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 All right! Other people are the most interesting and beautiful entities you’ll ever see. Honor them, talk to them, accept them, love them. I’m sorry I’ve been hung up on this book so long, Sylvia. Let’s play. This moment is the only moment you have. I’m turning off the machine and going camping. Good bye! [ And then I did go to the beach, and thought better of this ending and wrote a new one on August 28, 2004, with some advice and this closing.] Here I am preaching platitudes at you. How pompous, how fatuous, how annoying, how old. I sound like my father. Well, let’s say the advice is actually aimed at me. I need it. I forget the simplest things. And yes, I realize it’s a stretch to say that these slogans follow from my book’s long class four chain of reasoning. But they’re a nice place to end up. So now we’re done. And thanks for riding along. It’s been fun. [And then I saw this line in a book review of David Mitchell’s Cloud Atlas in the New York Times: “Self-mockery as self-protection is a very old gambit, certainly but it is beneath a writer as brilliant a Mitchell.” And I thought, whoah, I don’t want to use selfmockery if it’s beneath a brilliant writer!] Caption to Post Problem Illo If the PCE were true, the region labeled “Universal” could be expanded to include almost everything except the contents of the “Solvable” region. But this is impossible, as we know there are very many unsolvable non-universal computations. Remark on Gerry Sacks As a historical aside, let me remark that Gerald Sacks is an utterly charming man, and the only really sharp-dressed mathematician I’ve ever known. Recursiveness as “Having a Solvable Halting Problem” We sometimes use the name recursive for a computation that has a solvable decision problem. Definition. The computation P is said to be recursive iff it has a solvable halting problem. The reasons for the name “recursive” are historical. In the 1930s, logicians such as Alonzo Church, Kurt Gödel, Jacques Herbrand, and Stephen Kleene ⎯ as well as Allan Turing ⎯ were investigating various ways to define elementary computations. One class of such computations was called the general recursive functions. General recursive functions map integers into integers, but they can have the property of failing to return any output at all for certain inputs. This is because the general recursive functions are allowed to use unbounded searches in their definitions ⎯ and sometimes these searches fail. When some particular input sets off an unsuccessful endless search, that means there’s never going to be an output. p. 100
Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker In the case where a general recursive function does happen to return an output for each input, we call it a total recursive function. When I say a computation is recursive in the sense of having a solvable halting problem, we really mean that it resembles a total recursive function ⎯ in the sense that, by combining P and its halt detector HP, we can create a computation P* which halts with an output for every input. P* acts the same as P on P’s halting inputs, and it returns False for P’s non-halting inputs. P* uses HP to decide which case to follow. The Unsolvable Production Problem Suppose I have a computation P and I’d like to find a way, given an In and an Out, to find out if P(In) ever produces Out. As with the halting question, we can’t distinguish the two possibilities simply by running P(In) and waiting for Out to appear. P(In) produces Out. In this case, we’ll be successful, and we know after a finite time that, yes, Out appears among the states produced by P(In). P(In) doesn’t produce Out. In this case, running P(In) never leads to an answer, because we’ll never be sure if we’ve waited long enough for Out to appear. The cases where P(In) doesn’t produce Out lead to an unsuccessful search. We’d like to find a way to short circuit the endless wait for Out to turn up among the states produced by P(In). That is, we’d like to have an non-production detector computation PFailsToProduce. Definition. Given a computation P, we say the computation PFailsToProduce is a non-production detector if PFailsToProduce has a special state True, and the following two conditions are equivalent PFailsToProduce (In, Out) produces True P(In) doesn’t produce Out. Definition. A computation P has a solvable production problem iff it has a nonproduction detector. Otherwise we say P has an unsolvable production problem. Turing’s Theorem (Variation 2). If U is a universal computation, then U has an unsolvable production problem. Degrees of Unsolvability It’s convenient to use the notion of emulation to compare the power of computations. Definition. Let P and Q be computations. If Q can emulate P, we say P has an emulation degree less than or equal to Q, and write P ≤ e Q. If Q can emulate P and P can emulate Q, we say P has the same emulation degree as Q, and we write P = e Q. If Q can emulate P and P can’t emulate Q, we say P has a smaller emulation degree than Q, and we write P < e Q. It’s not hard to prove that having the same degree is a transitive relationship; that is, If P = e Q and Q = e R, then P = e R. This means we can use = e to divide the set of all possible computations into equivalence classes of computations such that all the computations in a given class can emulate each other. One can also prove that = e preserves the property of having a solvable halting p. 101
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Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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