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Intrinsic Universality in Seeded Active Tile Self-Assembly

Tim Gomez, Elise Grizzell, Asher Haun, Ryan Knobel, Tom Peters, Robert Schweller, Tim Wylie

TL;DR

Intrinsic universality in seeded Tile Automata establishes a non-committal intrinsically universal TA with about $4600$ states, capable of simulating any TA's final assemblies, construction dynamics, and internal state transitions. The approach uses large macrotiles called supertiles that encode complete target tiles via a lookup table and transition gadgets, enabling nondeterministic dynamics to be captured in a single-step mapping while preserving all possible outcomes. The work proves negative results for passive or bounded-state-change models (no non-committal IU) and provides a robust positive construction for seeded TA, including temperature-1 universality at scale $O(|\Sigma|^3)$ and temperature-simulation bounds, with a transfer to 2D asynchronous cellular automata (pairwise ACA) in about $2600$ states. These results bridge self-assembly and cellular automata, showing that seeded TA can universally simulate a broad class of CA models, with implications for laboratory realizations and future multicellular computation paradigms.

Abstract

The Tile Automata (TA) model describes self-assembly systems in which monomers can build structures and transition with an adjacent monomer to change their states. This paper shows that seeded TA is a non-committal intrinsically universal model of self-assembly. We present a single universal Tile Automata system containing approximately 4600 states that can simulate (a) the output assemblies created by any other Tile Automata system G, (b) the dynamics involved in building G's assemblies, and (c) G's internal state transitions. It does so in a non-committal way: it preserves the full non-deterministic dynamics of a tile's potential attachment or transition by selecting its state in a single step, considering all possible outcomes until the moment of selection. The system uses supertiles, each encoding the complete system being simulated. The universal system builds supertiles from its seed, each representing a single tile in G, transferring the information to simulate G to each new tile. Supertiles may also asynchronously transition states according to the rules of G. This result directly transfers to a restricted version of asynchronous Cellular Automata: pairwise Cellular Automata.

Intrinsic Universality in Seeded Active Tile Self-Assembly

TL;DR

Intrinsic universality in seeded Tile Automata establishes a non-committal intrinsically universal TA with about states, capable of simulating any TA's final assemblies, construction dynamics, and internal state transitions. The approach uses large macrotiles called supertiles that encode complete target tiles via a lookup table and transition gadgets, enabling nondeterministic dynamics to be captured in a single-step mapping while preserving all possible outcomes. The work proves negative results for passive or bounded-state-change models (no non-committal IU) and provides a robust positive construction for seeded TA, including temperature-1 universality at scale and temperature-simulation bounds, with a transfer to 2D asynchronous cellular automata (pairwise ACA) in about states. These results bridge self-assembly and cellular automata, showing that seeded TA can universally simulate a broad class of CA models, with implications for laboratory realizations and future multicellular computation paradigms.

Abstract

The Tile Automata (TA) model describes self-assembly systems in which monomers can build structures and transition with an adjacent monomer to change their states. This paper shows that seeded TA is a non-committal intrinsically universal model of self-assembly. We present a single universal Tile Automata system containing approximately 4600 states that can simulate (a) the output assemblies created by any other Tile Automata system G, (b) the dynamics involved in building G's assemblies, and (c) G's internal state transitions. It does so in a non-committal way: it preserves the full non-deterministic dynamics of a tile's potential attachment or transition by selecting its state in a single step, considering all possible outcomes until the moment of selection. The system uses supertiles, each encoding the complete system being simulated. The universal system builds supertiles from its seed, each representing a single tile in G, transferring the information to simulate G to each new tile. Supertiles may also asynchronously transition states according to the rules of G. This result directly transfers to a restricted version of asynchronous Cellular Automata: pairwise Cellular Automata.
Paper Structure (56 sections, 24 theorems, 37 figures, 1 table)

This paper contains 56 sections, 24 theorems, 37 figures, 1 table.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Cellular Automata.
  4. Passive Self-Assembly.
  5. Simulation between Tile Assembly and CA.
  6. Our Contributions
  7. Preliminaries
  8. The Seeded Tile Automata Model
  9. Simulation
  10. Impossibility for Passive or Bounded State Change Systems
  11. Overview of Intrinsic Universality in TA
  12. Temperature-1 Seeded TA is Intrinsically Universal
  13. Supertiles.
  14. Attachments.
  15. Transitions.
  16. ...and 41 more sections

Key Result

Lemma 9

For any system $Y$ that simulates $X_n$ under mapping $R$, and for any valid assembly sequence $\langle A_{\pi_1},\ldots, A_{\pi_m} \rangle$ of $Y$ such that for all $1\leq i \leq m$, $R^*(A_{\pi_i}) = s$, either:

Figures (37)

  • Figure 1: Example of a Tile Automata system with 6 states, a system temperature of 4, affinities of strengths 1, 2, and 3 vertical and horizontal transitions, and a seed assembly. The assembly sequence to a terminal assembly is also shown with the changes highlighted. Due to the affinity strengthening restriction, there is no detachment.
  • Figure 2: An overview of a supertile. (1) An agent inside of the supertile. (2) Wires connecting supertiles from each edge to the lookup table. West wires are drawn individually. (3) The lookup table storing the information about the system being simulated. (4) A row containing the information about the state of the east neighbor of the supertile. (5) The active column, representing the current state of the supertile. (6) A group of datacells storing all information for the north side. (7) A single datacell, in this case, storing the affinities and transitions for when both this supertile and the East supertile are in state 1. (8) The table control edge, with an agent waiting to enter. (9) Transition selection gadget at each edge, dictating the transition of this supertile with its east neighbor.
  • Figure 3: The temperature-1 system that simulates the system in Figure \ref{['fig:example_system']}.
  • Figure 4: The construction process that the Tile Automata in Figure \ref{['fig:sim_system']} builds, representing the same attachments and transitions as in Figure \ref{['fig:example_system']}
  • Figure 5: The door in action. Once an agent asks to pass the door, the door first confirms with its handle, after which it goes into an open state. The agent can then pass the door. The door goes into an orange warning state, after which it is only allowed to swap with a wire tile to go back to its original position.
  • ...and 32 more figures

Theorems & Definitions (31)

  • Definition 1: Equivalent Productions
  • Definition 2: Follows
  • Definition 3: Non-Committally Models
  • Definition 4: Non-Committal Simulation
  • Definition 5: Non-committal Intrinsic Universality.
  • Definition 6: $k$-burnout, bounded, unbounded, passive, freezing
  • Definition 8
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • ...and 21 more