Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1608.03036

Universal Simulation of Directed Systems in the abstract Tile Assembly Model Requires Undirectedness

Jacob Hendricks·Matthew J. Patitz·Trent A. Rogers
View original source ↗Read source document ↗Export metadata ↓

Abstract

As a mathematical model of self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) is a remarkable platform for studying the behaviors and powers of self-assembling systems. Capable of Turing universal computation, the aTAM allows algorithmic self-assembly, in which the components can be designed so that the rules governing their behaviors force them to inherently execute prescribed algorithms as they combine. Adding to its completeness, the aTAM was shown to also be intrinsically universal, which means that there exists a single tile set such that for any arbitrary input aTAM system, that tile set can be configured into a seed structure which will then cause self-assembly using that tile set to simulate the input system, capturing its full dynamics modulo only a scale factor. However, the universal simulator previously given makes use of nondeterminism in terms of tile types placed in several key locations when different assembly sequences are followed, even when simulating a directed system, meaning one that has exactly one unique terminal assembly. The question then became whether or not that nondeterminism is fundamentally required. Here, we answer that in the affirmative: the class of directed systems in the aTAM is not intrinsically universal, meaning there is no universal simulator for directed systems which itself is always directed. This provides insight into the role of nondeterminism in self-assembly, which is itself a fundamentally nondeterministic process. To achieve this result we leverage powerful results of computational complexity hierarchies, including tight bounds on both best and worst-case complexities of decidable languages, to design systems with precisely controllable space resources available to embedded computations. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers.

Explore related subjects

cs.CGcs.ET

Keep this discovery

BibTeXRIS

Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers. 2016-08-10. Universal Simulation of Directed Systems in the abstract Tile Assembly Model Requires Undirectedness. https://arxiv.org/abs/1608.03036

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Almost Linear Universal Point Sets for Planar Graphs

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

cs.CG↗
arXiv2026

Some results on Archdeacon's conjecture for rotation systems

A rotation system on $n$ elements assigns to each element a cyclic order of the other $n-1$ elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of $K_4$. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on $n$ elements has at least $H(n)=\frac{1}{4} \lfloor\frac {n}{2}\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for $n\leq 10$ and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on $n$ elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.

cs.CG↗
arXiv2026

The Hyperbolic Surface Distance, Diameter, and Dirichlet Problems

Despite the prominence of hyperbolic surfaces in mathematics, basic algorithmic questions about them, even computing the distance between two points, have remained open, leaving many features of these surfaces inaccessible. The classical machinery assumes a polyhedral structure absent on a smooth surface. We remove these obstacles. We begin with an efficient $O(g^2)$ algorithm for the distance between two points, where $g$ is the genus of the surface. Building on it, we obtain an $O(g^2 \log g)$ method for answering distance queries from a fixed source and, as a consequence, for recentering a Dirichlet domain around an arbitrary point. This understanding of distances on the surface then lets us approximate the diameter to within any $\eps$ in time $O(g^3 \log g / \eps^2)$. We further show that the diameter, a single real number encoding a great deal about the surface, is exactly computable. Its hyperbolic cosine is an algebraic number over the field encoding the coefficients of the hyperbolic isometries defining the surface.

cs.CG↗