Hexagonal Edge Traversal (HET)

• Hexagonal Edge Traversal (HET) is a coordination problem where two autonomous robots perpetually alternate moves on a hexagon’s edges using phase-sensitive rules.
• The fully synchronous luminous algorithm employs four light colors to encode a four-phase cycle that ensures exact lock-step edge traversals.
• HET delineates a solvability boundary showing that persistent state and scheduler synchrony enable coordination, while semi-synchronous and silent models fail due to phase ambiguity.

Searching arXiv for the specified paper to confirm bibliographic details and citation. Hexagonal Edge Traversal (HET) is a perpetual coordination problem for two autonomous mobile robots on a regular hexagon, introduced in "Separation of Three or More Autonomous Mobile Models under Hierarchical Schedulers" (Naito et al., 27 Aug 2025). The problem is formulated to isolate how synchrony, persistent visible state, and symmetry breaking interact in graph-based mobile robot systems. In HET, two robots start on opposite vertices of a hexagonal graph and must continue forever in a prescribed alternating pattern of edge traversals on the two edges incident to each robot’s initial vertex. The problem is notable because its solvability boundary is neither a simple consequence of synchrony alone nor of visibility alone: it is solved exactly in the strong luminous-memory models $\mathcal{LUMT}^{F}$ and $\mathcal{LUMT}^{R}$, and fails under semi-synchronous activation and in the silent models considered in the paper (Naito et al., 27 Aug 2025).

1. Formal statement of the traversal task

HET is defined on the graph $H=(V,E)$ with vertex set

$V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$

and edge set

$$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$

Thus, the graph consists exactly of the sides of a regular hexagon (Naito et al., 27 Aug 2025).

Two robots, denoted $r$ and $q$, are initially placed on opposite vertices: $r(0)=v_i,\quad q(0)=v_{i+3\bmod6}$ for some unknown $i$. The requirement is not eventual rendezvous, exploration, or termination, but an infinite edge-traversal cycle. For robot $r$, there must exist strictly increasing times

$\mathcal{LUMT}^{R}$0

such that, for every integer $\mathcal{LUMT}^{R}$1, $\mathcal{LUMT}^{R}$2 stays at $\mathcal{LUMT}^{R}$3, traverses the edge $\mathcal{LUMT}^{R}$4 to $\mathcal{LUMT}^{R}$5, stays there, and then traverses back to $\mathcal{LUMT}^{R}$6. Robot $\mathcal{LUMT}^{R}$7 is required to execute the analogous perpetual pattern from $\mathcal{LUMT}^{R}$8, alternately going to $\mathcal{LUMT}^{R}$9 and returning, then to $H=(V,E)$0 and returning, and repeating (Naito et al., 27 Aug 2025).

Informally, each robot repeatedly walks back and forth on its two adjacent edges in alternation, never leaving the hexagon, and must sustain this pattern forever. The requirement is therefore a perpetual, phase-sensitive coordination task rather than a one-shot decision problem.

2. Robot model and scheduler assumptions

HET is studied in the luminous, full-communication, full-memory setting under multiple scheduler assumptions. The internal capability emphasized in the detailed exposition is $H=(V,E)$1, described as “mutual lights, with memory.” Each robot has a persistent light $H=(V,E)$2 taking values from a constant finite color set $H=(V,E)$3, and during each Look operation the robot observes the positions and light colors of all robots, including itself. During Compute, it may update its own light, and the lights persist indefinitely (Naito et al., 27 Aug 2025).

The movement assumption is rigid motion along straight-line segments between nodes. The robots share chirality, meaning they agree on clockwise versus counterclockwise orientation. At the same time, they are disoriented in the sense that each robot’s local coordinate frame may be rotated or reflected arbitrarily, and may change between Look operations (Naito et al., 27 Aug 2025).

The paper parameterizes HET by scheduler class:

• $H=(V,E)$4: fully synchronous, where in each discrete round all robots execute Look–Compute–Move atomically.
• $H=(V,E)$5: round-robin, where after a possible finite prefix of fully synchronous rounds, each subsequent round activates exactly one new robot in cyclic order, with no overlapping activations.
• $H=(V,E)$6: semi-synchronous, where each round activates a nonempty subset of robots, but those activated robots act synchronously (Naito et al., 27 Aug 2025).

The paper also considers silent models, specifically $H=(V,E)$7 and $H=(V,E)$8, in which the robots are described as silent and oblivious. For HET, these models serve to separate the role of persistent state from the role of scheduler strength.

3. Exact solvability boundary

The HET results are summarized by five lemmas and a final theorem. The paper states:

• Lemma 3.1: HET is solvable in $H=(V,E)$9.
• Lemma 3.2: HET is solvable in $V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$0.
• Lemma 3.3: HET is not solvable in $V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$1.
• Lemma 3.4: HET is not solvable in $V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$2.
• Lemma 3.5: HET is not solvable in $V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$3 (Naito et al., 27 Aug 2025).

These statements culminate in Theorem 3.6: $V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$4 with both $V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$5 and $V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$6 strictly weaker than $V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$7 (Naito et al., 27 Aug 2025).

A compact summary of the classification is as follows:

Model	HET
$V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$8	Solvable
$V=\{v_0,v_1,v_2,v_3,v_4,v_5\},$9	Solvable
$$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$0	Not solvable
$$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$1	Not solvable
$$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$2	Not solvable

The paper further states that no nontrivial time or space-complexity bounds arise in these impossibility proofs; the relevant issue is whether the model can break the initial symmetry and maintain enough state to coordinate the cyclical edge traversals. This identifies HET as a separation problem in the sense of distributed computability rather than a complexity-optimization problem.

