Sideways on the highways

TL;DR

The work addresses whether the highway conjecture for Langton's ant extends to generalised ants and demonstrates counterexamples using two specific rules. By constructing the $LLRRRL$ and $LLRLRLL$ ants and analyzing their traces and trajectories with widget-based patterns, the paper shows multiple emergent behaviours beyond diagonal highways, including non-diagonal highways, increasing-rectangle, and cone behaviours, as well as an infinite family of diagonal highways. These results reveal that even finite initial configurations can yield rich, diverse asymptotic dynamics, challenging the universality of the highway conjecture. The authors propose a recurrence-based conjecture: for any non-trivial ant and initial configuration, the set of infinitely visited positions is either empty or all of $\mathbb{Z}^2$, guiding future research on generalised turmites.

Abstract

We present two generalised ants (LLRRRL and LLRLRLL) which admit both highway behaviours and other kinds of emergent behaviours from initially finite configurations. This limits the well known Highway conjecture on Langton's ant as it shows that a generalised version of this conjecture generically does not hold on generalised ants.

Figures (7)

• Figure 1: The configuration reached by Langton's ant from the uniform configuration after $13\,000$ steps. We can observe a periodic pattern leaving the initial seemingly chaotic pattern.
• Figure 2: A simple example of the behaviour of the $LLRL$ ant. Left turning states are shown in shades of green, and the right turning state is in red. Non-zero states are additionally labelled with their state number.
• Figure 3: The pattern $P_\beta$ (left) on which the ant $LLRRRL$ starts an increasing rectangle behaviour, and the configuration reached from $P_\beta$ (right, scaled down) after $39480$ steps ($20$ full revolutions).
• Figure 4: The pattern $P_\alpha$ (with bold boundary) on which the ant $LLRRRL$ starts a highway of period $300$ and drift $(-2,0)$. To the left of $P_\alpha$ we see the residue left behind by the highway.
• Figure 5: The pattern $P_\gamma(k)$ consisting of a $B_\gamma$ widget, $k$ aligned $L_\gamma$ widgets and a $M_\gamma$ widget on which the ant $LLRRRL$ starts a highway of period $800+96k$ and drift $(2,2)$.

Theorems & Definitions (13)

• Definition 1: Internal states
• Definition 2: Generalised ant of rule word $w$
• Definition 3
• Definition 4: Interlaced word
• Definition 5: Previsible trajectory
• Definition 6: Emergent behaviour
• Definition 7: Increasing rectangle behaviour
• Definition 8: Cone behaviour
• Theorem 1
• proof : Sketch