It looks very intriguing and promising .It would be nice if following could be considered.

1.Would it be feasible to give a simple example

2. Is it possible to invert it ? ie if we start with a known sequence

can we get back the original sequence

3. Does it require a seed ?

4. What happens if applied sequentially?

5. Can we think of it as a sort of integer  chaotic result ? ie complex integer sequence?

6. Does maximally disordered mean same as uniform distribution?( max entropy )

It seems that with the above we could see many applications such as

cryptography , image enhancements , artificial creativity etc..

Cheers

I can' say that I have use for the software, but as far as biological sequences are concerned, they may naturally show some "disorderedness". If we assume a genome is required to cover a maximal number of functions with minimal length and that each function is a unique combination of nucleotides, then we are very close to the kind of strings you are talking about. Of course, this is a model situation. In natural genomes we often see several families of repetitive sequences that go against these qualities. In genomics and bioinformatics people often mask the repetitive sequences - see Repeat Masker for example. So your definitions could be perhaps useful to isolate the "strictly non-repetitive part" of a natural genome from parts with various degrees of repetitivness.

Re No.6 from @Narasim: If I understand it well, maximally disordered sequence will be uniform not only for single letters of the alphabet, but also for all words of length up to the main shustring length for the entire sequence.

In my previous posts, I give some example sequences.  Here, we will look at a few characters of a 100 digit UDS.

Answers to your questions 1 and 6 seem appropriate; the other questions are probably not suitable to the sequences under discussion.

Our sequence:

0102030405060708091121314151617181922324252627282933435363738394454647484955657585966768697787988990

The above sequence is of 2-tiles on the integers {0, 1, ... 9}.  Notice that each pairing of digits (00, 01, 02, ... 09, 10, 11 ... 19, 20 ... 29, ... 99) appears in the sequence just one time; the sequence is in the form of a ring, and so 00 straddles the ends of the sequence.

Hence, all shustrings are of length 2.

We may manipulate the sequence, as shown in the following example.  Note the digit pairs 9 and 2.  find any two regions of the sequence that are bounded by this pair, and swap those regions.  So, the first pairing of 9 and 2, from the left of the sequence, is ...9112..., and the second such pairing is ...1922...

Thus, we move the 11 from the first to the second location.  This makes the pair 92 appear earlier in the sequence, and the pairs 91, 11 and 12 appear later in the sequence, as shown:

010203040506070809.11.213141516171819.22324252627282933435363738394454647484955657585966768697787988990

010203040506070809.213141516171819.11.22324252627282933435363738394454647484955657585966768697787988990

We now do the same for pair 2 and 8.

0102030405060708092.13141516171.819112.23242526272.82933435363738394454647484955657585966768697787988990

0102030405060708092.23242526272.819112.13141516171.82933435363738394454647484955657585966768697787988990

NB - while the locus indicated by digit pairs (shustrings) have change in some cases, all of the shustrings remain of length 2.

It is very hard to compute one of these sequences, even as it is fairly easy to verify that a sequence of digits is in fact a Uniformly Disordered Sequence.

The problem with this name, uniform disorder, is that it does not imply maximal disorder.  That is, one may have uniform disorder that is nevertheless somewhat less than maximally disordered.  So, in truth, the sequences I have identified are both maximally disordered and uniformly disordered.

@ Matej: Remember, the RAND Corporation produced a sequence of one million highly random digits, so somebody uses this kind of informational structure.  It might be difficult to find those people by posting to ResearchGate but, testing the waters is better than is not testing the waters.  :)

As to the number of functions, remember that an informational structure is meaningful only in the context of some system of interpretation.  Hence, if there are multiple interpreters, then a single message may have multiple meanings.  I have deep interest in systems of interpreters, each ascribing different meaning to a common informational structure; how forces guiding process longevity may act to mold a message acceptable to a set of otherwise disparately behaving systems.

It will be instructive to note that while all shustrings are of length 2 (for the above examples), it is nevertheless true that all the '0' symbols are found in a subsequence just 18 symbols in length.  There is some order to maximal disorder.  This is a point of Shannon's uncorrected paper, from the quote on page 11, in the discussion of telegraphy is mentioned that these

"...sequences, however, are not completely random."

Thus, we are now at the point where only one question matters.

How can the disorder of the sequences be increased, such that they have higher entropy?

Either this subset of the permutations on the symbol set display the highest entropy possible, or they do not.  We should like to know, which case is it?

• I suspect the clustering of digits in your string is a side-product of your algorithm. Perhaps there is another way to generate such string that would not show such clustering. For example, when your string ends with a '5' and you have a choice of adding '52' and '57'. Do you always add the lower tile? I think the permutations you showed us and some choice like this in the algorithm should be equivalent.
• Also, it completely makes sense that it is hard to compute. If you put the tiles in a graph and make your string a path through this graph, it turns out you are essentially solving the travelling salesman problem!

