Online Tessellation Automata

• Online tessellation automata are 2D picture models that update cell states in raster order using only the state above, the left, and the current symbol.
• The model is equivalently expressed through DAG automata on grid encodings, preserving intrinsic two-dimensional neighborhood structures.
• Extensions include irregular graph-based and stochastic tessellation systems, highlighting versatile applications in symbolic dynamics and geometry.

Searching arXiv for recent and foundational papers on online tessellation automata and closely related tessellation-based automata. Online tessellation automata are local, forward-propagating computational systems whose evolution is defined over two-dimensional pictures or, in several related lines of work, over tessellated geometric domains. In the classical formal-language sense, a two-dimensional online tessellation automaton (2OTA, 2DOTA) assigns states to picture cells in raster order, with each update depending only on the state above, the state to the left, and the current symbol, and it accepts when the bottom-right state is final (Meeres et al., 15 Sep 2025). Related work places this locality-and-tessellation pattern in broader settings, including row-by-row graph growth, Voronoi-based graph cellular automata, multiplication tessellation systems, and continuous growth tessellations, which together suggest a wider family of rule-driven tessellation processes (Kaszanyitzky, 2018, Grattarola et al., 2021, Kopra, 2022, Ewers et al., 2024).

1. Classical two-dimensional online tessellation automata

In the standard picture-LLM, a picture over an alphabet $\Sigma$ is a mapping

$$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$

and its boundary picture $\overline{P}$ is obtained by surrounding $P$ with the border symbol $\#$. A nondeterministic or deterministic two-dimensional online tessellation automaton is the five-tuple

$A=(\Sigma,Q,q_0,F,\delta),$

where

$\delta:Q\times Q\times \Sigma \to 2^Q$

in the nondeterministic case and

$\delta:Q\times Q\times \Sigma \to Q$

in the deterministic case. A run assigns a state $q_{i,j}$ to every position of the boundary picture, initializes the first row and first column to $q_0$, and computes interior states by

$$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$0

or, nondeterministically,

$$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$1

Acceptance requires $$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$2 (Meeres et al., 15 Sep 2025).

The defining operational feature is the “online” character of the computation. The automaton proceeds cell-wise, in raster order, over the interior of the boundary picture. At each position $$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$3, the new state depends only on the state above, the state to the left, and the current symbol. There is no lookahead, no backward movement, and no auxiliary stack or tape. The model is therefore a local, forward-propagating tiling device over rectangular symbol arrays rather than a two-dimensional analogue of a Turing machine.

This locality is directly tied to expressive power. The picture languages recognized by 2OTA form the class REC of recognizable picture languages, and REC coincides with the projections of local picture languages, namely languages defined by constraints on all $$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$4 subpictures (Meeres et al., 15 Sep 2025). In that precise sense, online tessellation automata occupy the regular-like tier of two-dimensional language recognition.

2. DAG encodings and the unifying characterization

A recent unifying account recasts online tessellation automata as DAG automata operating on a particular picture-to-DAG encoding. A directed acyclic graph over $$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$5 is a tuple

$$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$6

and a DAG automaton is a triple $$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$7 whose rules have the form $$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$8, with edge states in $$P : [1..m] \times [1..n] \to \Sigma, \quad P(i,j)=P_{i,j},$$9. A run labels edges so that every vertex satisfies a local compatibility rule determined by its incoming states, label, and outgoing states (Meeres et al., 15 Sep 2025).

For online tessellation automata, the relevant encoding is the input-agnostic grid encoding of a picture as a 2D DAG. Each picture position is a vertex; each interior vertex has two incoming edges, from above and from the left, and two outgoing edges, to below and to the right. This encoding preserves the intrinsic two-dimensional neighborhood structure rather than serializing the picture into a path.

The central equivalence theorem states that online tessellation automata and nondeterministic DAG automata are equivalent with respect to this grid encoding. Formally, if $\overline{P}$0 is the class of picture languages recognized by online tessellation automata and $\overline{P}$1 is the class recognized by DAG automata on the grid encoding, then

$\overline{P}$2

The mutual simulation is structural. In one direction, a 2OTA state stores the outgoing edge labels that a DAG automaton would assign at the corresponding grid vertex. In the other, DAG rules reproduce exactly the 2OTA dependence on the symbol and the two already-processed neighbors (Meeres et al., 15 Sep 2025).

