Leave Place Convergence (LP–Cv)

• Leave Place Convergence (LP–Cv) is defined as a two-phase process where robots initially move outward from their positions and then converge to a common point.
• The task illustrates how internal memory, light observability, and scheduler patterns (e.g., Rsync, asynchronous with lights) dictate feasible coordination.
• Impossibility results in fully or semi-synchronous settings emphasize that observable state alone cannot overcome symmetry issues in achieving correct phase transitions.

Searching arXiv for the specified LP-Cv source paper and closely related robot-model context. Leave Place Convergence (LP–Cv) is a two-phase task for two autonomous mobile robots in the classical Look–Compute–Move framework. It requires the robots, starting from arbitrary non-overlapping positions in $\mathbb{R}^2$, first to move away from one another along the line joining them and only afterward to solve convergence. The formulation is used to expose how internal memory, light observability, and scheduler structure interact in canonical robot models: the task is impossible in some settings that appear stronger in synchrony, yet solvable in others that provide either patterned activation or visible lights (Naito et al., 27 Aug 2025).

1. Formal task specification

LP–Cv is defined for two robots $r$ and $q$, modeled as points in the Euclidean plane $\mathbb{R}^2$. The robots are anonymous, identical, homogeneous, disoriented, and equipped with chirality. Movements are rigid: when a robot computes a destination, it traverses the corresponding straight segment until arrival. The problem specification is expressed in global coordinates, which are unavailable to the robots but valid for the external task predicate.

The formal definition is:

$$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$

This predicate decomposes into two phases. In the LP phase, there exists a time $T$ such that both robots remain on the line segment determined by their instantaneous positions, both have left their initial positions by time $T$, and their displacements are outward relative to the standardized coordinate convention: $r$ moves in the negative direction and $q$ in the positive direction. In the Cv phase, after some $T' \geq T$, both robots converge to an arbitrary point $r$0 in the sense that they can be made arbitrarily close to a common point.

A central feature of LP–Cv is that the phase ordering is part of the specification. Convergence cannot begin before both robots have left their initial positions. The paper also states that, unlike LP–MLCv or MLCv, LP–Cv imposes no monotonicity requirement on the convergence distance (Naito et al., 27 Aug 2025).

2. Robot models, observability, and scheduler assumptions

The task is studied in two standard robot models. In the FSTA model, each robot has a persistent internal state or light visible only to itself, and cannot see the other robot’s color. In the FCOM model, each robot can see the other robot’s light but not its own, and therefore has communication via visible lights but no internal memory in the FSTA sense.

The schedulers considered for LP–Cv are $r$1, $r$2, $r$3, and $r$4. FSYNC ($r$5) activates all robots every round and executes one atomic Look–Compute–Move cycle. SSYNC ($r$6) activates some nonempty subset each round, also atomically. Rsync ($r$7) is a special semi-synchronous scheduler: after a finite fully synchronous prefix, subsequent rounds activate nonempty subsets such that any two consecutive subsets are disjoint. Intuitively, if a robot is activated again, the other robot must have been activated in the meantime. The asynchronous scheduler $r$8 satisfies M-atomicity:

$r$9

This means that no robot takes a snapshot while another robot is moving; movements are atomic with respect to Look.

These distinctions are decisive for LP–Cv because the critical event is not merely movement, but recognition that both robots have completed the LP phase. In FSTA, a robot can remember whether it has moved, but cannot directly observe whether the other robot has moved. In FCOM, a robot can observe the other robot’s progress through visible light states. The task is therefore a particularly sharp probe of what can and cannot be inferred from local state, geometry, and activation structure alone (Naito et al., 27 Aug 2025).

3. Operational intuition and phase-transition difficulty

Informally, LP–Cv asks the robots to perform two behaviors in sequence. First, each robot must leave its initial place by moving away from the other along their line of separation. Second, after both have done so, the robots must execute convergence toward some common limit point. A successful execution therefore requires not only outward displacement and later convergence, but also a correct transition condition between the two phases.

The difficulty lies in deciding when the system may safely switch from LP to Cv. A robot with only its own internal state can record that it has already performed the outward move, but that does not reveal whether the peer has done the same. Under adversarial activation patterns, one robot can be activated again before the other has moved at all. At that point, starting convergence may violate the LP predicate, while waiting may lead to indefinite stalling. The problem is thus not geometric in a narrow sense; it is fundamentally about observability of global progress.

This suggests why LP–Cv is useful as a separation problem. The geometric motion required in each phase is simple: move outward by a prescribed fraction of the observed distance, then move inward by a prescribed fraction. What is nontrivial is synchronizing the phase change without shared coordinates, without names, under disorientation, and under schedulers that may preserve symmetry rather than break it. The paper explicitly uses LP–Cv to illustrate the limitations of internal memory in symmetric settings (Naito et al., 27 Aug 2025).

4. Impossibility results

The paper states two core impossibility lemmas for FSTA-based settings:

$q$0

and

$q$1

The first impossibility is presented as Lemma 27. The accompanying argument is that robots cannot determine from observation whether they have left their initial positions. Because the robots are anonymous and disoriented, the adversary can choose coordinate systems so that snapshots before and after moves look identical from a robot’s local perspective. The task’s phase discipline therefore cannot be enforced solely from geometric observation.

The second impossibility, Lemma 28, concerns SSYNC. Under semi-synchronous activation, some robots may be activated more often than others. If robot $q$2 has already moved away and is activated again while $q$3 may or may not have moved, then $q$4 faces an irresolvable choice. If it starts convergence, it may begin Cv while $q$5 is still at the initial position; if it waits, it may wait forever because it never observes a configuration that definitively certifies that $q$6 has completed LP. The paper identifies this as a symmetry and indistinguishability barrier: internal memory alone does not encode enough information to break the uncertainty about who moved and when (Naito et al., 27 Aug 2025).