4. Fully synchronous luminous algorithm

For $$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$3, the paper gives a complete algorithm using four light colors $$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$4 to encode the phase of the traversal cycle (Naito et al., 27 Aug 2025). The phases correspond to the following recurrent pattern:

• from the start vertex, move to the left neighbor;
• return to the start vertex;
• move to the right neighbor;
• return to the start vertex.

In the pseudocode, each robot stores its own light state and, in each synchronous round, updates both its light and its destination. In state $$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$5, the robot sets its light to $$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$6 and moves toward the counterclockwise adjacent vertex. In state $$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$7, it sets its light to $$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$8 and returns to its initial vertex. In state $$E=\bigl\{\{v_i,v_{i+1 \bmod 6}\}\mid 0\le i<6\bigr\}.$$9, it sets its light to $r$0 and moves toward the clockwise adjacent vertex. In state $r$1, it resets its light to $r$2 and returns to its initial vertex. Movement is rigid, and the destination is computed using the agreed chirality through the operation neighbor(v,k).

A critical ingredient is myStartPos, defined as the unique vertex observed in the very first snapshot. The detailed explanation in the paper attributes the availability of this datum to the fully synchronous start condition: all robots begin in exactly one round, allowing them to anchor the cycle to the same initial reference point (Naito et al., 27 Aug 2025).

The significance of the algorithm is not merely that it uses four phases, but that full synchrony keeps both robots in exact lock-step. After each round, both are at the same phase number, so the local meaning of “next edge” remains globally consistent throughout the infinite execution. This suggests that, for HET, the essential algorithmic resource is not motion planning in the geometric sense, but persistent phase coherence under symmetry.

5. Why weaker assumptions fail

The paper’s impossibility arguments isolate two distinct failure modes: loss of lock-step coordination under semi-synchrony, and loss of persistent phase information in silent models (Naito et al., 27 Aug 2025).

For $r$3, the adversary may activate the robots in different subsets of a round, such as activating only $r$4 and then only $r$5 in successive rounds. Under variable disorientation, each robot may be unable to determine whether it is currently in the outward move or the return move when activated alone, because the visible configuration can remain “the other on the opposite vertex.” The paper states that no finite light-memory suffices to break this ambiguity, so the cycle can become permanently out of phase and HET fails.

For $r$6, the scheduler is as strong as possible, but the robots are silent and oblivious. The paper emphasizes that synchrony does not by itself encode the four-step cycle. At each round, a robot sees the other on the opposite vertex and cannot distinguish “just started phase 1” from “just started phase 3,” so there is no way to store the phase index internally. Hence no oblivious algorithm exists.

For $r$7, the situation is strictly worse than in $r$8. The adversary activates exactly one robot per round, so the robots never obtain a simultaneous Look and never break the initial $r$9 symmetry. The paper therefore concludes that HET is impossible in this model as well.

Two incorrect extrapolations are explicitly ruled out by these results. First, stronger synchrony alone does not suffice, because $q$0 still fails. Second, visible persistent state alone under weaker synchrony does not suffice, because $q$1 still fails. HET therefore separates phase memory from phase coherence rather than collapsing them into a single resource.

6. Proof ideas and structural role in the hierarchy

The proof sketches in the paper revolve around symmetry invariants and adversarial scheduling. Whenever both robots occupy opposite vertices of the regular hexagon, the global configuration is centrally symmetric. If the robots’ lights and memories do not encode the round number modulo $q$2, then every valid Look snapshot is invariant under a $q$3 rotation, and no deterministic rule can select which adjacent edge should be traversed next (Naito et al., 27 Aug 2025).

The scheduler-based arguments sharpen this symmetry obstruction. In $q$4, the adversary interleaves single-robot activations so that each robot effectively “thinks” it is still at the start of phase $q$5. In silent models, the first Look is symmetric and the paper states that no subsequent Look provides new distinguishing information. These arguments are not quantitative lower bounds; they are impossibility criteria based on indistinguishability and persistent ambiguity.

Within the paper’s broader “Separation Map,” HET occupies an intermediate band. It is solvable exactly by $q$6 and $q$7, but not by $q$8 (Naito et al., 27 Aug 2025). The paper places HET alongside two related problems:

• ETE lies strictly above HET, because ETE is solvable only in $q$9 and not even in $r(0)=v_i,\quad q(0)=v_{i+3\bmod6}$0.
• TAR($r(0)=v_i,\quad q(0)=v_{i+3\bmod6}$1)* is exactly co-located with HET, being solvable in $r(0)=v_i,\quad q(0)=v_{i+3\bmod6}$2 and $r(0)=v_i,\quad q(0)=v_{i+3\bmod6}$3 but not in $r(0)=v_i,\quad q(0)=v_{i+3\bmod6}$4.

This positioning is important for the paper’s higher-order separation program. The authors state that these results extend the known separation map of 14 canonical robot models and reveal structural phenomena visible only through comparisons involving three or more models rather than pairwise separations alone. A plausible implication is that HET functions as a canonical witness problem for a specific mid-level separation: persistent luminous state combined with either full synchrony or round-robin regularity is sufficient, whereas semi-synchronous activation reintroduces enough adversarial freedom to destroy perpetual phase coordination.