This equivalence isolates what is specific about the model: not merely locality, but locality on a fixed input-agnostic grid DAG. The same paper shows that once the encoding becomes input-driven, DAG automata become strictly more expressive. In particular, there exists a picture language

$\overline{P}$3

recognized by a deterministic DAG automaton with input-driven encoding but by neither an online tessellation automaton nor a sgraffito automaton. This establishes the strict inclusion

$\overline{P}$4

(Meeres et al., 15 Sep 2025).

3. Relation to other picture automata and one-dimensional restrictions

The same DAG framework clarifies the position of online tessellation automata relative to returning finite automata (RFA) and boustrophedon automata (BFA). RFAs and BFAs correspond to deterministic DAG automata on specific input-agnostic encodings that serialize the picture according to their scanning strategies: left-to-right with return edges for RFA, and snake-like alternating row order for BFA (Meeres et al., 15 Sep 2025).

This contrast is important. Under RFA or BFA encodings, the picture is effectively reduced to a string DAG, so the automaton behaves like a one-dimensional device following a total order. Under the grid encoding used for 2OTA, the automaton retains the two-dimensional dependency pattern through simultaneous access to the above and left contexts. The distinction is topological rather than merely notational: different DAG structures induce different language classes.

For one-row pictures, the distinction disappears. When restricted to inputs $\overline{P}$5, RFAs, BFAs, 2OTA, and deterministic 2OTA all recognize exactly the regular string languages. Likewise, when strings are encoded as simple directed paths, DAG automata recognize exactly $\overline{P}$6 (Meeres et al., 15 Sep 2025). The formal picture-automata notion of online tessellation automaton is therefore a genuinely two-dimensional extension of finite-state recognition, but not a super-regular model in one dimension.

A common misconception is that “tessellation” in the name implies arbitrary polygonal tilings. In the classical automata-theoretic usage, it does not: the underlying object is a rectangular picture, and the tessellation is the induced cell decomposition together with its local state assignment. More general geometric tessellations enter elsewhere in the literature.

4. Geometric and graph-based generalizations of the online paradigm

Several works outside classical picture-language theory use the same combination of locality, incremental update, and tessellated space in more geometric settings. These are not identical to 2OTA, but they provide natural extensions of the idea.

One line of work treats graph cellular automata on irregular tessellations. A graph cellular automaton is defined as a 4-tuple $\overline{P}$7, where $\overline{P}$8, $\overline{P}$9 is the state set, $P$0, and the local rule is

$P$1

The Voronoi experiment in this framework uses a Voronoi tessellation of $P$2 random points in the unit square, with the Delaunay triangulation as adjacency graph. States are binary, and the local rule is the outer-totalistic threshold update

$P$3

with “edge of chaos” behavior around $P$4 and a non-chaotic, non-trivial regime reported for $P$5 (Grattarola et al., 2021). Here the tessellation is genuinely irregular, and the update remains synchronous and local.

A second line uses sectional-Voronoi tessellations as the substrate for rule-driven tessellation evolution. In that framework, a cell is

$P$6

with weighted-Delaunay dual, reciprocal duality, and Minkowski-sum constructions for crease patterns. In the adhesion model for the cosmic web, the weights satisfy

$P$7

so time evolution can be represented by updating $P$8, recomputing the weighted tessellation, and deriving new structures from it (Neyrinck, 2018). The associated Python package sectional-tess is web-deployable via Binder and supports interactive manipulation of generators and weights, which makes a browser-based, rule-driven tessellation system technically immediate. This suggests a geometric notion of online tessellation automaton in which the “state” is a weighted tessellation rather than a rectangular picture.

5. Tessellation-generating automata beyond fixed grids

Other automaton families make the tessellation itself the visible output of the dynamics rather than merely the substrate of recognition.

