Hexagonal Boustrophedon Finite Automata

• General hexagonal boustrophedon finite automata are finite-state recognizers that traverse bordered hexagonal pictures using alternating directional scans defined by 12 symmetry modes.
• They employ a dual-phase system with forward and backward transition rules, ensuring deterministic behavior through fixed-direction semantics and geometric symmetry.
• The model extends traditional rectangular picture automata by introducing novel hexagonal traversals and showing equivalence with hexagonal returning automata in language acceptance.

Searching arXiv for the specified papers and closely related context. arxiv_search(query="General hexagonal Boustrophedon finite automata Hexagonal Picture Scanning Automata (D et al., 11 Aug 2025)", max_results=5) General hexagonal Boustrophedon finite automata are finite-state recognizers for hexagonal pictures that extend boustrophedon scanning from rectangular grids to hexagonal geometry. Introduced in "Hexagonal Picture Scanning Automata" (D et al., 11 Aug 2025), the model reads a bordered hexagonal picture in alternating directions across successive hexagonal sides or strips, replacing each consumed symbol by an erased marker and accepting when the traversal terminates in a final state. The model is paired with a hexagonal returning variant and is analyzed through fixed-direction semantics, rotational and reflectional symmetries, and equivalence results. A later unifying treatment of picture automata via DAG encodings is directly relevant as methodological context, but it is formulated for rectangular pictures and does not itself define hexagonal boustrophedon automata (Meeres et al., 15 Sep 2025).

1. Formal definition

A General Hexagonal Boustrophedon Finite Automaton (GHBFA) is defined as the 8-tuple

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

where $Q$ is a finite state set partitioned into $Q_f$ and $Q_b$, $\Sigma$ is the input alphabet, $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$ is a finite transition relation written as $qa \rightarrow p$, $s \in Q_f$ is the initial state, $F$ is the set of final states, $\# \notin \Sigma$ is the boundary symbol, $Q$0 is the erased symbol, and $Q$1 is the set of allowed scanning directions on the hexagonal grid (D et al., 11 Aug 2025).

The partition $Q$2 is structural rather than cosmetic. If $Q$3 and $Q$4, then a rule $Q$5 is permitted only when $Q$6; these rules form the forward transition set $Q$7. If $Q$8 and $Q$9, then a rule $Q_f$0 is permitted only when $Q_f$1; these rules form the backward transition set $Q_f$2. Border transitions belong to $Q_f$3 and have the form $Q_f$4, with the condition that $Q_f$5 iff $Q_f$6. The boundary symbol therefore mediates alternation between forward and backward phases rather than serving only as padding.

The term boustrophedon retains its classical meaning: the scan alternates direction from one pass to the next, analogously to ox-plowing. In the hexagonal setting, however, the alternation is not organized by rectangular rows but by the geometry of the hexagon and its admissible side-to-side traversals.

2. Hexagonal pictures and traversal geometry

A hexagonal picture over $Q_f$7 is a finite hexagonal arrangement of symbols from $Q_f$8. The family of all such pictures is denoted

$Q_f$9

and the nonempty hexagonal pictures are

$Q_b$0

A picture has size $Q_b$1 when its three side lengths are $Q_b$2, with opposite sides equal. Its bordered version $Q_b$3 has size $Q_b$4 (D et al., 11 Aug 2025).

Scanning is performed on the bordered picture. Boundary cells carry the symbol $Q_b$5, and each scanned interior symbol is replaced by $Q_b$6. This erasure convention is part of the operational semantics: acceptance is defined over a progressively modified picture rather than over a static input.

The traversal discipline is one of the main differences from rectangular boustrophedon automata. GHBFA uses a set $Q_b$7 of 12 possible direction modes. Each mode is encoded by a $Q_b$8 matrix specifying the starting corner $Q_b$9, the initial reading direction, and the alternating movement between successive hexagonal sides of the picture. The paper gives 12 such matrices explicitly and interprets them as the admissible boustrophedon scans on the hexagonal grid (D et al., 11 Aug 2025).

One representative mode is written in the paper as

$\Sigma$0

with the intended meaning: start at the designated corner, scan the first side in the indicated direction, move to the next side when the boundary symbol is encountered, reverse traversal direction, and continue alternating until the entire picture has been scanned. The underlying principle is the same as in classical boustrophedon scanning, but the geometric unit of alternation is adapted to hexagonal sides or strips rather than horizontal rows.

3. Configurations, transitions, and acceptance

A configuration consists of the current state together with the current picture, some of whose cells may already have been erased. The transition relation distinguishes forward, backward, and border steps. If $\Sigma$1 and $\Sigma$2 differ only in that the cell $\Sigma$3 changes from $\Sigma$4 to $\Sigma$5, then

$\Sigma$6

and similarly

$\Sigma$7

At a boundary position, the automaton may change phase:

$\Sigma$8

The reflexive transitive closure is denoted $\Sigma$9 (D et al., 11 Aug 2025).

Acceptance is parameterized by a direction mode $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$0. A picture $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$1 is accepted if, for the chosen mode, the initial configuration reaches a final configuration:

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

where $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$3 is final. The language accepted with direction $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$4 is denoted $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$5, and the full family is

$R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$6

This direction-indexed semantics matters because the model is not defined by a single canonical traversal. The theory first studies fixed-direction language families and then relates them by geometric symmetries. A common misunderstanding is to treat the 12 scans as operational variants with potentially unrelated expressive behavior. The paper instead organizes them into symmetry-equivalent classes and derives explicit transfer results between them.

