Re: The Rapidly-Accelerating Computer
Hal Finney wrote: It's not clear to me what is wrong with letting a standard TM operate for a transfinite number of steps. We usually give the TM an infinite tape, and we feel free to imagine the infinite tape initialized with an infinite number of symbols. Only with oracle machine. That in itself implies, in a sense, an infinite amount of time or effort to initialize the TM. As Wei Dai says that would be equivalent to the UTM with an oracle for the halting problem. (at least with the right symbol on the tape). Note that there are still unsolvable problems for such an UTM. There is an infinite lattice of degrees of unsolvability. A transfinite alpha-machine with alpha stritly bigger than omega could do much more than an omega machine. Why would we hesitate to accept the infinite stream of 1's but not the infinite stream of 0's? Because the zero are supposed to be there by default. To put the ones you need omega times. Wei Dai wrote: I'll point out that this model actually makes it clear that normal observer + RAC is not equivalent to CSO + normal computer. A CSO + a never stopping computer can emulate the halting problem in the sense that he can guess that a non-stopping program will never stop ... until he changes his mind (after seeing a program which halts). In this way he will always be wrong, but he will be less and less wrong with time. In the limit he will be correct. It is ambiguous if the limit can be said to exist in our case. (There is a well know theorem in Recursion theory explaining this: the Schoenfield modulus lemma). Bruno
Re: The Rapidly-Accelerating Computer
On Fri, Oct 13, 2000 at 08:25:39PM -0700, [EMAIL PROTECTED] wrote: I'm not sure it would be zero. The program for the CSO is not particularly complex compared to other observer programs. If you have the program for a constant-speed observer then you only need to simulate the program, inserting ever increasing delays between simulated clock cycles. Now all you have to do is wait infinity+1 ticks of the UTM and you will have your CSO at subjective time 1, and the program to create him was not particularly long or unlikely. Did you really have in mind something like the following? for (i=0; iinfinity+1; i++) ComputeNextStepOfUniverse(); OutputObserverMoment(); I'm not sure how you can define a model of computation that involves transfinite time (i.e., one where the above program would have a well-defined output). If it is possible, I have a feeling that the model might be equivalent to an UTM that has access to an oracle for the halting problem. This sounds correct; it's hard to imagine a problem which takes an infinite amount of computation to solve, but whose solutions could be tested in finite time. Is this a theorem of computational theory? If the solutions can be tested in finite time, then you can solve the problem by testing all possible solutions, and this process would halt in a finite amount of time. (I'm assuming that no solution counts as a solution.) On the other hand there might be theoretical reasons to believe in the RAC; for example, if the laws of physics appear to be such as to allow for infinitely fast computation, then it might be that we believe in the RAC due to our understanding of the details of its construction. It's like our belief today in the correctness of large proofs that can only be verified by computer. It boils down to how to define the measure of observer moments. If you define it with a standard UTM, then nothing can convince you that RACs exist. If the laws of physics appear to allow infinitely fast computation, you'll just assume that you don't have a complete understanding of those laws.
Re: The Rapidly-Accelerating Computer
Wei Dai wrote: On Thu, Sep 14, 2000 at 01:13:35PM -0400, [EMAIL PROTECTED] wrote: It may be impossible to construct such a machine in our universe, but can we achieve the same results by slowing down the consciousness of the observer observing a conventional computer? In other words, each observer's clock cycle (assuming a computer model for the observer) would double in duration in relation to the computer clock. Could there be such an observer in our universe? I suspect that there can't be because the construction of the observer's clock woud require smaller and smaller energy packets in the presence of constant background noise. Even if a continually slowing observer (CSO) could exist, it's relationship to a normal computer would not be the same as that of a normal observer to a RAC. To a normal observer, there is some finite subjective time in the future when the RAC will have gone through an infinite number of clock cycles, but to the CSO there is no finite subjective for when a normal computer will have gone through an infinite number of clock cycles. This is obvious when you consider that any finite subjective time for the CSO is also a finite objective time. But if as a function of objective time t the subjective time s(t) is such that Lim t --- Infinity s(t) = 1 then after a subjective time of 1 second an infinite amount of objective time will have passed, unless you assume that the CSO can only exist at times s(t) 1 . Saibal
RE: The Rapidly-Accelerating Computer
It is meaninless fr5om an objective point of view, but from a 'classical universe perspective' it has meaning. There is no objective meaning other that 'everything exists'. -Original Message- From: [EMAIL PROTECTED] [SMTP:[EMAIL PROTECTED]] Sent: Friday, September 15, 2000 5:06 PM To: [EMAIL PROTECTED]; [EMAIL PROTECTED]; [EMAIL PROTECTED] Subject: RE: The Rapidly-Accelerating Computer James writes: There is no distinction. No observer-moments are related. No observations are related to events. But of course, all observer-moments exists, and all events exist, so you could argue that all observations are accurate. So in this formulation, the question of whether a RAC could exist is meaningless? That doesn't sound like a very useful approach. Hal
