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)
Version 5
: Received: 7 November 2024 / Approved: 8 November 2024 / Online: 8 November 2024 (07:28:40 CET)
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.v5
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.v5
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.v5
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.v5
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.v5
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 $N_{(N-1)}=b^2+b+1$ and by $N_{(N-k)}=b^2+b+2k$ for $2 \le k \le 9$. We also provide particular forms of such strings. Knowing the latter bound, we conjecture that the maximum assembly index of a string of length $N_{(N-2)} \le N \le N_{\text{max}}$ is given by $a_{\text{max}}^{(N,b)} = \lfloor N/2 \rfloor + b(b+1)/2$, where $N_{\text{max}} = 4b^4$ if $b$ is even and $N_{\text{max}} = 4(b^4+1)$ otherwise. 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 each $k$ copies of an $n$-plet contained in a string decrease its assembly index at least by $k(n-1) - a$, 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, the assembly depth of a maximum assembly index string satisfies $d_{a_{\text{max}}}^{(N,b)} \ge \left\lceil \log_2(N) \right\rceil$. Since these results are, in general, 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.