The GraftalLace Cellular Automaton (GLCA) is a one-dimensional cellular automaton on the regular square lattice whose configurations are naturally viewed as evolving tessellations of the plane by arcs and junctions. Each junction has up to three incoming and three outgoing arcs; root and branch patterns are binary triplets encoded as octal digits in $P$9. The update combines branch bits from neighboring cells according to

$\#$0

and the global mapping is $\#$1 (Kaszanyitzky, 2018). The paper explicitly describes this row-by-row growth as a hallmark of an online tessellation automaton.

Multiplication tessellation systems provide a different construction. A multiplication cube set $\#$2 yields a tessellation space $\#$3, and the main diagonal coding establishes a conjugacy between tessellation shifts and multiplication cellular automata. For $\#$4,

$\#$5

Thus shifts on the tessellation implement multiplication by positive numbers in suitable bases, and macrotile operations induce conjugacies and factor maps between systems in different bases (Kopra, 2022).

A continuous-space counterpart appears in the rectangular Gilbert tessellation. Seeds are points of a Poisson process in $\#$6; each seed chooses horizontal or vertical direction with probability $\#$7; two rays grow at speed $\#$8 in opposite directions and stop when they hit another ray or the boundary. The final state is a random planar quadrangulation. The paper derives exponential tail bounds

$\#$9

proves exponential decay of correlations, and shows that the expected number of escaping rays satisfies

$A=(\Sigma,Q,q_0,F,\delta),$0

for large $A=(\Sigma,Q,q_0,F,\delta),$1 (Ewers et al., 2024). In this setting, the automaton is asynchronous and continuous in space and time, but still local and incremental.

Model family	Local update structure	Tessellation role
2OTA	raster-order update from above, left, symbol	rectangular picture grid
GLCA	row-by-row recombination of branch bits	arc-and-junction tessellation
Multiplication tessellation systems	local face matching and shifts	Wang-hypercube tessellation
Rectangular Gilbert tessellation	grow-until-collision rays	random quadrangulation

These systems are not identical, but they share a common architecture: local rules generate or propagate structure over a tessellated domain, and global organization emerges from repeated local compatibility.

6. Conceptual synthesis

The formal core of online tessellation automata is narrow and precise: two-dimensional picture automata whose state propagation is local, forward, and finite-state, with expressive power exactly REC under the standard grid encoding (Meeres et al., 15 Sep 2025). Around that core, the literature shows several ways of relaxing the substrate while retaining the same organizing principles.

One axis of variation is geometry. The regular rectangular picture of 2OTA becomes an irregular Voronoi or Delaunay graph in graph cellular automata, a sectional-Voronoi or weighted-Delaunay pair in origami and cosmic-web constructions, a Wang-hypercube tiling in multiplication systems, or a random quadrangulation in Gilbert growth models (Grattarola et al., 2021, Neyrinck, 2018, Kopra, 2022, Ewers et al., 2024). Another axis is dynamical structure: synchronous finite-state scanning in 2OTA, synchronous graph updates in GCA, row-by-row branching in GLCA, topological shifts in multiplication tessellations, and asynchronous geometric stopping rules in Gilbert tessellations (Kaszanyitzky, 2018, Ewers et al., 2024).

This suggests a useful general distinction. In the strict automata-theoretic sense, “online tessellation automaton” denotes a classical picture automaton. In a broader geometric sense, the phrase can also denote systems in which a tessellation is generated or updated incrementally by local rules. The first sense is formally characterized by equivalence to DAG automata on the grid encoding; the second is a family resemblance grounded in locality, incremental propagation, and emergent large-scale structure.

A plausible implication is that future work may continue to move between these senses. The DAG perspective already separates input-agnostic from input-driven topology in two-dimensional recognition (Meeres et al., 15 Sep 2025). The graph-cellular and sectional-Voronoi perspectives show how local update rules can be learned, parameterized, or deployed online on irregular tessellations (Grattarola et al., 2021, Neyrinck, 2018). The multiplication and Gilbert constructions show that arithmetic and stochastic growth can both be realized as local tessellation dynamics (Kopra, 2022, Ewers et al., 2024). Taken together, these results place online tessellation automata at the intersection of picture languages, symbolic dynamics, cellular automata, stochastic geometry, and computational tessellation design.