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Bill Gosper | A Wisdom Archive on Bill Gosper | | Bill Gosper A selection of articles related to Bill Gosper | |
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| ARTICLES RELATED TO Bill Gosper | | | |  Bill Gosper: Encyclopedia II - Two's complement - The weird numberWith only one exception, when we start with any number in two's complement representation, if we flip all the bits and add 1, we get the two's complement representation of the negative of that number. Negative 12 becomes positive 12, positive 5 becomes −5, zero becomes zero, etc.
The most negative number in two's complement is sometimes called "the weird number" because it is the only exception.
The two's complement of the minimum number in the range will not have the desired effect of negating the number. For example, the two's complement of -128 results in the same binary number:
1000 0000 (−12 ...
See also:Two's complement, Two's complement - Calculating two's complement, Two's complement - Addition, Two's complement - Subtraction, Two's complement - The weird number, Two's complement - Sign extension, Two's complement - Multiplication, Two's complement - Why it works, Two's complement - Two's Complement and The Machine of the Universe Read more here: » Two's complement: Encyclopedia II - Two's complement - The weird number |
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| | | | | Bill Gosper: Encyclopedia II - Conway's Game of Life - Examples of patterns There are all sorts of different patterns that occur in the Game of Life, including static patterns ("still lifes"), repeating patterns ("oscillators" - a superset of still lifes), and patterns that translate themselves across the board ("spaceships"). The simplest examples of these three classes are shown below, with live cells shown in black, and dead cells shown in white.
The "block" and "boat" are still lifes, the "blinker" and "toad" are oscillators, and the "glider" and "lightweight spaceship" ("LWSS") are spaceships w ...
See also:Conway's Game of Life, Conway's Game of Life - Origins, Conway's Game of Life - Rules of Life, Conway's Game of Life - Description, Conway's Game of Life - The game, Conway's Game of Life - Iteration, Conway's Game of Life - Examples of patterns, Conway's Game of Life - Algorithms, Conway's Game of Life - Variations on Life, Conway's Game of Life - Patterns, Conway's Game of Life - 125/36, Conway's Game of Life - 245/3 245/36, Conway's Game of Life - Bibliography, Conway's Game of Life - External Article Links, Conway's Game of Life - Patterns and Pattern Collections, Conway's Game of Life - Life Program Links, Conway's Game of Life - External Cellular Automata Links Read more here: » Conway's Game of Life: Encyclopedia II - Conway's Game of Life - Examples of patterns |
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| | | | | Bill Gosper: Encyclopedia II - Conway's Game of Life - Algorithms The earliest results in the Game of Life were obtained without the use of computers. The simplest still-lifes and oscillators were discovered while tracking the fates of various small starting configurations using graph paper, blackboards, physical game boards and pieces, and the like. During this early research, Conway discovered that the R-pentomino failed to stabilize in a small number of generations.
These discoveries inspired computer programmers over the world to write programs to track the evolution of Life patterns. Most of th ...
See also:Conway's Game of Life, Conway's Game of Life - Origins, Conway's Game of Life - Rules of Life, Conway's Game of Life - Description, Conway's Game of Life - The game, Conway's Game of Life - Iteration, Conway's Game of Life - Examples of patterns, Conway's Game of Life - Algorithms, Conway's Game of Life - Variations on Life, Conway's Game of Life - Patterns, Conway's Game of Life - 125/36, Conway's Game of Life - 245/3 245/36, Conway's Game of Life - Bibliography, Conway's Game of Life - External Article Links, Conway's Game of Life - Patterns and Pattern Collections, Conway's Game of Life - Life Program Links, Conway's Game of Life - External Cellular Automata Links Read more here: » Conway's Game of Life: Encyclopedia II - Conway's Game of Life - Algorithms |
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| | | | | Bill Gosper: Encyclopedia II - Conway's Game of Life - Iteration From an initial pattern of living cells on the grid, you will find, as the generations tick by, the population constantly undergoing unusual, sometimes beautiful and always unexpected, change. In a few cases the society eventually dies out (all living cells vanishing), although this may not happen until after a great many generations. Most initial patterns either reach stable figures - Conway calls them "still lifes" - that cannot change or patterns that oscillate forever. Patterns with no initial symmetry tend to become symmetrical. Once this happens the symmetry ca ...
