Hexagonal Returning Finite Automata

• General hexagonal returning finite automata are finite-state devices that recognize hexagonal picture languages using returning or unimodal scanning patterns and sixfold symmetry.
• They process hexagonal grids by performing repeated directional sweeps, marking visited cells, and ensuring systematic state transitions across the array.
• The model is robust under hexagonal group operations and is equivalent in expressive power to hexagonal Boustrophedon automata, providing a solid framework for two-dimensional language analysis.

General hexagonal returning finite automata are finite-state devices for recognizing hexagonal picture languages, that is, two-dimensional arrays of symbols arranged on a hexagonal grid in which each cell has six neighbors. They were introduced together with general hexagonal Boustrophedon finite automata in "Hexagonal Picture Scanning Automata" (D et al., 11 Aug 2025) as a hexagonal analogue of returning finite automata on rectangular grids, with traversal constrained to returning, or unimodal, sweep patterns. The model extends classical finite automaton ideas from strings to hexagonal pictures, while making traversal direction and hexagonal symmetry central to its formalization and analysis (D et al., 11 Aug 2025).

1. Formal model

A general hexagonal returning finite automaton (GHRFA) is defined as an 8-tuple

$M=(Q,\Sigma,R,s,F,\#,\square,D')$

where $Q$ is a finite set of states, $\Sigma$ is a finite input alphabet, $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$ is a set of transition rules, $s \in Q$ is the initial state, $F \subseteq Q$ is the set of final states, $\# \notin \Sigma$ is a special boundary symbol, $\square$ is an erased or visited symbol, and $D'$ is the set of possible scanning directions used for picture traversal (D et al., 11 Aug 2025).

The input domain is a hexagonal picture rather than a string. In this setting, the grid geometry is not incidental: the scanning discipline must respect the sixfold symmetry and neighbor structure of the hexagonal lattice. The paper therefore treats direction not merely as an implementation detail, but as part of the automaton specification through the component $D'$ (D et al., 11 Aug 2025).

A defining distinction from the general hexagonal Boustrophedon finite automaton is that the GHRFA uses returning traversal patterns. In the terminology of the paper, these are “returning” or unimodal scans, meaning that the device repeatedly sweeps in a single direction and returns to a designated starting side before the next pass begins (D et al., 11 Aug 2025).

2. Returning traversal on hexagonal pictures

The operational mechanism of a GHRFA consists of repeated directional sweeps over the hexagonal picture. A run begins at a prescribed corner or side in the initial state $Q$0, proceeds according to a chosen mode in $Q$1, reads the symbol in each visited cell, updates the state according to a rule of the form $Q$2, and replaces the scanned symbol with $Q$3 to mark it as processed (D et al., 11 Aug 2025).

The set $Q$4 contains 12 traversal modes. These modes correspond to choices of direction and orientation along axes or diagonals of the hexagonal grid. One example given in the paper is top-down column scanning from left to right,

$Q$5

which means that scanning starts at the top left corner, proceeds top-to-bottom along a column, then moves to the next column on the right, and repeats (D et al., 11 Aug 2025). The full set also includes bottom-up, leftward, rightward, and other modes induced by the symmetries of the hexagonal shape.

Boundary behavior is handled explicitly through the distinguished symbol $Q$6. When the automaton reaches a boundary, it uses special transitions associated with the movement mode to continue the sweep protocol. Acceptance occurs when the entire picture has been scanned and the automaton is in a final state (D et al., 11 Aug 2025).

This scanning discipline is structurally close to one-dimensional finite automata in that computation is mediated entirely by a finite control and local symbol reading. The extension lies in the two-dimensional, hexagonal organization of the input and in the nontrivial dependence of traversal on geometric orientation.

3. Relation to classical and rectangular models

The model generalizes classical finite automata by replacing linear inputs with hexagonal pictures. The paper identifies three principal extensions of the one-dimensional setting: the grid structure is two-dimensional, the scanning mode is adapted to hexagonal geometry, and state evolution depends on both the currently scanned symbol and traversal history within the picture (D et al., 11 Aug 2025).

The closest antecedent named in the paper is the returning finite automaton on rectangular grids, cited as Fernau et al. (2015). GHRFA inherit the returning-scan idea from that line of work, but the change from rectangular to hexagonal geometry necessitates new definitions of admissible movements and new equivalence arguments because the underlying symmetry group is different (D et al., 11 Aug 2025).

