Tile Automata: Active Self-Assembly

• Tile Automata are active self-assembly models that allow tiles to change state post-attachment, enabling dynamic computation and programmable shape formation.
• They generalize the passive aTAM by incorporating asynchronous, local updates with precise bonding and transition rules, offering both nondeterministic and deterministic frameworks.
• TA systems reduce state complexity for assembling shapes, support universal simulation, and bridge connections between cellular automata, topology, and programmable matter.

Tile Automata (TA) constitute a formal model for active self-assembly systems, generalizing concepts from cellular automata and classic tile assembly models by permitting local, asynchronous state changes within an existing assembly. TA stands in contrast to passive frameworks such as the abstract Tile Assembly Model (aTAM), where each tile’s identity is fixed upon attachment and no further computation or state evolution subsequently occurs. TA is characterized by its capacity for both attachment and active post-attachment transition, providing a rich substrate for investigating the limits of programmable self-assembly, universal simulation, and algorithmic growth of complex structures.

1. Formal Structure and Dynamics of Tile Automata

A seeded Tile Automata system is formally specified as a 6-tuple:

$\Gamma = (\Sigma, \Lambda, \Pi, \Delta, s, \tau)$

• $\Sigma$: finite state set, each tile carries a state from $\Sigma$
• $\Lambda \subseteq \Sigma$: initial states allowed for attachment
• $$\Pi : \Sigma \times \Sigma \times \{\perp, \vdash\} \rightarrow \{0, 1\}$$: bond or glue strength between two states; $\perp$ and $\vdash$ denote vertical and horizontal orientation, respectively
• $\Delta$: set of transition rules of the form $((\sigma_1, \sigma_2), (\sigma_3, \sigma_4), d)$, meaning adjacent tiles in states $(\sigma_1, \sigma_2)$ and orientation $d$ may both simultaneously change state to $(\sigma_3, \sigma_4)$
• $s$: seed assembly (typically a single tile)
• $\tau$: temperature threshold for attachment (usually taken as 1)

Production steps are of two types:

• Attachment: A new tile in state $\sigma \in \Lambda$ may attach to an empty position $p$ iff the total bond strength to its neighbors in the current assembly is at least $\tau$
• Active transition: Any pair of adjacent, already-attached tiles in states $(\sigma_1, \sigma_2)$ may simultaneously transition to $(\sigma_3, \sigma_4)$ according to some rule in $\Delta$

The TA model generalizes the aTAM by permitting such transitions post-attachment, introducing local “active memory” and allowing individual assembly sites to perform continued computation and coordination (Alaniz et al., 2022).

2. Variants and Classification

TA systems are classified based on the complexity and determinism of their transition rules:

• General (nondeterministic) TA: Multiple possible transitions for any input pair; the system may nondeterministically choose among overlapping enabled rules.
• Single-transition TA: Each transition rule may change only one of the two tile states per step (either $\sigma_1 = \sigma_3$ and $\sigma_2 \neq \sigma_4$, or vice versa). Still nondeterministic, multiple rules per input pair may exist.
• Deterministic TA: At most one transition rule for each pair $(\sigma_1, \sigma_2)$ and orientation $d$, guaranteeing unique evolution from a given assembly configuration.

The table below summarizes the essential features:

Variant	Transition Rule Semantics	Determinism
General TA	Multiple transitions per pair	Nondeterministic
Single-transition TA	One state changes per rule	Nondeterministic
Deterministic TA	At most one transition per pair	Deterministic

These restrictions have direct implications for the computational and state complexity of shape assembly processes (Alaniz et al., 2022).

