Braids on the Stranded Cellular Automata Model

Alexa Renner

Home/Archive/arXiv

arXiv · 2609.08091

Braids on the Stranded Cellular Automata Model

Alexa Renner

View original source ↗Read source document ↗Export metadata ↓

Abstract

The Stranded Cellular Automata (SCA) model is a grid of cells such that each cell can contain 0, 1, or 2 strands, together with two cellular automata that control when and how strands turn and cross. It was developed to study patterns occurring in fiber arts. We define a notion of what it means for a braid, in the sense of an element of a braid group, to be represented by an SCA pattern, and provide several algorithms to determine when a braid has an SCA representation with certain additional properties.

Explore related subjects

math.GR cs.FL

Keep this discovery

BibTeX RIS

Alexa Renner. 2026-09-08. Braids on the Stranded Cellular Automata Model. https://arxiv.org/abs/2609.08091

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

Discover connections

cs.FL subject math.GR subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related papers

arXiv2026

Cycle Counting and Character Expectations Using Alternating Structures

Recently, two related papers [arXiv:2412.13941, arXiv:2409.03626] found a connection between two subjects: the w-cycle theorem, which is a theorem about counting appearances of cycles reading out a word w in certain graphs, and character expectations on word measures. The w-cycle theorem was proven independently by [arXiv:1410.2540] using stackings and by [arXiv:1410.2579] using bislim structures. In the current work, we generalize stackings and bislim structures to alternating stackings and alternating bislim structures. We show how this significantly strengthens the w-cycle theorem for words admitting such alternating structures, and as a result, also strengthens the recent results of [arXiv:2412.13941] and [arXiv:2409.03626]. We show that generic words admit alternating bislim structures, and therefore, the strengthened results hold for generic words. Using our new machinery, we address conjectures of Wilton, of Hanany-Puder and of Puder-Shomroni. We prove that all three conjectures hold for generic words, but we also find counterexamples for the first two.

math.GR↗

arXiv2026

Finite-Monoid Compression in Syntactic Concept Lattices: Arity Hierarchies and a Pseudovariety Trichotomy

Clark's syntactic concept lattice (SCL) records two-sided distributional structure, and Wurm extended it to tuples of arbitrary finite arity. We study $\operatorname{cmp}_f(L)$, the minimum image size of a finite-monoid observation that preserves guarded tuple substitution through arity $f$ on the principal layer. For regular languages, we characterize $\operatorname{cmp}_f(L)$ exactly as the least cardinality of the codomain of an $f$-separating relational morphism from the pointed syntactic monoid. Let $\operatorname{ch}(\mathbf V)$ denote the least arity at which these compression numbers stabilize uniformly over a pseudovariety $\mathbf V$. Our main result is the following trichotomy of possible uniform heights: $\operatorname{ch}(\mathbf V)\in\{1,2,\infty\}$, with $\operatorname{ch}(\mathbf V)=\infty$ if and only if $\operatorname{Synt}(\{ab\})\in\mathbf V$. Thus no finite uniform compression height $3,4,\ldots$ occurs. The infinite case is sharp: inside $\langle\operatorname{Synt}(\{ab\})\rangle$, every boundary $d\to d+1$ admits unbounded compression gaps, and arbitrary finite strict prefixes of the arity hierarchy are realizable. On the finite side, commutative monoids and bands stabilize at arity one, while every completely regular syntactic monoid stabilizes by arity two; finite group kernels show that the binary bound is sharp. At unary arity, every nonempty finite simple graph is realized by an explicit length-three language, yielding an exact chromatic-number formula and NP-completeness of deciding $\operatorname{cmp}_1(L)\le 3$ for explicitly listed length-three languages. The structural boundary between compression heights one and two remains open.

cs.FL↗

arXiv2026

Adaptive Strategies for GR(1) Games

We consider two-player GR(1) games on graphs, where the system player Eve must satisfy \[ \Box\Diamond A_1\land\cdots\land\Box\Diamond A_m \;\implies\; \Box\Diamond G_1\land\cdots\land\Box\Diamond G_n \] against the environment player Adam. Here $A_1,\ldots,A_m$ are assumptions on the environment, $G_1,\ldots,G_n$ are guarantees the system must provide, and $\Box\Diamond S$ denotes ``always eventually $S$''. Traditional static strategies are overly conservative: they may actively violate assumptions to trivially satisfy the implication, or abandon all guarantees when any assumption is violated. Existing methods to prevent such behaviors incur doubly exponential blowup. We introduce an adaptive framework treating Adam as a non-adversarial agent with unknown objectives. Eve monitors which assumptions Adam actually meets and adapts her strategy at runtime to maximize satisfied guarantees. Central to our approach is a novel algorithm for monitoring liveness properties $\Box\Diamond S$, enabling Eve to maintain real-time likelihood estimates of which assumptions will be fulfilled. Eve pre-computes strategies optimal for different assumption subsets, deploying a probability distribution over them that dynamically adjusts based on monitor outputs. We prove that when assumptions are violated, Eve's randomized adaptive strategy converges asymptotically to the deterministic strategy maximizing guarantees. A prototype demonstrates effectiveness and superior computational performance compared to the state of the art.

cs.LO↗