Binary Vector Theory

S.Roof 2-1-94

Introduction to the Realm of Binary Vectors

This text attempts to probe some of the fundamental properties of vector bit-space. My hope is to shed some light on the spatio-temporal requirements of information as representation. Computer programmers are well familiar with the fact that all representations of information take up space. Most us, in fact, encounter this requirement of information when the magazines,mail,books or other paperwork piles up. We are perhaps less aware while watching TV. The same limitations apply here also, however, in a more subtle form. Indeed we even pay the spatio-temporal cost of information in the abstact realms. In this realm of 'pure thought' , these costs are amortized as computational complexity. 'Information overload' is a real concern whenever* the content of information exceeds the ability of some information system ( or 'mind' ) to adequately represent it in a timely fashion.

The subject I shall be converging on in this paper will be representational complexity. It is my belief that higher dimensional representations encode information in ways which allow a simplification of information handling, and possibly even allow controlled ways of abstracting only the salient features of said information. From a cognitive viewpoint, I also think that mental representations* may be a complexification by expansion into higher dimensionality while, long term memory may be a flattening of dimensionality into reduced but salient* content. Neural networks may model some of the ideas I am working on. Recursive espalier may be a succint way of accomplishing such dimensional folding and flattening.

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Consider some binary representation of a number 'N'. This number will have some finite number of bits 'b'. Such a form of representation allows the embodiment of any one of 2^b possible 'scalar' values.

i.e. 0 <= N <= 2^b

The above is just simple binary positional notation. In general we have 0 <= N <= B^b where B:= base of number system

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Now consider some n-dimensional vector V.

We have V = { v0, v1, v2, ... v(n-1) } where each v(i) is a value in some dimension i up to n total dimensions.

by definition the magnitude 'M' of V is given by

M = |V| = [ v0^2 + v1^2 + ... vn^2 ]^1/2

or |V| = square root of the sum from i = 0 to n of v-sub-i squared.

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Assuming the above standard ways of representing scalars and vectors ( note: a scalar is a vector of dimension n = 1 ), notice that each dimensional component of a vector may be represented to any degree of accuracy by a binary number. In fact we may interpret any arbitrary string of bits ( ex. 000110101111011000110000 ) as either a simple scalar with some magnitude, or as a vector with n dim components. The dimensionality is arbitrary from zero up to some maximum given by the total number of bits B. For simplicity, lets consider only homogeneous vectors i.e. all n dim components have the same bit-width 'b'. Also we will assume a natural number interpretation of binary strings ( i.e. negative integers, and floating point numbers excluded ).

Next I wish to establish some boundary or limit conditions of such vectors.

In general, we have total number of bits B = n * b. Taking B as fixed for a given bit string, means that we may partition B into n different groupings of b bits each. If a vector be so defined, then the maximum and minimum values that each component can represent is given by: max component value: v(i)max = (2^b-1) min componentvalue: v(i)min = 0

Therefore the maximum magnitude of any vector V is given by

M = |V|max = square root of [ sum of v(i)^2 ]

|V|max = square root of [ n * (v(i)max)^2 ] |V|max = v(i)max * square root of n |V|max = (2^b-1) n^1/2

In some vector space of possible vectors, the above expression represents the largest distance between any two points in said space. Geometry is the algebraic intertwining of position and size within space. Here we have the prime pythagorean method of spatial reckoning.

But there are two more boundary conditions we wish to consider. The number of bits in each component* may range from a minimum of 1 to a maximum of B. Likewise the number of possible dimensions can range from 1 to B. therefore we have two cases: I) if b = 1, n = B II) if b = B, n = 1

The first boundary condition is for the interpretation where the bit-string is a B-dimensional vector and every dimension can take only one of two values ( 0 or 1 ). The second interpretation is for a simple one-dimensional or 'scalar' binary number. Either interpretation ( among endless others ), may be applied to the same set of bits or pattern in space-time. Most computer programs generally operate on the interpretive level of case II).

OK. Going back to our expression for |V|max, let's apply our two boundary cases.

case I) B-dimensional* vector: |V|max = (2^1-1) * (B)^1/2 |V|max = (2-1)*square root of B |V|max = squre root of B

case II) 'scalar' with B bits: |V|max = (2^B-1)* (1)^1/2 |V|max = 2^B-1

definition: let 'm' = bitmass, where bitmass is the total number of one or 'on' bits in some binary number.

Applying the idea of bitmass to case I) we observe that the absolute magnitude of the maximum pure binary-vector is merely the square root of the bitmass! This is easy to see because, a number with maximum bitmass will have all bits set to 'one' hence the bitmass will be given by m = B * 1 = B. Case II) on the other hand, just reduces to the maximum* value of an ordinary binary number.

Now, the astute reader may notice that I have mixed up both interpretations with my generalized formula above. The two boundary* cases represent* extremes of purity of interpretation. In between there is a whole world of possible* ways of vectoring. What's going on in a way is some kind of informational tradeoff. We can look at things in terms of pure magnitude or value. We can also look at things as pointing to or representing some direction as well as value. By giving up some of our directional info we can maximize magnitude info, or by sacrificing pure magnitude info we can maximize* directional info. Note that for any reasonable value B, the magnitude of a pure b-vector ( square root of B ) is very much smaller than that of the scalar interpretation ( 2^B -1 ). That is because its directional* info has been maximized.

I find it very interesting that the same patterns in space-time can convey varying degrees and kinds of information merely by shifting the focus of interpretation*. This shifting of information from one sort of representation to another is apparently quantifiable, and may provide clues about how to manage information wisely. The above approach is based on vectors*. I believe all the theorems and techniques of vector analysis apply with some restrictions. Matrix math could have been used also. Perhaps I should note here that most of the mathematics I have encountered treats numbers as having some abstract platonic value, without any regard to the physical or mental costs of representation. I believe this is a fatal oversight in mathematics which semiotics and information* theory can perhaps relieve.

The next section or phase of my theoretical exploration will attempt to mathematically correlate the above exchanges between magnitude & direction with the theorems from my revious paper on uniqueness vs. redundancy. The concept of 'bitmass*' will provide a viable bridge between* ideas about spatial-representation* and ideas about newness of informational content*. Entropy figures in also. There might also be a viable concept of 'bit-energy'.