The paper also places GHRFA alongside Boustrophedon automata and picture-walking automata. Boustrophedon automata use alternating row-wise or sweep-wise motion, whereas GHRFA maintain returning sweeps. Picture-walking automata are more general in that they may permit arbitrary or less constrained traversals. In that comparison, GHRFA are presented as a special but powerful subclass in which regularity of scanning is built into the model (D et al., 11 Aug 2025).

A plausible implication is that GHRFA occupy a methodological middle ground: more geometrically specialized than general picture-walking formalisms, but broad enough to support nontrivial structural results about language classes and symmetry.

4. Language families and directional equivalence

For a fixed direction $Q$7, the language accepted by a GHRFA is denoted $Q$8, the set of all hexagonal pictures accepted under traversal in direction $Q$9. The full language family of the model is

$\Sigma$0

This decomposition makes directional choice explicit at the language-theoretic level (D et al., 11 Aug 2025).

A major result of the paper is that many directional modes define equivalent language families up to the symmetry operations of the hexagon. For suitable directions $\Sigma$1 and $\Sigma$2,

$\Sigma$3

where $\Sigma$4 is a rotation or reflection induced by the group of hexagon symmetries (D et al., 11 Aug 2025). The paper states that Theorems 8 and 9 formalize such identities using rotations $\Sigma$5 and reflections $\Sigma$6, and that directions can be partitioned into subsets whose associated language classes are mutually related by these group actions.

The paper further states that, up to symmetry, the complete family can be represented using three distinguished directions: $\Sigma$7 Although written there for the Boustrophedon family, the surrounding equivalence results are used to relate the corresponding returning classes as well (D et al., 11 Aug 2025).

These results show that directional variation does not generate essentially new expressive behavior once hexagonal rotations and reflections are taken into account. The paper therefore treats symmetry not only as a geometric property of the input domain but also as an organizing principle for the classification of accepted picture languages.

5. Equivalence with hexagonal Boustrophedon automata

One of the central theorems states that

$\Sigma$8

that is, the language class accepted by general hexagonal Boustrophedon finite automata coincides with the language class accepted by general hexagonal returning finite automata (D et al., 11 Aug 2025). The proof strategy described in the summary is bidirectional construction: for every GHBFA an equivalent GHRFA can be constructed, and conversely.

The construction is said to use state information that tracks the current state, the desired state at the end of a column, and the orientation of the scan, in analogy with a subset-construction style argument from one-dimensional automata theory (D et al., 11 Aug 2025). This is significant because the two models impose visibly different traversal disciplines—alternating sweeps in one case, returning sweeps in the other—yet they recognize the same family of hexagonal picture languages.

The paper also reports the corollary that deterministic and nondeterministic variants coincide in expressive power, again in parallel with the classical case (D et al., 11 Aug 2025). Within the scope of the reported results, traversal flexibility therefore does not enlarge the recognized language family.

A common misconception in this area is that more elaborate scan control necessarily yields strictly greater recognition power. The equivalence theorem directly counters that expectation for the hexagonal returning and hexagonal Boustrophedon models considered in the paper: the extra freedom of one systematic sweep regime over another does not produce a larger class of accepted hexagonal picture languages (D et al., 11 Aug 2025).

6. Closure properties, distinctive features, and research directions

The paper attributes two distinctive structural features to GHRFA. First, hexagonal geometry introduces sixfold symmetry and a neighbor structure different from that of rectangular grids. Second, the resulting language families are directionally robust: changing the traversal direction within $\Sigma$9 does not alter expressive power except through rotation or reflection of the accepted pictures (D et al., 11 Aug 2025).

This is formalized by the closure statement that $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$0 is closed under all hexagonal group operations. For any such operation $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$1,

$R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$2

The same summary identifies this as a consequence of the equivalence analysis over directional classes (D et al., 11 Aug 2025). In effect, the accepted-language family is stable under the full group of rotations and reflections of the hexagon.

The paper presents several future directions. These include extending deterministic GHRFA to probabilistic or nondeterministic variants for randomly traversed pictures, determining the complexity of decision problems such as emptiness, universality, and membership, exploring applications in computer vision, image processing, and formal verification, and comparing GHRFA with more general or more restricted two-dimensional automaton models (D et al., 11 Aug 2025).

These directions suggest that the principal contribution of GHRFA is foundational rather than application-specific. The model provides a rigorous framework for studying regular classes of picture languages on hexagonal grids, while the symmetry and equivalence theorems indicate that hexagonal traversal constraints can be analyzed in a systematic algebraic way (D et al., 11 Aug 2025).