Subshifts defined by nondeterministic and alternating plane-walking automata

TL;DR

This work extends plane-walking automata to nondeterministic and alternating settings to define an alternating hierarchy of two-dimensional subshifts. It proves that subshifts recognized by $\\Sigma_1$ and $\\Pi_1$ plane-walking automata are incomparable and strictly larger than the deterministic class, while the overall alternating class remains a strict subset of sofic subshifts. Concrete examples, such as the Sunny Side Up subshift and the Cone Labyrinth, demonstrate incomparability at the first level and provide explicit witnesses for membership in $\\Pi_1$ or $\\Sigma_1$ respectively. The paper situates these results within the broader SFT/sofic landscape, outlines a non-collapsing hierarchy, and discusses open questions and alternative approaches (e.g., Kari–Moore rectangles) for extending incomparability to higher levels.

Abstract

Plane-walking automata were introduced by Salo & Törma to recognise languages of two-dimensional infinite words (subshifts), the counterpart of $4$-way finite automata for two-dimensional finite words. We extend the model to allow for nondeterminism and alternation of quantifiers. We prove that the recognised subshifts form a strict subclass of sofic subshifts, and that the classes corresponding to existential and universal nondeterminism are incomparable and both larger that the deterministic class. We define a hierarchy of subshifts recognised by plane-walking automata with alternating quantifiers, which we conjecture to be strict.

Figures (3)

• Figure 1: On the left, an example of an alternating plane-walking automaton. On the right, a finite pattern of a configuration accepted by the automaton. The associated subshift is the subshift where all vertical and horizontal runs of $1$s are either of even size or infinite.
• Figure 2: A $\forall$-automaton accepting $X_{ssu}$. A branch visiting the central state has no next move, so it rejects.
• Figure 3: Left: the first case when all vectors in $P_0$ are colinear to $v$. Right: the second case when $\{v_a, v_b\}\subset P_0$. In both cases, $P_1 = \{v_1, v'_1\}$ and the circles represent positions $0$ and $p$. The proof shows that each initial position admits a path that never visits $0$ or $p$ and accepts in $y$ or $\sigma^p(y)$.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Proposition 3
• Definition 4: Subshifts defined by plane-walking automata
• Definition 5: Branch, Footprint
• Proposition 6
• Proposition 7
• Theorem 8
• Definition 9
• Lemma 10