Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter

Tortoise: You can put it that way if you want, but think of this: the fact that two numbers which are bigger than 29 can't have ...

FIGURE 71. Order and Chaos, by M. C. Escher (lithograph, 1950). Achilles: Chaos, part of perfection? Order and chaos make a plea ...

Tortoise: The first type of search-the non-chaotic type-is exemplified by the test involved in checking for the Goldbach propert ...

Achilles: Is 15 wondrous, or unwondrous? Let's see: 15 IS ODD, so I make 3n + 1: 46 46 IS EVEN, so I take half: 23 23 IS ODD, so ...

a single entity a property which it has only by virtue of being embed- ded in a particular context. Achilles: I suppose you're r ...

there is a problem. Suppose your padding were very obvious-such as a lot of blanks, or pages covered with X's or random letters. ...

your number-theoretical entertainment has been a perfect antidote to my usual tossing and turning. And who knows-perhaps I may e ...

Tortoise: Achilles: Good idea. (Scrambles in behind the dresser.) Who's there? Voice: Open up-it's the cops. Achilles: Come in, ...

CHAPTE.R XIII BlooP and FlooP and GlooP Self-Awareness and Chaos BLOOP, FLOOP, AND GLOOP are not trolls, talking ducks, or the s ...

can only make a record called "I Cannot Be Played on Record Player X" provided that Record Player X is really a record player! T ...

possibly lead to never-ending searches among strings. The discovery of Godel-numbering showed that any search for a string havin ...

understood. Except that each message appears to be random until we estab- lish a code to read it. This code takes the form of an ...

Loops and Upper Bounds If we try to formulate a test for, say, primality in terms of such steps, we shall soon see that we have ...

scrutiny. A procedure is defined, having one input parameter, N; its output is the desired value. This procedure definition has ...

DEFINE PROCEDURE "MINUS" [M,N]: BLOCK 0: BEGIN IF M < N, THEN: QUIT BLOCK 0; LOOP AT MOST M + t TIMES: BLOCK t: BEGIN IF OUTP ...

learned motions, etc.-and the nestedness, or chunkedness, can go back many layers until you hit primordial muscle-actions. And t ...

example of a full BlooP program would be the definition of the procedure TWO-TO-THE-THREE-TO-THE, followed by the call TWO-TO-TH ...

r ~~:edu~'1 I definition I I #1 I FIGURE 72. The structure of a call-less BlooP program. For this program to be self-contained, ...

FIBO [N] = the Nth Fibonacci number (e.g., FIBO [9] = 34) PRIME-BEYOND [N] = the lowest prime beyond N (e.g., PRIME-BEYOND [33] ...

FALSE? [N] = YES if N, seen as a TNT-string, IS a false statement of number theory; otherwise NO. (e.g., FALSE? [666111666] = NO ...