Version 1
: Received: 19 September 2024 / Approved: 20 September 2024 / Online: 20 September 2024 (05:20:03 CEST)
Version 2
: Received: 30 September 2024 / Approved: 1 October 2024 / Online: 3 October 2024 (08:55:32 CEST)
Version 3
: Received: 11 October 2024 / Approved: 12 October 2024 / Online: 14 October 2024 (03:33:03 CEST)
Version 4
: Received: 22 October 2024 / Approved: 22 October 2024 / Online: 23 October 2024 (13:05:56 CEST)
How to cite:
Bieniawski, W.; Masierak, P.; Tomski, A.; Łukaszyk, S. On the Salient Regularities of Strings of Assembly Theory. Preprints2024, 2024091581. https://doi.org/10.20944/preprints202409.1581.v4
Bieniawski, W.; Masierak, P.; Tomski, A.; Łukaszyk, S. On the Salient Regularities of Strings of Assembly Theory. Preprints 2024, 2024091581. https://doi.org/10.20944/preprints202409.1581.v4
Bieniawski, W.; Masierak, P.; Tomski, A.; Łukaszyk, S. On the Salient Regularities of Strings of Assembly Theory. Preprints2024, 2024091581. https://doi.org/10.20944/preprints202409.1581.v4
APA Style
Bieniawski, W., Masierak, P., Tomski, A., & Łukaszyk, S. (2024). On the Salient Regularities of Strings of Assembly Theory. Preprints. https://doi.org/10.20944/preprints202409.1581.v4
Chicago/Turabian Style
Bieniawski, W., Andrzej Tomski and Szymon Łukaszyk. 2024 "On the Salient Regularities of Strings of Assembly Theory" Preprints. https://doi.org/10.20944/preprints202409.1581.v4
Abstract
Using assembly theory of strings of any natural radix $b$ we find some of their salient regularities. In particular, we show that the upper bound of the assembly index depends quantitatively on the radix $b$ and the longest length $N$ of a string that has the assembly index of $N-k$ is given by $N_{(N-k)}=b^2+b+3k-2$ for $k=\{1,2\}$ and by $N_{(N-3)}=b^2+b+6$. We also provide particular forms of such strings. Knowing the latter bound we conjecture that the maximum assembly index of a string of length $N>N_{(N-2)}$ is given by $a_{\text{max}}^{(N,b)} = \lfloor N/2 \rfloor + b(b+1)/2$. For $k=1$ such odd length strings are nearly balanced and there are four such different strings if $b=2$ and seventy-two if $b=3$. We also show that a string containing $k$ copies of an $n$-plet has the assembly index of at most $N-1 - (n-1)k - a$, where $a$ is the assembly index of this $n$-plet. Finally, we show that the assembly depth of a minimum assembly index string is equal to the assembly index of this string, while the assembly depth of a maximum assembly index string has a value between $\left\lceil \log_2(N) \right\rceil$ and the assembly index of this string. Since these results are also valid for $b=1$, assembly theory subsumes information theory.
Keywords
assembly theory; information theory; complexity measures; information entropy; mathematical physics
Subject
Physical Sciences, Mathematical Physics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