3. State Complexity and Square Construction

The principal quantitative results for TA shape assembly concern asymptotic bounds on the minimal number of states required to uniquely assemble an $n \times n$ square (with $\tau=1$ and no detachment):

• General TA: $$\lvert\Sigma\rvert = \Theta\left((\log n)^{1/4}\right)$$
• Single-transition TA: $$\lvert\Sigma\rvert = \Theta\left((\log n)^{1/3}\right)$$
• Deterministic TA: $$\lvert\Sigma\rvert = \Theta\left(\left(\frac{\log n}{\log \log n}\right)^{1/2}\right)$$

Matching lower bounds are obtained by encoding arguments, viewing TA systems as bit-strings encoding $\Sigma, \Pi, \Delta$ and using pigeonhole principles against the vast number of possible $n\times n$ squares. Upper bounds are achieved via explicit constructions combining active string printing, base conversion counters, low-height rectangle growth, and frame-chaining to create the square outline, followed by flooding the interior (Alaniz et al., 2022).

For comparison, the classical aTAM at $\tau=2$ requires tile-type complexity $\Theta(\log n / \log \log n)$; allowing active state transitions drastically reduces the necessary state set, underscoring the efficiency benefits of active post-attachment computation.

4. Universality and Model Simulation

TA is proven to be intrinsically universal for seeded active self-assembly: there exists a fixed universal rule set (≈4600 states) and a scheme for encoding any TA system’s rules, initial configuration, and affinity table onto a seed, such that the universal system non-committally simulates both the output assemblies and the full attachment/transition dynamics of any TA system at a well-characterized scale (Gomez et al., 16 Jul 2024).

The universality construction employs supertiles (macroblocks), each encoding the simulated monomer, local rule tables, and state-transmission wires. Both attachment and state transitions are simulated via assembly and active gadgets, with nondeterminism deferred non-committally until locally committed by transitions. This mechanism extends to block-pairwise 2D asynchronous cellular automata, establishing TA as a substrate capable of simulating a broad class of distributed local-update automata and programmable matter algorithms.

5. Connections to Cellular Automata, Topology, and Programmable Matter

TA generalizes cellular automata (CA) in that locally adjacent tiles perform pairwise updates, supporting asynchronous, distributed computation over the spatial assembly. It is shown that TA can simulate models such as amoebots (for programmable matter) via macrotiles, where each simulated particle is a $100 \times 100$ subassembly equipped with protocols for signal exchange, movement (expansion, contraction), and lock/unlock mechanisms—thereby bridging self-assembly and reconfigurable particle systems (Alumbaugh et al., 2019).

TA further supports the automata-driven generation of topological spaces. In the context of fractal geometry and self-affine tile theory, automata (including TA) generate equivalence relations among infinite address sequences inducing self-similar, compact Hausdorff quotient spaces. Algorithmic procedures allow construction of automata for $k$-tuple equivalences and finite topological approximations, with realization via iterated function systems (IFS) shown to uniquely correspond to particular automata (2312.01486).

6. Open Problems and Future Directions

Key open questions include:

• Whether equivalent state-complexity bounds persist under further restrictions, such as requiring transitions to act only on already bonded pairs (“locality constraints”)
• Extension of TA universality to more general asynchronous models (e.g., surface-CRNs, Nubots)
• Whether optimal bounds generalize from square construction to arbitrary algorithmically defined shapes or support full Turing-universal assembly at $\tau=1$
• Quantitative analysis of assembly kinetics and running times in active TA constructions (Alaniz et al., 2022, Gomez et al., 16 Jul 2024).

A plausible implication is that advances in TA encoding, scale factor reduction, and signal efficiency could enable efficient, practical design of programmable matter at the nanoscale, as well as more versatile assembly frameworks for topological and geometric computation.

7. Significance in Self-Assembly Theory and Computation

Tile Automata interpolate between passive tile assembly models and cellular automata, providing a minimal, algorithmically expressive active self-assembly formalism. By leveraging local active memory, TA achieves exponential reductions in tile-type complexity for shape formation, implements universality for programmable self-assembling computation, and offers formal connections to automata-generated topologies and programmable matter systems. The TA model exposes a fundamental trade-off between the richness of local state evolution and the global complexity required for algorithmic shape construction.

References: (Alaniz et al., 2022, Gomez et al., 16 Jul 2024, Alumbaugh et al., 2019, 2312.01486)