See also:Conway's Game of Life, Conway's Game of Life - Origins, Conway's Game of Life - Rules of Life, Conway's Game of Life - Description, Conway's Game of Life - The game, Conway's Game of Life - Iteration, Conway's Game of Life - Examples of patterns, Conway's Game of Life - Algorithms, Conway's Game of Life - Variations on Life, Conway's Game of Life - Patterns, Conway's Game of Life - 125/36, Conway's Game of Life - 245/3 245/36, Conway's Game of Life - Bibliography, Conway's Game of Life - External Article Links, Conway's Game of Life - Patterns and Pattern Collections, Conway's Game of Life - Life Program Links, Conway's Game of Life - External Cellular Automata Links Read more here: » Conway's Game of Life: Encyclopedia II - Conway's Game of Life - Iteration |
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| | | | | Bill Gosper: Encyclopedia II - Conway's Game of Life - Rules of Life Conway chose his rules carefully, after a long period of experimentation, to meet three criteria:
There should be no initial pattern for which there is a simple proof that the population can grow without limit.
There should be initial patterns that apparently do grow without limit.
There should be simple initial patterns that grow and change for a considerable period of time before coming to an end in the following possible ways:
Fading away completely (from overcrowding or from be ...
See also:Conway's Game of Life, Conway's Game of Life - Origins, Conway's Game of Life - Rules of Life, Conway's Game of Life - Description, Conway's Game of Life - The game, Conway's Game of Life - Iteration, Conway's Game of Life - Examples of patterns, Conway's Game of Life - Algorithms, Conway's Game of Life - Variations on Life, Conway's Game of Life - Patterns, Conway's Game of Life - 125/36, Conway's Game of Life - 245/3 245/36, Conway's Game of Life - Bibliography, Conway's Game of Life - External Article Links, Conway's Game of Life - Patterns and Pattern Collections, Conway's Game of Life - Life Program Links, Conway's Game of Life - External Cellular Automata Links Read more here: » Conway's Game of Life: Encyclopedia II - Conway's Game of Life - Rules of Life |
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| | | | | Bill Gosper: Encyclopedia II - Hacker - Recognized hackers Due to the overlapping nature of the hacker concept space, many of these individuals could be included in more than one category. See also Hacker (computer security), which has a list of people in that category, including criminal and unethical hackers.
Hacker - Recognized programmers.
Mel Kaye — Near-legendary figure and the archetypal Real Programmer. He was credited with doing "the bulk of the programming" for the Royal McBee LGP-30 drum-memory computer in the 1950s. In the 1980s, Ed Nather, ano ...
See also:Hacker, Hacker - Categories of hacker, Hacker - Hacker: Computer and network security, Hacker - Hacker: Highly skilled programmer, Hacker - Hacker: Hardware modifier, Hacker - Recognized hackers, Hacker - Recognized programmers, Hacker - Security Experts, Hacker - Hardware modifiers, Hacker - Hacker media personalities, Hacker - Related books Read more here: » Hacker: Encyclopedia II - Hacker - Recognized hackers |
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| | | | | Bill Gosper: Encyclopedia II - Conway's Game of Life - Variations on Life Since Life's original inception, new rules have been developed. The standard Game of Life, in which a cell is "born" if it has exactly 3 neighbors, stays alive if it has 2 or 3 alive neighbors, and dies otherwise, is symbolized as "23/3". The first number, or list of numbers, is what is required for a cell to continue. The second set is the requirement for birth. Hence "16/6" means "a cell is born if there are 6 neighbours, and lives on if there are either 1 or 6 neighbours". HighLife is therefore 23/36, because having 6 neighbors, in ...
See also:Conway's Game of Life, Conway's Game of Life - Origins, Conway's Game of Life - Rules of Life, Conway's Game of Life - Description, Conway's Game of Life - The game, Conway's Game of Life - Iteration, Conway's Game of Life - Examples of patterns, Conway's Game of Life - Algorithms, Conway's Game of Life - Variations on Life, Conway's Game of Life - Patterns, Conway's Game of Life - 125/36, Conway's Game of Life - 245/3 245/36, Conway's Game of Life - Bibliography, Conway's Game of Life - External Article Links, Conway's Game of Life - Patterns and Pattern Collections, Conway's Game of Life - Life Program Links, Conway's Game of Life - External Cellular Automata Links Read more here: » Conway's Game of Life: Encyclopedia II - Conway's Game of Life - Variations on Life |
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| | | | | Bill Gosper: Encyclopedia II - Conway's Game of Life - Patterns
Conway's Game of Life - 125/36.
Conway's Game of Life - 245/3 245/36.
...