A further summary in the source places oblivious robots outside the solvability region entirely: LP–Cv is impossible everywhere for robots with no memory and no lights. A plausible implication is that the LP phase itself is not the obstacle in isolation; rather, the obstacle is certifying completion of LP under symmetry-preserving observations.

5. Solvability under Rsync and under $q$7 with lights

The solvable side of the classification is given by Lemma 29 and Lemma 30:

$q$8

and

$q$9

Under Rsync with FSTA, the paper gives Algorithm 13. Each robot maintains an internal two-state variable $\mathbb{R}^2$0, initially $\mathbb{R}^2$1. On the first activation in state $\mathbb{R}^2$2, the robot switches to $\mathbb{R}^2$3 and performs the LP move: it sets its destination to the point at distance $\mathbb{R}^2$4 opposite to $\mathbb{R}^2$5. On subsequent activations in state $\mathbb{R}^2$6, it performs the Cv move: it sets its destination to the point at distance $\mathbb{R}^2$7 toward $\mathbb{R}^2$8. The scheduler is the crucial ingredient. Because any two consecutive activation sets are disjoint, a robot that is activated for the second time can infer that the other robot has been activated at least once in the meantime. Scheduler structure thereby substitutes for direct communication.

Under $\mathbb{R}^2$9 with FCOM, the paper gives Algorithm 14 using three light colors $$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$0. The Compute phase depends on $$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$1:

$$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$2

Here visible lights substitute for the activation guarantees provided by Rsync. Because each robot can observe the partner’s light and because M-atomicity prevents snapshots during movement, the robots can coordinate the transition from LP to Cv even in an asynchronous setting. The source summarizes the boundary as follows: LP–Cv is solved in both $$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$3 and $$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$4, but not in any of $$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$5 or $$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$6 (Naito et al., 27 Aug 2025).

Model / scheduler	Status for LP–Cv	Basis
$$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$7	impossible	Lemma 27
$$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$8	impossible	Lemma 28
$$\begin{aligned} LP\text{--}Cv \equiv \big[ &\exists T \geq 0 : \big[ \{ \forall t \leq T : r(0), q(0) \in \overline{r(t)q(t)} \} \ &\land \{ r(T) \neq r(0), \, q(T) \neq q(0) \} \ &\land \{ \forall t \leq T : r(t) - r(0) \leq 0, \, q(t) - q(0) \geq 0 \} \big] \ &\land \big[ \exists l \in \mathbb{R}^2, \, \epsilon \geq 0, \, \exists T' \geq T, \, \forall t \geq T' : |r(t) - l| + |q(t) - l| \leq \epsilon \big] \big] \end{aligned}$$9	solvable	Lemma 29 / Algorithm 13
$T$0	solvable	Lemma 30 / Algorithm 14

6. Separation role, structural significance, and related tasks

LP–Cv appears in the paper’s separation map as one of the witness problems used to distinguish models that would be difficult to separate by pairwise comparisons alone. In Section 5.2, LP–Cv and VTR are grouped under “Synchrony Is Necessary for FSTA (LP–Cv, VTR).” In the global classification, LP–Cv lies outside the region solvable by oblivious robots and by FSTA robots under FSYNC or SSYNC, but inside the region solvable by FSTA under Rsync and by FCOM under $T$1.

The task is especially notable because it exhibits a non-monotone relation between synchrony notions and computational power. The paper states that this is a hierarchical scheduler separation: $T$2 is strictly more powerful than $T$3 and $T$4 for FSTA on LP–Cv, even though these schedulers sit nearby in the synchrony hierarchy. This directly counters a common simplification that “more synchrony is more powerful.” For LP–Cv, what matters is not synchrony in the abstract, but whether the scheduler imposes the specific activation pattern needed to make local state semantically informative.

LP–Cv is closely related to LP–MLCv, a stricter variant requiring monotonic non-increasing distance in convergence, and to Cv itself, which omits the initial LP phase. The source states that Cv is solvable in more models than LP–Cv because it does not require phase ordering or recognition of whether the robots have already left. It also relates LP–Cv to VTR, ZCC, and VEC as part of a broader landscape of trade-offs between memory, communication, atomicity, and scheduler guarantees (Naito et al., 27 Aug 2025).

7. Conceptual implications and common misunderstandings

LP–Cv is a canonical example of the limitations of internal memory in symmetric settings. In FSTA, each robot can remember facts about its own history but cannot see the partner’s state. Under SSYNC, an adversarial scheduler can always activate one robot “ahead” of the other, creating a persistent mismatch that internal memory alone cannot repair. This shows that memory and communication are not interchangeable under all schedulers.

A second misconception is that full synchrony should dominate scheduler variants such as Rsync. The LP–Cv classification demonstrates the opposite phenomenon: FSYNC with FSTA is insufficient, whereas Rsync with FSTA is sufficient. The paper explains this by noting that Rsync’s disjoint consecutive activation sets encode coordination information that robots can exploit. Patterned scheduler structure is therefore a computational resource in its own right.

A third misconception is that LP–Cv is merely a variant of convergence. The addition of the LP prefix changes the problem qualitatively because it introduces a phase-ordering requirement: “first leave, then converge.” The impossibility results arise precisely from the inability to certify completion of the first phase under symmetry and adversarial timing. This suggests a broader principle emphasized by the paper: for tasks with explicit phase structure, the decisive issue is often not the complexity of the geometric move itself, but whether observability and scheduling together make phase transitions locally decidable (Naito et al., 27 Aug 2025).