Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker can emulate any system, it can in particular emulate any system that is trying to outrun it. And from this it follows that nothing can systematically outrun the universal system. For any system that could would in effect also have to be able to outrun itself.” A New Kind of Science, p. 742. Wolfram’s idea seems to be that if we (thesis) have a predictor computation that emulates our universal computation with faster runtime than that of actually running the universal computation, then (antithesis) having the universal computation emulate the predictor should also be faster than the universal computation, and then (synthesis) having the predictor emulate the universal computation’s emulation of the predictor should be faster yet, and now we can use the synthesis as a new thesis, and get a yet faster computation, and so on forever. But it would be impossible to have an endless sequence of smaller and smaller run times. Therefore a universal computation like can’t be predicted. But if you think through some specific cases, we can see that this particular argument doesn’t work. The descending chain will stop as soon as one crosses a cost-benefit condition whereby the overhead of simulation cost swamps the gain of the successive levels of simulation. Put differently, the antithetic step won’t work very often. Let me refer back to the U and DumbU example in section 6.3 of the main text. If we try and apply Wolfram’s argument, we see that DumbU can emulate U. But, since DumbU is wired to carry out the wasteful tally doubling operation between each step, DumbU-emulating-U will run slower than DumbU, so the very first antithetic step fails and we never even take the first step of Wolfram’s proposed infinite regress. Falling back, we now ask if there is any kind of counting argument that might suggest that unpredictable universal computations are more common than predictable ones? One thinks of Chaitin’s argument that we can find arbitrarily long strings of zeroes and ones are random in the sense of not having a description shorter than themselves. To be more precise, Chaitin supposes that we fix a particular universal Turing machine U and that we say that string NameK is a name for string K if U with input NameK produces the string K and halts. For any string length n, there are 2 n strings of length n, but only 2 n - 1 strings of length less than n, so at least one string of length n has no name shorter than n. And if we say string K of length n is predictable if it has a name of length at most log(n), then very many strings are unpredictable. But this style of argument doesn’t apply to universal Turing machines because when I say U is unpredictable, I’m saying there is no Turing machine V of any size such that V emulates U with a runtime eventually on the order of the log of U’s runtime. Unlike in the Chaitin argument, we’re not limited to looking at predictor Turing machines of size smaller than the object to be predicted. Nevertheless, with Wolfram, I too, want to believe the Universal Unpredictability Lemma to be applicable to all of the interesting cases that I’ve discussed ⎯ including the dynamics of physical systems, the processes of the human mind, and the workings of human society. Definition of Oracle • Definition. The computation OP is an oracle for the computation P iff (i) OP takes the same inputs as P, is everywhere defined, and has a special extra output state ó. (i) If P(a) halts at b, then OP(a) halts at b. p. 98
Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker (ii) If P(a) doesn’t halt, then OP(a) halts at ó. And we say that P has an oracle iff there is some OP which is an oracle for P. I won’t use oracles for anything in this appendix, but as the notion of oracle seems more intuitive than the notion of a solvable halting problem, I use it in the main text. Before moving on, I’ll show that the two notions are really equivalent. Proposition. P has an oracle iff P has a solvable halting problem. To see the truth of this proposition in the forward direction, note that if P has an oracle OP, you can use OP as a black box inside a new computation HP such that HP(a) outputs 0 as soon as OP(a) halts at ó, and HP(a) outputs 1 as soon as OP(a) halts at any other value. To see the truth in the reverse direction, you can cobble together an OP by putting both HP and P inside a black box. Given an input a, you first feed it to HP. If HP(a) = 0, let OP(a) return ó. If HP(a) = 1, then run the computation P(a) until it halts at some b and let OP(a) = b. Drafts For the Ending [Aug 25, 2004. I was going to call this last section “Wake Up”.] Although it may not seem like it, one of my goals in writing this book has been to liberate myself from computers. So why have I been writing about them at such length? I wasn’t quite able to formulate my logic until I found the following remark by Marshall McLuhan. Though most of us imagine McLuhan to be a cheerleader for progress, the opposite was the case. “I am resolutely opposed to all innovation, all change, but I am determined to understand what’s happening. Because I don’t choose just to sit and let the juggernaut roll over me. Many people seem to think that if you talk about something recent, you’re in favor of it. The exact opposite is true in my case. Anything I talk about is almost certainly something I’m resolutely against. And it seems to me the best way to oppose it is to understand it. And then you know where to turn off the buttons.” In his later life, McLuhan rephrased his adage “the medium is the message” by saying that the best way to understand the effects of a new technology is to look not at the figure, but at the ground. Thus, rather than talking about what people use computers for, we might look at how they change people’s behavior. Think of the time you spend upgrading the software on your machine. Think of the toll the hours at the screen take on your wrists and your back. Go into a coffee shop and look at the people isolated behind their laptop screens. Walk down a street and see blank-faced people pecking at their wireless gizmos. Imagine a world where your time was your own. When I say that everything in the world can be viewed as a computation, I’m not saying that PCs are as good as reality. Far from it. Yes, universal automatism teaches us that there’s a common ground by which to compare nature to PCs. But on this common ground, we can readily see that the natural world is incalculably more powerful and interesting than the odd flickering boxes we’re wedded to in the era Y2K. The air is a gnarly ocean; the leaves dance on the trees. Have pity on your tired eyes and aching back. Do something nice with your body. Who am I to tell you what to do? Well, actually this advice is for me. And look into your head. Underneath the planning and resenting and wanting and worrying is the river of thought. Look at it like you’d watch the ripples in a stream. It’s beautiful. Let go of plans and expectations. p. 99
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Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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