computer. [more "William R. Buckley" <wrb(a)wrbuckley.com>om>, another message] This makes it appear that you do _not_ think that being able to alter code is necessary for something to be considered a computer. I'm becoming confused about what your stance actually is here. Perhaps your position is that it is necessary for there to exist some level of abstraction at which code-that-writes-code is possible for something to be considered a computer? If so, it's not at all clear ["Brian L. Stuart" <blstuart(a)bellsouth.net>] Well, not when writing about computation, at least. :-) This is probably because it's a very subtle distinction, one which I'm not convinced even exists for some models of infinity, and one which makes no difference whatever for (almost?) all uses. Hm, so you consider "analog computer" to be an oxymoron? (or most other models of computation) (a) Because we're theoreticians. (b) Because they're useful idealizations. Very nicely put - and aside from leaving analog computers out in the cold, I think I basically agree with it. Only after you make the shift from the universal Turing machine itself (which has fixed code) to the specific Turing machine being simulated by the uTm - and not then, even, if the Turing machine simulator is simulating the sort of Turing machine I studied, which has fixed code, set when its state transition table is designed. (The simulated machine's code (= state transition table) may change from one run of the simulator to the next, but not during a run.) Yes, and the tape may change - but that doesn't constitute changes in the code being executed by the simulated Turing machine. [Back to "William R. Buckley" <wrb(a)wrbuckley.com>] Not if there are computations that can be performed on one but not the other. And there are computations that can be performed on a Turing machine with infinite - or unbounded - tape which cannot be performed on any existing, or foreseeable, computer. (As an example, consider computing Ackerman's function with arguments 100,100.) Therefore they are not computationally equivalent. [more "William R. Buckley" <wrb(a)wrbuckley.com>] This is not a prticularly interesting notion, as what computations can be expressed depends on your expressive notation. For example, if I choose the notation of real analysis and calculus, there are computations that can be expressed fairly easily which cannot be carried out by a Turing machine - because the Turing machines cannot work with real numbers, only discrete approximations to them. (Any numerically unstable iterated computation, applied to a transcendental number like pi, will do fine.) /~\ The ASCII der Mouse \ / Ribbon Campaign X Against HTML mouse(a)rodents.montreal.qc.ca / \ Email! 7D C8 61 52 5D E7 2D 39 4E F1 31 3E E8 B3 27 4B