相关论文: Templated Assembly Theory: An Extension of the Can…

The assembly index of assembly theory quantifies the minimal number of composition steps required to construct an object from elementary components. The study proves that the decision version of the assembly index problem is NP-complete,…

Assembly theory (AT) quantifies selection using the assembly equation and identifies complex objects that occur in abundance based on two measurements, assembly index and copy number, where the assembly index is the minimum number of…

We prove the full equivalence between Assembly Theory (AT) and Shannon Entropy via a method based upon the principles of statistical compression renamed `assembly index' that belongs to the LZ family of popular compression algorithms (ZIP,…

We present assembly-theory, a Rust package for computing assembly indices of covalently bonded molecular structures. This is a key complexity measure of assembly theory, a recent theoretical framework quantifying selection across diverse…

A common approach for studying a solid solution or disordered system within a periodic ab-initio framework is to create a supercell in which a certain amount of target elements is substituted with other ones. The key to generating…

Programmable self-assembly enables the construction of complex molecular, supramolecular, and crystalline architectures from well-designed building blocks. We introduce a hypergraph-based formalism, Blocks & Bonds (B&B), that generalizes…

Assembly Theory (AT) was developed to help distinguish living from non-living systems. The theory is simple as it posits that the amount of selection or Assembly is a function of the number of complex objects where their complexity can be…

We devise a unified framework for the design of canonization algorithms. Using hereditarily finite sets, we define a general notion of combinatorial objects that includes graphs, hypergraphs, relational structures, codes, permutation…

We prove a Pumping Lemma for the noncooperative abstract Tile Assembly Model, a model central to the theory of algorithmic self-assembly since the beginning of the field. This theory suggests, and our result proves, that small differences…

We introduce a notion of compatibility between constraint encoding and compositional structure. Phrased in the language of category theory, it is given by a "composable constraint encoding". We show that every composable constraint encoding…

We present a generic construction of finite realisations of amalgamation patterns. An amalgamation pattern is specified by a finite collection of finite template structures together with a collection of partial isomorphisms between them. A…

Inspired by traditional handmade crafts, where a person improvises assemblies based on the available objects, we formally introduce the Craft Assembly Task. It is a robotic assembly task that involves building an accurate representation of…

An interval in a combinatorial structure S is a set I of points which relate to every point from S I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a…

Term Coding asks: given a finite system of term identities $\Gamma$ in $v$ variables, how large can its solution set be on an $n$--element alphabet, when we are free to choose the interpretations of the function symbols? This turns familiar…

We introduce the Insertion Chain Complex, a higher-dimensional extension of insertion graphs, as a new framework for analyzing finite sets of words. We study its topological and combinatorial properties, in particular its homology groups,…

A parameterized string (p-string) is a string over an alphabet $(\Sigma_{s} \cup \Sigma_{p})$, where $\Sigma_{s}$ and $\Sigma_{p}$ are disjoint alphabets for static symbols (s-symbols) and for parameter symbols (p-symbols), respectively.…

A combinatorial theory of associative $n$-categories has recently been proposed, with strictly associative and unital composition in all dimensions, and the weak structure arising as a combinatorial notion of homotopy with a natural…

Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata. The…

Assembly theory has received considerable attention in the recent past. Here we analyze the formal framework of this model and show that assembly pathways coincide with certain minimal hyperpaths in B-hypergraphs. This makes it possible to…

The study of interlocking assemblies is an emerging field with applications in various disciplines. However, to this day, the mathematical treatment of these assemblies has been sparse. In this work, we develop a comprehensive mathematical…