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Ants on the highway

Anahí Gajardo, Victor Lutfalla, Michaël Rao

TL;DR

The paper investigates highway phenomena in generalized Langton's ants, addressing whether highways can be unbounded in number and how their frequencies distribute across rule families. It combines formal definitions (patterns, traces, and highway concepts) with large-scale simulations to classify highway behavior, and it provides constructive, analytic results. Key findings include universal highway incidence for the $L^+R$ family, a dominant-but-rare highway structure in $LLRL$, and the existence of infinitely many highways in $LLRLRL$, including a widget-based construction for an entire infinite family of highways. The work advances understanding of emergent computation in cellular automata, offering both rigorous and constructive insights into the richness of asymptotic behaviours and their practical detectability.

Abstract

We perform intensive computations of Generalised Langton's Ants, discovering rules with a big number of highways. We depict the structure of some of them, formally proving that the number of highways which are possible for a given rule does not need to be bounded, moreover it can be infinite. The frequency of appearing of these highways is very unequal within a given generalised ant rule, in some cases these frequencies where found in a ratio of $1/10^7$ in simulations, suggesting that those highways that appears as the only possible asymptotic behaviour of some rules, might be accompanied by a big family of very infrequent ones.

Ants on the highway

TL;DR

The paper investigates highway phenomena in generalized Langton's ants, addressing whether highways can be unbounded in number and how their frequencies distribute across rule families. It combines formal definitions (patterns, traces, and highway concepts) with large-scale simulations to classify highway behavior, and it provides constructive, analytic results. Key findings include universal highway incidence for the family, a dominant-but-rare highway structure in , and the existence of infinitely many highways in , including a widget-based construction for an entire infinite family of highways. The work advances understanding of emergent computation in cellular automata, offering both rigorous and constructive insights into the richness of asymptotic behaviours and their practical detectability.

Abstract

We perform intensive computations of Generalised Langton's Ants, discovering rules with a big number of highways. We depict the structure of some of them, formally proving that the number of highways which are possible for a given rule does not need to be bounded, moreover it can be infinite. The frequency of appearing of these highways is very unequal within a given generalised ant rule, in some cases these frequencies where found in a ratio of in simulations, suggesting that those highways that appears as the only possible asymptotic behaviour of some rules, might be accompanied by a big family of very infrequent ones.
Paper Structure (8 sections, 6 theorems, 10 equations, 14 figures, 1 table)

This paper contains 8 sections, 6 theorems, 10 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalised ants
  3. Simulations and classification
  4. The $L^+R$ ants behave nicely.
  5. The $LLRL$ ant has an overwhelmingly dominant highway, and very rare highways.
  6. The $LLRLRL$ ant has infinitely many highways.
  7. Multiple highways for simple family of ants
  8. Infinitely many highways for a single ant

Key Result

Lemma 1

Let $a,b,c,d \in \mathbb{N}$ such that $2k \geq a \geq b,c,d \geq 0$. Let $b' = b+(2k-a)$, $c' = c+(2k-a)$, $d' = d+(2k-a)$. Let $P$ and $P'$ be the patterns of Fig. fig:l2kr_elementary_cycle. Then $T_{L^{2k}R}^{4(2k-a)}(P) = P'$.

Figures (14)

  • Figure 1: Each cell has two fixed entering sides: either vertical or horizontal.
  • Figure 2: Zoology of emergent behaviours of generalised ants: (a) rule LR, the original rule; (b) rule LRRRRRLLR builds a textured square over which highways repetitively born and crash; (c) rule LLRRRLRRRLLL builds a growing triangle that travels on the plane; (d) rule LRRRRLLLRRRL builds a square with a logarithmic spiral inside.
  • Figure 3: Configuration at iteration 817888 of rule LLRR starting with the initial configuration where all the cells are white (0).
  • Figure 4: After 104 iterations the ant has moved by the vector $(-2,2)$, and the configuration which produced that movement is also shifted by $(-2,2)$. If the cells in direction $(-2,2)$ from this configuration have all the symbol 0, then this movement will repeat itself forever.
  • Figure 5: An elementary cycle of the $L^{2k}R$ ant.
  • ...and 9 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Example 1
  • Lemma 1: $L^{2k}R$ cycles
  • proof
  • Lemma 2: $L^{2k}R$ almost highways
  • proof
  • Theorem 1: Fundamental and harmonic highway for $L^{2k}R$
  • proof
  • Theorem 2: Variants of the harmonic highway for $L^{2k}R$
  • ...and 7 more