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 the string n is found on the Turing machine tape. Definition. The integer n is Chaitin-Kolmogorov incompressible if for no e < n do we have T e (0) = n. Rather than working with arbitrary kinds of computations, the proof of Chaitin’s theorem focuses on Turing machines with two symbols. The key notion behind the proof is that both the programs and the outputs of these Turing machines can both be thought of as strings of 0s and 1s. Because of this, a program can itself be regarded as possibly being the output of a smaller program. For any P, let P* be the program such that if P(0) halts with output Q, Q is treated as a program and Q(0) is run. There will be some fixed number overhead such that size(P*) = size(P) + overhead. Now Chaitin argues that if of, once again, the unsovability of the halting problem. know that a string is a name for another string, you need to know that, when fed into the universal computer, the name produces a computation that writes the target string and halts. So proving that a number is Chaitin complex involves proving that certain computations don’t halt and therefore fail to name the string. And, as we know by now, some computations never halt even though you can’t prove this to be so. A more transparent way to get at the proof of Chaitin’s theorem is to use the following line of thought. If that if Chaitin’s theorem weren’t true, then T could prove there is a number called BerryT which is definable as “the first number that doesn’t have a definition shorter than itself.” And then T would be in trouble as this proof provides a short name for BerryT. People Who Email Me About Gödel’s Proof I’ve written about Gödel’s proof before, like in Infinity and the Mind and in Mind Tools, and I get email about it every few months, usually from people who think the proof is wrong. Guys wanting to tell me they’re smarter than Kurt Gödel. “Oooooh kay.” Adding Quantifiers The simple sentences G I’ve been discussing can be characterized as having a single kind of quantifier ⎯ where a quantifier is a phrase like “for all n” or “for some n”. Although we’ve speaking of G as having the form “For all n, g[n] is false,” we could also express it as the negation of a sentence of the form “There is an n such that g[n].” Adding more quantifiers of the same kind doesn’t change much. But if we allow alternating quantifiers we get into a richer zone of undecidability. Course-of-values Simulation A slightly stronger form of simulation is what one might call course-of-values simulation. Say that Big with translation function tr is a course-of-values-simulation of Small iff, for any s0, if the computation Small(s0) produces, at some increasing sequence of times, the output states s1, s2, s3, etc., then the Big(tr(s0)) will also produce, for some increasing sequence of times, the states tr(s1), tr(s2), tr(s3), etc. References David Deutsch, The Fabric of Reality: The Science of Parallel Universes — and Its p. 106
Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker Implications (Penguin Books, 1997). Great, but way flaky. Books To Look At Steven Strogatz, Synch. Has 3D Zhabotinsky rules. Steven Johnson, Emergence. Seems important. Matthew Derby. Super Flat Times. SF stories as mainstream. Cohen and Stewart, The Collapse of Chaos. Emanuel Rosen, The Anatomy of Buzz. Hubs. Notes on Linked by Albert-László Barabási (Plume, New York 2003) Considerable bragging and self-aggrandizing over essentially one single non-obvious result, viz. that building a network by adding nodes with each new node linking to existing nodes with probability based on the existing nodes’ links leads to a power-law network with exponent -3. (1) Intro: preliminary strutting and wild promises (2) Erdös studied random networks generated by taking a set of N nodes, listing all pairs of nodes, and using a randomizer to connect each pair with some fixed probability p. A cluster is a set S of nodes so that for ever a and b in S, there is a series of links leading from a to b. Erdös and Rényi discovered a “percolation” effect, or phase transition, whereby if each node has, on the average, at least one link, there will be a cluster including most of the nodes. That is, as the average number of links per node increases, the number of nodes left out of the giant cluster decreases exponentially. (3) Consider the network whose nodes are people and whose links are the “knows personally” relationship. Empirically, it seems to take less than six links to get from any node to another in P. Or the network W with pages as nodes and hyperlinks as links. Usually takes less than 19 links to get from one page to another. (4) Say a node n is linked to k nodes. These neighbor nodes have a maximum possibility of (k choose 2) links among them. If the actual number of links among the neighbor nodes is m, we say that the clustering coefficient at node n is m/(k choose 2). Watts and Strogatz observed that if you have a model with relatively high clustering, but with some constant number of relatively few links at each node, then adding a few long distance links decreases the average link distance between nodes. Duh! (5) On the web, a few nodes have many more links than the others. Duh! These are called hubs. (6) Vilfredo Pareto formulated the 80/20 rule. 80 percent of the peas come from 20 percent of the peapods. 80 percent of the publications come from 20 percent of the faculty. 80 percent of the links on the Web point to 15 percent of web pages. A power law function relates a dependent variable y to an independent variable x in a certain way. Let u and y be the base ten logarithms of x and y, that is, suppose that x = 10 u and y = 10 v . A typical power law has the form y = 10 v = 10 (b + alpha*u) = 10 (b + alpha*log(x)) . If you plot it on log log paper, you get v = b + alpha*u, a line. So really a power law arises simply when y varies as some power of x. Mandelbrot writes the most general form as y = y 0 (x + xshift) alpha on p. 240 of his Fractals (1977). The xshift can be thought of as a shift along the x axis, and we normally prefer to simply work with x’ = x + xshift. p. 107
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Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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