Assembly Multi-Magma: Theory & Applications

• Assembly Multi-Magma is an algebraic structure that extends classical addition chains to arbitrary discrete gluing systems through unique, iterative building block decompositions.
• It provides a rigorous framework for quantifying assembly complexity in systems such as strings, graphs, and polyominoes with clearly defined optimal assembly chain bounds.
• The theory underpins innovative assembly protocols in computational and physical contexts, offering practical insights for efficient design in diverse combinatorial applications.

An Assembly Multi-Magma is an algebraic structure that generalizes classical addition chains from the semigroup $(\mathbb{Z}^+,+)$ to arbitrary discrete gluing systems $(S, \circ, BB)$, where $S$ is an object-set, $\circ$ is a binary multi-operation, and $BB$ is a designated set of building blocks. This abstraction enables the definition and analysis of optimal assembly protocols for objects including strings, graphs, and polyominoes, providing a rigorous framework for the study of assembly addition chains, their lengths, and associated combinatorial invariants (Cronin et al., 19 Dec 2025).

1. Algebraic Structure of Assembly Multi-Magma

A multi-magma on a set $S$ is a binary set-operation

$\circ:2^S\times2^S\to2^S$

such that $(2^S,\circ)$ is not required to be associative or commutative. For most purposes, only the restriction to singletons is needed: for $x,y\in S$, the result $\{x\}\circ\{y\}\subseteq S$ specifies allowed ways to glue objects.

An Assembly Multi-Magma (AMM) is a triple $(S, \circ, BB)$ where $BB\subset S$ is the set of elementary building blocks. The defining axiom is that every non-block $O\in S\setminus BB$ admits a unique (up to permutation) decomposition sequence:

$$O_1\in\{\beta_0\}\circ\{\beta_1\}, \quad O_2\in\{O_1\}\circ\{\beta_2\},\ldots,O_r=O\in\{O_{r-1}\}\circ\{\beta_r\}$$

with each $\beta_i\in BB$, and that the seed multiset $\{\beta_0, \ldots, \beta_r\}$ is unique. This requirement ensures a well-defined minimality and assembly-index theory analogous to classical addition chains.

An Assembly Space is an AMM admitting a well-defined size function $s : S\rightarrow\mathbb{N}$, satisfying $s(b)=1$ for $b\in BB$, and $s(x) = s(x′)+s(x′′)$ whenever $x\in\{x′\}\circ\{x′′\}$. This enables the quantification of assembly complexity as the total number of building blocks.

2. Assembly Addition Chains: Definition and Properties

Given an Assembly Space $(S, \circ, BB)$ and $O\in S\setminus BB$, an assembly addition chain (AAC) for $O$ is a sequence $(O_1,O_2,\ldots,O_r=O)$ satisfying:

1. $O_1 \in BB\circ BB$.
2. For each $i\geq 2$, $O_i\in\{O_j\}\circ\{O_k\}$ for $j,k<i$ and $O_j, O_k\in\{O_1,\ldots,O_{i-1}\}\cup BB$.

The length $L(C)$ of a chain $C$ is the number of steps, and the assembly index $a(O)=\min\{L(C) : C\in AAC(O)\}$ analogues the minimal length in classical addition chains. Any chain achieving $L(C)=a(O)$ is an optimal assembly addition chain for $O$.

For object $O$ of size $s=s(O)$, the classical lower/upper bounds for AACs are:

$\lfloor\log_2 s\rfloor \leq a(O) \leq s-1$

The lower bound comes from the fact that each binary assembly step can at most double the assembly size.

There exists an injective map from AACs of $O$ to classical addition chains of $s(O)$, which sharpens the lower bound to $\ell(s(O))\leq a(O)$, where $\ell(s)$ is the minimal length of addition chains for integer $s$ (Cronin et al., 19 Dec 2025).

3. Improved Bounds via Structural Decomposition

Stronger upper bounds arise when $O$ exhibits structural decomposability:

Binary-Decomposable Objects: If $O$ can be decomposed into parts of sizes $2^{n_j}$, writing $s=\sum_{j=1}^H 2^{n_j}$, then the AAC length obeys

$$a(O)\leq (H-1) + \sum_{i=1}^{n_1} \min\left\{\sum_{j=1}^H 2^{n_j-i},\; \#S(2^i)\right\}$$

where $\#S(2^i)$ is the number of objects of size $2^i$. The explicit chain builds assemblies of increasing power-of-two sizes, carefully not exceeding the available objects at each level, thereby paralleling the classical Schönhage bound for integers.