4. Determinism, directional families, and symmetry closure

The deterministic variant is called GHBDFA. It is deterministic when for every $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$7 and $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$8, there is at most one $R \subseteq Q \times (\Sigma \cup \{\#\}) \times Q$9 such that

$qa \rightarrow p$0

For each fixed direction mode $qa \rightarrow p$1, the paper proves

$qa \rightarrow p$2

with the proof idea attributed to the usual subset construction for finite automata equipped with the prescribed scanning discipline (D et al., 11 Aug 2025).

A major part of the theory concerns the 12 directional modes and the symmetries of the hexagon. The paper defines rotations $qa \rightarrow p$3 for $qa \rightarrow p$4, where $qa \rightarrow p$5 is clockwise rotation by $qa \rightarrow p$6, and reflections $qa \rightarrow p$7 for $qa \rightarrow p$8, where $qa \rightarrow p$9 is reflection about a line through the center. These 12 operations form a group. The paper also records identities such as

$s \in Q_f$0

Using these symmetries, the paper proves that the language family for one canonical direction $s \in Q_f$1 determines the other 11 directional families. It gives explicit equations expressing each noncanonical $s \in Q_f$2 as a composition of $s \in Q_f$3, $s \in Q_f$4, and their iterates applied to $s \in Q_f$5. It also derives corollaries such as

$s \in Q_f$6

$s \in Q_f$7

and, because $s \in Q_f$8,

$s \in Q_f$9

The strongest closure statement is that

$F$0

Thus the full GHBFA family is closed under all 12 hexagonal symmetries (D et al., 11 Aug 2025). The significance is not merely geometric elegance: it shows that the apparent proliferation of scanning modes does not fragment the theory into unrelated cases.

5. Equivalence with General Hexagonal Returning Finite Automata

The closely related model of General Hexagonal Returning Finite Automata (GHRFA) is introduced as the hexagonal analogue of returning finite automata. Its canonical direction is

$F$1

described as starting from the top-left corner and scanning top-down, then moving to the next column in the indicated rotational order, again top-down. The full direction set $F$2 also contains 12 modes, partitioned into $F$3, $F$4, and $F$5 (D et al., 11 Aug 2025).

The paper states that the rectangular identity

$F$6

has a hexagonal adaptation, again yielding

$F$7

for the corresponding hexagonal models. The proof constructs an equivalent GHRFA $F$8 from a given HBFA $F$9 by expanding the state space to

$\# \notin \Sigma$0

The intended interpretation is explicit: the first component stores the actual current state, the second stores the state to be restored or reached after completing a column, and the third is a direction flag, with $\# \notin \Sigma$1 meaning top-to-bottom scan and $\# \notin \Sigma$2 meaning bottom-to-top scan.

The transition set is given in the paper as

$\# \notin \Sigma$3

$\# \notin \Sigma$4

$\# \notin \Sigma$5

$\# \notin \Sigma$6

$\# \notin \Sigma$7

The final-state clause is reported as

$\# \notin \Sigma$8

and the description notes that the notation appears to contain a typo in the recursive reference to $\# \notin \Sigma$9. Even so, the intended content is unambiguous: the construction is presented as preserving language acceptance (D et al., 11 Aug 2025).

This equivalence places GHBFA and GHRFA within the same recognition family despite their different traversal disciplines. The contrast between alternating-direction scanning and returning scanning is therefore operational rather than language-theoretic at the level established in the paper.

6. Relation to rectangular picture automata and DAG encodings

General hexagonal boustrophedon automata extend a line of work on rectangular picture automata. In the rectangular setting, boustrophedon finite automata (BFA) and returning finite automata (RFA) are finite-state machines over boundary pictures with different scanning orders; the unifying paper on DAG encodings proves

$Q$00

and also shows that deterministic and nondeterministic versions coincide (Meeres et al., 15 Sep 2025). Restricted to one-row inputs, these models recognize exactly the regular string languages. The same paper characterizes them through deterministic DAG automata under appropriate picture-to-DAG encodings.

For the hexagonal topic, the key point is negative as well as positive. The DAG-encoding framework is stated for rectangular pictures on a square grid and explicitly does not define hexagonal boustrophedon automata, triangular-grid automata, or non-rectangular picture models as separate formal frameworks (Meeres et al., 15 Sep 2025). Accordingly, none of its theorems transfers verbatim to GHBFA.

At the same time, the framework is suggestive. It treats scanning order as a primary abstraction and separates geometry from automaton behavior through the choice of encoding. This suggests that a hexagonal boustrophedon scan could, in principle, be represented by a DAG whose vertices are hexagonal cells and whose edges encode the chosen traversal or neighborhood dependencies. A plausible implication is that GHBFA could eventually be recast within a more general encoding-based theory of picture automata. The currently established results, however, are those of the dedicated hexagonal model, not those of the rectangular DAG formalism.

The theoretical significance of GHBFA therefore lies in two simultaneous extensions. First, it transfers classical boustrophedon and returning scanning ideas from rectangular to hexagonal pictures. Second, it organizes the resulting language families by the symmetry group of the hexagon rather than by ad hoc case analysis. The resulting picture is that richer geometry changes the traversal structure and symmetry behavior, while nondeterminism still collapses to determinism for each fixed direction and the boustrophedon and returning variants remain language-equivalent (D et al., 11 Aug 2025).