You are correct in the suspicion that algorithmic ordering (the order of operations) yields the demonstrated case of clustering of digits in a sequence that is otherwise maximally disordered.  My algorithm starts with the 0...0 tile, and chooses the next one available in numerical order; i.e., 0...1.  And, it is true that I am finding a special path from among all possible paths over a set of data points; TSP does indeed come to mind..

Actually, my algorithm selects a subset of the permutations on the [0..n) interval.

Another important point is that the algorithm is iterative.  You call it the first time, you get the zeroth UDS.  Call the algorithm again, and you get the next UDS in the sequence.  Here are the first few members of the set of 2-tile decimal UDSs:

0102030405060708091121314151617181922324252627282933435363738394454647484955657585966768697787988990

0102030405060708091121314151617181922324252627282933435363738394454647484955657585966768697787998890

0102030405060708091121314151617181922324252627282933435363738394454647484955657585966768697788798990

0102030405060708091121314151617181922324252627282933435363738394454647484955657585966768697788799890

0102030405060708091121314151617181922324252627282933435363738394454647484955657585966768697788987990

While iterating the 2-tile decimal UDS set, and printing only one UDS out of every ten million generated, we get this list:

0102030405060708091121314151617181922324252627282933435363738394454647484955659757799678768985886690

0102030405060708091121314151617181922324252627282933435363738394454647484955667885996875797769865890

0102030405060708091121314151617181922324252627282933435363738394454647484955675857786979876596688990

0102030405060708091121314151617181922324252627282933435363738394454647484955678986579977585966887690

0102030405060708091121314151617181922324252627282933435363738394454647484955687988669776757858965990

0102030405060708091121314151617181922324252627282933435363738394454647484955697798867875859657668990

0102030405060708091121314151617181922324252627282933435363738394454647484955756889798587677866965990

0102030405060708091121314151617181922324252627282933435363738394454647484955765867759799668878985690

0102030405060708091121314151617181922324252627282933435363738394454647484955769658566786875977988990

0102030405060708091121314151617181922324252627282933435363738394454647484955776966758865985689787990

0102030405060708091121314151617181922324252627282933435363738394454647484955785877566596897698867990

0102030405060708091121314151617181922324252627282933435363738394454647484955789668658875677976985990

0102030405060708091121314151617181922324252627282933435363738394454647484955798997856776865887596690

0102030405060708091121314151617181922324252627282933435363738394454647484955857797678988756966865990

0102030405060708091121314151617181922324252627282933435363738394454647484955866879778896567576985990

0102030405060708091121314151617181922324252627282933435363738394454647484955875785689669776598867990

0102030405060708091121314151617181922324252627282933435363738394454647484955879688676598566975778990

0102030405060708091121314151617181922324252627282933435363738394454647484955886987685665967577897990

0102030405060708091121314151617181922324252627282933435363738394454647484955896799759887657786685690

0102030405060708091121314151617181922324252627282933435363738394454647484955965758667798568897876990

0102030405060708091121314151617181922324252627282933435363738394454647484955969765885667868757798990

0102030405060708091121314151617181922324252627282933435363738394454647484955978575686588776799669890

0102030405060708091121314151617181922324252627282933435363738394454647484955987685778996975658866790

0102030405060708091121314151617181922324252627282933435363738394454647484955996975677866876585798890

0102030405060708091121314151617181922324252627282933435363738394454647484956557675899885966869778790

0102030405060708091121314151617181922324252627282933435363738394454647484956559876677868975885796990

0102030405060708091121314151617181922324252627282933435363738394454647484956579897768867558785966990

0102030405060708091121314151617181922324252627282933435363738394454647484956596698687678855897577990

The above list took about one minute to generate.  It demonstrates the mechanism of churning.

Further, when one may shuffle and churn a UDS in any order and for any number of iterations.  The yield is always another UDS.

Shuffling allows for mixing of shustrings at any position, while the iterative search through the allowable permutations restricts mixing to the right end of the sequence.  The shuffle operator is the last operator to be implemented in software; earlier I give examples of use.

It seems the algorithm would be useful in identifying special paths in WSN (wireless sensor network). Is it possible to illustrate the example on a two dimensional plot as a path as in the TSP . It looks highly promising.

Cheers

The shuffle algorithm is now properly functioning, and mixes the shustrings very quickly,  The result is a UDS without lumps; instead, the UDS is rather smooth, with combinations of all symbols to be found sprinkled throughout the sequence.  Again, a UDS is a UDS, all are equivalently disordered.  It is just that some UDSs have better visual appeal as compared with other UDSs.