Two-Piece-Decomposable Objects: When $O$ is iteratively assembled from pairs (after an optional block-removal), the length bound becomes

$$a(O)\leq \min\{\lfloor s/2\rfloor,\,\#S(2)\}+\lceil s/2 \rceil-1$$

This paradigm is natural for assembly spaces where binary pairings dominate, such as in the skeleton graphs underlying colored polyominoes or graphs.

4. Canonical Examples: Strings, Graphs, Polyominoes

The AMM formalism applies to a wide spectrum of combinatorial objects:

Strings over $j$-letter alphabets

• $S$ is the set of all (directed or undirected) strings, $BB$ is the set of single-letter strings.
• For directed strings of length $s=\sum 2^{n_j}$,

$$\#S_d(2^i) = j^{2^i},\quad a(O)\leq \min\left\{s-1,\; (H-1)+\sum_{i=1}^{n_1-c}j^{2^i} +\sum_{k=1}^{c}\min\left\{\sum_j2^{n_j-(n_1-c+k)},\;j^{2^{n_1-c+k}}\right\}\right\}$$

• The upper bound captures the combinatorial restriction imposed by the rapid growth of string count with power-of-two length.

Colored Connected Graphs (CCG)

• $BB$ is the set of one-edge monochromatic segments.
• Any connected graph on $e$ edges is two-piece-decomposable:

$$\#S(2)= \binom{\#\text{colors}}{2}+\#\text{colors},\quad a(O)\leq \min\{\lfloor e/2 \rfloor,\,\#S(2)\} + \lceil e/2 \rceil -1$$

• The actual chain length exhibits a monotonic quasi-linear saw-tooth dependence on the edge-count, reflecting the discrete jumps at level-set transitions.

Colored Polyominoes

• The structure reduces to CCG via skeleton graphs, with an extra color-choice cost:

$$a(O)\leq 1+\min\{\lfloor (s-1)/2\rfloor,\,\#S(2)\}+\lceil (s-1)/2\rceil-1$$

These explicit bounds allow detailed analyses of AAC complexity in combinatorial enumeration and algorithmic assembly.

5. Comparison with Classical Addition Chain Theory

The AMM framework generalizes the classic theory of integer addition chains:

• Lower Bound: Both settings share the $\log_2 s$ information-theoretic lower bound.
• Upper Bound (Schönhage): In the classical case,

$$\ell(n)\leq (H(n)-1)+\sum_{i=1}^{n_1}2^{n_i-(n_1-i)}\leq \log_2 n + H(n) - 1$$

• AMM Replacement: Each assembly step must observe the combinatorial ceiling $\#S(\cdot)$ for each sub-assembly size. This constraint is a direct consequence of nontrivial multi-object gluing—there always exists a level saturating $a(O)\leq \min\{s-1,\ldots\}$ due to size-set cardinality.

In the limiting case where $|BB|=1$ and each assembly combines unique objects, the AMM theory specializes exactly to classical results.

6. Applications and Theoretical Implications

The theory of Assembly Multi-Magma underpins algorithmic design for minimal assembly protocols in physical, computational, and combinatorial domains:

• In molecular self-assembly or DNA-tile computation, AMM encodes the unique gluing sequences and resource bounds for composite target structures (Cronin et al., 19 Dec 2025).
• For string, graph, or polyomino building tasks, the assembly index directly quantifies minimal synthetic complexity, guiding efficient protocol design.
• The uniqueness axiom (each object’s multiset decomposition is unique up to permutation) is essential for the definition of a size function and hence for analytic control over assembly chain length.

A plausible implication is that the AMM formalism can systematically characterize the quasi-linear “saw-tooth” growth of assembly index in both classical and generalized assembly settings, governed by the interplay of binary decomposition and level-set cardinality. This suggests a unifying combinatorial language for a wide array of discrete assembly problems, with direct applications in optimization, enumeration, and the theoretical analysis of algorithmic assembly.