My posts, regarding entropy, Shannon, Chaitin, Kolmogorov and Uniformly/Maximally Disordered Sequences have till now engaged a bit of dancing-round-the-fire. Given the development of my understanding of these sequences (and concepts like shustrings) and concomitant development of software with which to test this understanding, I am now in a position to be a bit more provocative with my questions.
It is generally understood that greater complexity of generated sequences (algorithmic output - one assumes this complexity is directly related to entropy - as if they are synonymous) is reflected within generating functions (production algorithms), following Solomonoff/Kolmogorov/Chaitin.
I have produced a rather simple algorithm that produces UDSs of any integral size (power-set of the symbols - it is a rather general solution), which UDSs nevertheless represent maximal entropy states.
How is this possible? See the attached graphs for examples.
NB - We use software requiring the following reference:
H-BloX: visualizing alignment block entropies
Zuegge, J., Ebeling, M., Schneider, G.
J. Mol. Graph. Model., 2001, 19(3-4), 303-305.
See the website:
http://gecco.org.chemie.uni-frankfurt.de/h-blox/hblox.html
Let me extend this argument just a bit. I have a program that performs an omnibus of operations upon symbol sequences, those genetic and decimal, producing UDS and non-UDS permutations, computing shustrings, histograms and a number of other tasks. It is just over 50K bytes in size. When compared to a 2-tile UDS on the decimal digits, it is clear that the program is larger than the sequence. Yet, this 50K byte program can produce a 100 million digit sequence that exhibits maximal entropy. Further, were this program to be implemented upon hardware that supported registers a full 1K bytes long (each register), then my software should generate maximally disordered sequences of substantially greater length, without substantially (nor even so great as proportionally) increasing the size of the program.
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