probabilities as in M; = as es" it would retain these values across an infinite
number of self multiplications such that Hy = .5 x log(.5) + .5 x log(.5) = 1 and Hr - Hy = 1.00 - 1.00 = 0.0.
Complexity is a more general and variously defined descriptive expression than that of the topological and metric entropies and as such brings with it many kinds of definitions and computational approaches. One choice that’s intuitively appealing assumes that the relative complexity of an expression representing, say an outcome of an observation or experiment, is reflected in the minimum length of the most compressed program (algorithm) from which, given a suitable dictionary of symbolic equivalencies, one can reconstitute the original expression. Increases in what some have called algorithmic complexity, AC, are reflected in the growth of this minimally descriptive symbol series length. Karen Selz’s approach to compression and AC, similar to one proposed by Paul Rapp, involves the identification and symbolic representation of repeated blocks of symbols called words. For example, given an_ arbitrary, exemplifying binary — series: 011011101010001010101001001010011, we first find the longest repeated word [1010100] and represent it with the symbol, a, yielding a shortening in the original series, 011011a010a1010011. The next longest repeated word is [011] is replaced with b, yielding a further compression, 66a010a1010b. The next remaining binary word is of length equal to the previous one, [010], which, when replaced by c results in the series bbacat1cb. This can be further compressed to the final representation with four symbols and for the sequentially repeated b, one exponent of degree two, b*aca1cb. From this representation and a dictionary of letter equivalent words, the original binary expression can be recovered. For a quantitative index of the algorithmic complexity, AC, of the compression, Selz computes the sum of the number of distinct symbols plus the sum of the natural logarithms of the exponents: 4 + log (2) = 4.6931. The binary representation of 729, 1011011001, discussed above, is compressed by making two [101]’s = a and two 0’s = b resulting in a71b*1. Having three distinct symbols, a,o, and 1, and two exponents of two, its algorithmic
complexity is equal to, AC = 3 + 2 x log(2) = 4.38.
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