[<<][math][>>][..]Wed May 6 13:52:21 CEST 2009

Transmitting binary digit-encoded values is problematic in the sense that not all digits have the same noise sensitivity in terms of the encoded integer value. I.e. binary 1100 = decimal 12 binary 1101 = decimal 13 (diff = 1) binary 0100 = decimal 4 (diff = 8) Is there a way to encode a number such that each flipped bit will cause the same (or similar) error in the integer value, and not too many extra bits are used. One way to do this with lots of redundancy is to use a PWM representation. I.e. for 4 bit values, this gives 16 bits representation and a lot of redundancy. However, can this principle be used in a way that isn't too costly? 0 000 1 100 010 001 2 011 101 110 3 111 Using bigger numbers the redundancy explodes quite fast. Is there a way to "flatten the bulge" in the redundancy curve? This is related quite a bit to the concept of entropy. Patterns where the numers of ones and zeros are equally distributed are more common because there are more variants. What about using 100 010 001 as representations of 1,2,3. That way bit flips will only jump between "bands" and no longer all over the place. 0 {000} {0} 1 {100 010 001} {1,2,3} 2 {011 101 110} {4,5,6} 3 {111} {7} This produces a lattice structure. Now, is there a way to do this recursively? Convert each number to transitions. 100 10 010 11 001 01 The same structure is found the the other section since it's simply the inverse. Hm.. I don't see it. What I see is that numbers near the middle bands are all equal in some way. It's as if this says that only black and white are really important (they are outliers), and the rest is not so special. It's the binomial expansion of (1 + x)^n 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Can this be used in control systems? If the errors are small, we don't really care much about their exact value, just their sign and approximate size. If the errors are large, we really want to know them well. Suppose a we have n wires carrying a single signal. If they're all 1 or all 0 that's significant. If they are equally distributed that's not significant. So, this is quite a natural representation of "normalness". Flipping one bit won't do much, but the ensemble has a meaning that can be interpreted as a number. Now, can we count with these numbers? Or at least treat them as values to which we can apply functions? Let's define this better. - A node is a binary function of time. - A number is a set of nodes. - The sum of two numbers is the union of their sets. - The weight of a number is the cardinality of its set. - A "reduction" translates a number of high weight m to one of low weight n, m > n. Computation seems is hidden in the reduction operation. Let's define one for cardinality 2->1 This should be a transition of classes, probably random to eliminate arbitrary choices. The "outer" classes will remain while the inner clas picks a random choice. 2 ----> 1 00 | 0 01 10 | 0 1 11 | 1 Similar for the next in line. 3 ----------> 2 000 | 00 100 010 001 | 01 10 011 101 110 | 01 10 111 | 11 Now, can reduction be implemented with an electronic device that has some kind of balanced internal noise source? Now.. 3->1 can be implemented as a deterministic majority gate. But it seems that the big idea in the proposed computer is its indeterminate nature. So, with completely fair n -> (n-1) transitions, what is the probability of an n->1 transition? Wait. The simplest gate is really to just drop one bit randomly. (A computer full of Kill Gates?) Maybe random killing isn't the trick, maybe random permuting is? Then killing is simply not using an output. With a primitive in the random transposition gate. Enough insanity.. There might be something hidden here, but it is really dependent on cheap connections. It probably needs a 3D structure to be implemented. Random transpositions don't seem so difficult.. It's basically a beam splitter.

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