Search Results for « Assembly Theory »

Assembly theory measures a string by the shortest history that builds it. We sharpen this picture in two directions. First, we attach a discrete Dirichlet energy to an assembly space and show that it splits into the size of the space and a secondary energy charging each step vertex the square of its secondary (pathway depth) increment. The assembly index does not imply minimum energy, and the assembly depth and the energy are independent complexity measures. Second, we ask how an ensemble T of strings co-assembles within a single space. Defining step strings within an assembly space solely by their formation histories yields three joint assembly spaces: the collectively (\( \omega_T \)), multiplicity (\( \mu_T \)), and singly (\( \pi_T \)) optimal ones, where the latter need not exist for certain ensembles and \( |\omega_T| \le |\mu_T| \le |\pi_T| \). All three sizes can differ, and multiplicity can lower the singly optimal size only for ensembles with \( |\pi_T| - |\omega_T| \ge 2 \). Collective optimization is NP-complete, while singly and multiplicity optimizations are NP-hard and lie a priori in the second level of the polynomial hierarchy.

Assembly theory measures a string by the shortest history that builds it. We sharpen this picture in two directions. First, we attach a discrete Dirichlet energy to an assembly space and show that it splits into the size of the space and a secondary energy charging each step vertex the square of its secondary (pathway depth) increment. The assembly index does not imply minimum energy, and the assembly depth and the energy are independent complexity measures. Second, we ask how an ensemble T of strings co-assembles within a single space. Individuating step vertices by their formation histories rather than by the strings they carry yields three joint assembly spaces: the collectively (\( \omega_T \)), multiplicity (\( \mu_T \)), and singly (\( \pi_T \)) optimal ones, where the latter need not exist for certain ensembles and \( |\omega_T| \le |\mu_T| \le |\pi_T| \). All three sizes can differ, and multiplicity can lower the singly optimal size only for ensembles with \( |\pi_T| - |\omega_T| \ge 2 \). Collective optimization is NP-complete, while singly and multiplicity optimizations are NP-hard and lie a priori in the second level of the polynomial hierarchy.

Assembly theory measures a string by the shortest history that builds it. We sharpen this picture in two directions. First, we attach a discrete Dirichlet energy to an assembly space and show that it splits into the size of the space and a secondary energy, charging each step vertex the square of its secondary (pathway depth) increment. The assembly index does not imply minimum energy, and the assembly depth and the energy are independent complexity measures. Second, we ask how an ensemble T of strings co-assembles within a single space. Individuating step vertices by their formation histories rather than by the strings they carry yields three joint assembly spaces: the collectively (ωT), multiplicity (μT), and singly (πT) optimal ones, where the latter need not exist for certain ensembles and ∣ωT∣ ≤ ∣μT∣ ≤ ∣πT∣. All three sizes can differ, and multiplicity can lower the singly optimal size only for ensembles with ∣πT∣ − ∣ωT∣ ≥ 2. Collective optimization is NP-complete, while singly and multiplicity optimizations are NP-hard and lie in the second level of the polynomial hierarchy.

Assembly theory bridges the gap between evolutionary biology and physics by providing a framework to quantify the generation and selection of novelty in biological systems. We formalize the assembly space as an acyclic digraph of strings with 2-in-regular assembly steps vertices and provide a novel definition of the assembly index. In particular, we show that the upper bound of the assembly index depends quantitatively on the number b of unit-length strings, and the longest length N of a string that has the assembly index of N − k is given by N(N−1) = b2 + b + 1 and by N(N−k) = b2 + b + 2k for 2 ≤ k ≤ 9. We also provide particular forms of such maximum assembly index strings. 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. We show that the minimum assembly depth satisfies d min(N) = ⌈log2(N)⌉, for all b, and is the assembly depth of a maximum assembly index string. We also provide the general formula for the lengths of the minimum assembly index strings having only one independent assembly step in their assembly spaces. Since these results are also valid for b = 1, assembly theory subsumes information theory.

Assembly Theory, as developed by Cronin and co-workers, assigns to an object an assembly index: the minimal number of binary join operations required to build at least one copy of the object from a specified set of basic building blocks, allowing reuse of intermediate components. For strings over a finite alphabet, the canonical assembly index can be defined in the free semigroup (Σ+, ·) with universal binary concatenation and a “no-trash” condition, and its exact computation has been shown to be NP-complete. In this paper we propose an extension of the canonical, string-based formulation which augments pure concatenation with templated assembly steps. Intermediate objects may contain a distinguished wildcard symbol ∗ that represents a compressible block. Templates are restricted to block-compressed substrings of the target string and can be instantiated by inserting previously assembled motifs into one or many wildcard positions, possibly in parallel. This yields a new complexity measure, the templated assembly index, which strictly generalises the canonical index while preserving its operational character. We formalise the model, clarify its relation to the canonical assembly index and to classical problems such as the smallest grammar problem, and discuss the computational complexity of determining the templated assembly index. Finally, we sketch potential applications in sequence analysis, modularity detection, and biosignature design.

Assembly theory bridges the gap between evolutionary biology and physics by providing a framework to quantify the generation and selection of novelty in biological systems. We formalize the assembly space as an acyclic digraph of strings with 2-in-regular assembly steps vertices and provide a novel definition of the assembly index. In particular, we show that the upper bound of the assembly index depends quantitatively on the number b of unit-length strings, and the longest length N of a string that has the assembly index of N − k is given by N(N−1) = b2 + b + 1 and by N(N−k) = b2 + b + 2k for 2 ≤ k ≤ 9. We also provide particular forms of such maximum assembly index strings. For k = 1, such odd-length strings are nearly balanced. 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. We show that the minimum assembly depth satisfies d min(N) = ⌈log2(N)⌉, for all b, and is the assembly depth of a maximum assembly index string. We also provide the general formula for the lengths of the minimum assembly index strings having only one independent assembly step in their assembly spaces. Since these results are also valid for b = 1, assembly theory subsumes information theory.

Assembly theory bridges the gap between evolutionary biology and physics by providing a framework to quantify the generation and selection of novelty in biological systems. We formalize the assembly space as an acyclic digraph of strings with 2-in-regular assembly steps vertices and provide a novel definition of the assembly index. In particular, we show that the upper bound of the assembly index depends quantitatively on the number b of unit-length strings, and the longest length N of a string that has the assembly index of N − k is given by N(N−1) = b2 + b + 1 and by N(N−k) = b2 + b + 2k for 2 ≤ k ≤ 9. We also provide particular forms of such maximum assembly index strings. For k = 1, such odd-length strings are nearly balanced. 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. We show that the minimum assembly depth satisfies d min(N) = ⌈log2(N)⌉, for all b, and is the assembly depth of a maximum assembly index string. We also provide the general formula for the lengths of the minimum assembly index strings having only one independent assembly step in their assembly spaces. Since these results are also valid for b = 1, assembly theory subsumes information theory.

We formalize the assembly space as an acyclic digraph of strings with 2-in-regular assembly steps vertices and provide a novel definition of the assembly index. In particular, we show that the upper bound of the assembly index depends quantitatively on the number $b$ of unit-length strings, 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 maximum assembly index strings. 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. We show that the minimum assembly depth satisfies $d_{\text{min}}^{(N)} = \left\lceil \log_2(N) \right\rceil$, for all $b$, and is the assembly depth of a maximum assembly index string. We also provide the general formula for the lengths of the minimum assembly index strings having only one independent assembly step in their assembly spaces. Since these results are also valid for $b=1$, assembly theory subsumes information theory.