See also:Conway's Game of Life, Conway's Game of Life - Origins, Conway's Game of Life - Rules of Life, Conway's Game of Life - Description, Conway's Game of Life - The game, Conway's Game of Life - Iteration, Conway's Game of Life - Examples of patterns, Conway's Game of Life - Algorithms, Conway's Game of Life - Variations on Life, Conway's Game of Life - Patterns, Conway's Game of Life - 125/36, Conway's Game of Life - 245/3 245/36, Conway's Game of Life - Bibliography, Conway's Game of Life - External Article Links, Conway's Game of Life - Patterns and Pattern Collections, Conway's Game of Life - Life Program Links, Conway's Game of Life - External Cellular Automata Links Read more here: » Conway's Game of Life: Encyclopedia II - Conway's Game of Life - Patterns |
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| | | | | Bill Gosper: Encyclopedia II - Two's complement - Multiplication The product of two n-bit numbers can potentially have 2n bits. If the precision of the two two's complement operands is doubled before the multiplication, direct multiplication (discarding any excess bits beyond that precision) will provide the correct result. For example, take 5 × −6 = −30. First, the precision is extended from 4 bits to 8. Then the numbers are multiplied, discarding the bits beyond 8 (shown by 'x'):
00000101 (5)
× 11111010 (−6)
=========
0
101
0
101
...
See also:Two's complement, Two's complement - Calculating two's complement, Two's complement - Addition, Two's complement - Subtraction, Two's complement - The weird number, Two's complement - Sign extension, Two's complement - Multiplication, Two's complement - Why it works, Two's complement - Two's Complement and The Machine of the Universe Read more here: » Two's complement: Encyclopedia II - Two's complement - Multiplication |
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| | | | | Bill Gosper: Encyclopedia II - Two's complement - Sign extension When turning a two's complement number with a certain number of bits into one with more bits (e.g., when copying from a 1 byte variable to a two byte variable), the sign bit must be repeated in all the extra bits. For example:
Some processors have instructions to do this in a single instruction. On other processors a conditional must be used followed with code to set the relevant bits or bytes.
Similarly, when a two's complement number is shifted to the right, the sign bit must be maintained. However when shifted to the left, a ...
See also:Two's complement, Two's complement - Calculating two's complement, Two's complement - Addition, Two's complement - Subtraction, Two's complement - The weird number, Two's complement - Sign extension, Two's complement - Multiplication, Two's complement - Why it works, Two's complement - Two's Complement and The Machine of the Universe Read more here: » Two's complement: Encyclopedia II - Two's complement - Sign extension |
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| | | | | Bill Gosper: Encyclopedia II - Two's complement - Subtraction Computers usually use the method of complements to implement subtraction. But although using complements for subtraction is related to using complements for representing signed numbers, they are independent; direct subtraction works with two's complement numbers as well. Like addition, the advantage of using two's complement is the elimination of examining the signs of the operators to determine if addition or subtraction is needed. For example, subtracting -5 from 15 is really adding 5 to 15, but this is hidden by the two's complement representation:
11110 000 (borrow)
0000 1111 (15)
− 11 ...
See also:Two's complement, Two's complement - Calculating two's complement, Two's complement - Addition, Two's complement - Subtraction, Two's complement - The weird number, Two's complement - Sign extension, Two's complement - Multiplication, Two's complement - Why it works, Two's complement - Two's Complement and The Machine of the Universe Read more here: » Two's complement: Encyclopedia II - Two's complement - Subtraction |
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| | | | | Bill Gosper: Encyclopedia II - Two's complement - Addition Adding two's complement numbers does not require special processing if the operands have opposite signs, and the sign of the result is determined automatically. For example, adding 15 and -5:
11111 111 (carry)
0000 1111 (15)
+ 1111 1011 (-5)
===========
0000 1010 (10)
This process is dependent upon the restriction to 8 bits of precision; a value of 1 is actually carried to the left, but this bit is ignored, resultin ...
See also:Two's complement, Two's complement - Calculating two's complement, Two's complement - Addition, Two's complement - Subtraction, Two's complement - The weird number, Two's complement - Sign extension, Two's complement - Multiplication, Two's complement - Why it works, Two's complement - Two's Complement and The Machine of the Universe Read more here: » Two's complement: Encyclopedia II - Two's complement - Addition |
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| | | | | Bill Gosper: Encyclopedia II - Two's complement - Why it works The 2n possible values of n bits actually form a ring of equivalence classes, namely the integers modulo 2n, Z/(2n)Z. Each class represents a set {j + k·2n | k is an integer} for some integer j, 0 ≤ j ≤ 2n − 1. There are 2n such sets, and addition and multiplication are well-defined on them.
If the classes are taken to represent the numbers 0 to 2n − 1, and overflow ignored, then these ar ...
See also:Two's complement, Two's complement - Calculating two's complement, Two's complement - Addition, Two's complement - Subtraction, Two's complement - The weird number, Two's complement - Sign extension, Two's complement - Multiplication, Two's complement - Why it works, Two's complement - Two's Complement and The Machine of the Universe Read more here: » Two's complement: Encyclopedia II - Two's complement - Why it works |
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