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On Representational Alignment among Embodied Agents

Fulvio Mastrogiovanni

arXiv:2610.02985v1cs.RO

TL;DR

The paper asks which differences among heterogeneous, asynchronous observer representations must be resolved for coherent interaction. It formalizes task-relative relational sufficiency and finds that identical hand-off evidence needs no additional anchor for relative synchronization but one for absolute timing.

  • Problem

    Embodied agents can maintain heterogeneous, asynchronous, observer-relative representations, raising the question of which representational differences must be resolved for coherent interaction.

    Source: §1 · pp. 1, 3

  • Method

    The paper defines relational sufficiency by whether residual ambiguity under interaction-specific evidence stays within an interaction-equivalence class, and uses sensorimotor anchors to separate task-relevant ambiguities.

    Source: §1 · pp. 3–4, §7 · p. 31

  • Results

    The same analytical hand-off scenario requires no additional scalar anchor for relative synchronization but one for absolute timing, solely because interaction invariances differ.

    Source: §1 · p. 4, §7 · p. 31

  • Takeaways & Limitations

    Representational alignment can stop at residual disagreement that is invariant with respect to the interaction, rather than requiring complete agreement or full state identification.

    Source: §1 · p. 4, §7 · p. 31

    The lower bound is local and first-order, does not establish global sufficiency in nonlinear spaces, and matches required anchor count only under the anchor-richness condition.

    Source: §6.3 · p. 30

Abstract

from arXiv · show

Embodied agents interacting with the same physical process may maintain heterogeneous, asynchronous, and observer-relative representations. Rather than assuming that such representations should always be globally aligned, we investigate which distinctions among them must actually be resolved for coherent interaction. We formalize this question through relational sufficiency: an interaction-specific relational-evidence map defines the representational ambiguity left unresolved by comparison, and sufficiency holds when each residual ambiguity fiber lies within an interaction-equivalence class. When this condition fails, additional sensorimotor anchors must separate precisely the ambiguities that can change the interaction-relevant outcome. For smooth systems, we derive a local first-order lower bound on the number of inde- pendent scalar anchors required, given by the dimension of directions that are invisible to relational evidence but visible to the interaction-relevant map. We instantiate the theory in two variants of an asynchronous hand-off scenario that share the same observer agents, representations, transports, and relational evidence. Relative synchronization requires no additional anchor, whereas hand-off at an absolute time requires exactly one. The analysis there- fore provides a task-relative stopping condition for representational alignment: residual disagreement need not be eliminated when it is invariant with respect to the interaction.

1. Introduction

Embodied agents may hold heterogeneous, asynchronous, observer-relative representations, so the paper asks which disagreements must be resolved for coherent interaction. It formalizes task-relative relational sufficiency and shows that the same hand-off evidence can require different anchoring depending on the task.

1. Introduction

  • Relational evidence is sufficient when every configuration it leaves indistinguishable is also indistinguishable to the interaction.Agents need not identify the uniquely correct representation when residual ambiguity cannot change the task-relevant response.Source: §1 · pp. 3–4
  • When relational evidence leaves task-distinguishable ambiguity, sensorimotor anchors must separate those ambiguities using locally obtainable interaction-derived constraints, not global ground truth.Examples include environmental cues, contact events, and landmark observations.Source: §1 · p. 4
  • The proposed stopping condition for alignment is task-relative: resolve only representational ambiguities that can change the interaction-relevant outcome.The paper also characterizes alignment as task-over-refined when it distinguishes configurations within the same interaction-equivalence class.Source: §1 · pp. 4–5
  • Relative synchronization needs no additional anchor, while absolute hand-off timing requires exactly one independent scalar anchor despite identical agents, representations, transports, and relational evidence.The difference arises from task invariances, not representational accuracy or sensing capability.Source: §1 · p. 4
  • The paper derives a local first-order lower bound on independent scalar anchors from the dimensional mismatch between evidence-invisible directions and task-invariant directions.The bound concerns directions invisible to relational evidence but visible to the interaction-relevant map.Source: §1 · pp. 4–5

2. Background and Motivating Scenario

The background distinguishes this work from approaches that seek globally compatible representations: it asks whether unresolved relational disagreement matters to a particular task. A hand-off example shows how the same common temporal shift can be harmless for relative synchronization but consequential for absolute timing.

2.1. Shared representations and relational sufficiency

  • Unlike functional observability and decentralized information approaches, this work studies task-relevant discrepancies between heterogeneous representations and restricts additional measurements to admissible sensorimotor anchors.It also differs from task-oriented communication by characterizing which ambiguities must be eliminated regardless of how evidence is obtained.Source: §2.1 · p. 7
  • The paper reframes alignment as a task-dependent requirement, asking which unresolved relations can alter interaction rather than how to make representations globally consistent.It characterizes the object of interest as ambiguities among heterogeneous observer-relative representations.Source: §2.1 · p. 8
  • Agreement, epistemic congruity, and interaction adequacy are distinct: agreement may contribute to congruity, and congruity may contribute to successful interaction, but they are not equivalent.Residual disagreement can be harmless under task invariance, while perfect agreement can still fail when both agents share a task-relevant error.Source: §2.1 · p. 8

2.2. An example scenario

  • The S1-R and S1-A scenarios keep the agents, sensing, representations, transports, and pairwise evidence fixed while changing whether common-mode ambiguity is invariant to the interaction.The common shift is invisible to both evidence and interaction in S1-R, but visible to the interaction in S1-A.Source: §2.2 · pp. 9–10
  • When unresolved ambiguity cannot change the interaction, further alignment is unnecessary; otherwise an environmental relation need only distinguish configurations that are behaviorally different.A sensorimotor anchor need not provide a complete or privileged representation of the environment.Source: §2.2 · p. 10

3. A Relational Model of Embodied Observer Agents

The formal model separates observer-relative representations, interaction-specific comparison, relational evidence, task response, and adequacy. It permits residual representational ambiguity when that ambiguity cannot alter the chosen interaction-relevant outcome.

3.1. Embodied observer agents and representation events

  • Each embodied observer agent has a constitution comprising embodiment, sensing capabilities, and available actions, which shape environmental coupling and future observations.Agents may differ in sensing, sampling times, noise, and action spaces.Source: §3.1 · p. 11
  • A representation event is a situated representation constructed from an observer’s sensorimotor history for a purpose, viewpoint, modality, or interaction context; agents need not share representational spaces.Equality between local representations is neither assumed nor generally meaningful.Source: §3.1 · p. 11

3.2. Representational transport and relational evidence

  • Representational transport maps local representations into an interaction-specific comparison context, producing a joint configuration that need not be centralized knowledge or a complete world model.The available relational evidence records properties ascertainable through the interaction’s comparison mechanisms, such as relative differences or agreement residuals.Source: §3.2 · p. 12

3.3. Task response, relational ambiguity, and sensorimotor anchors

  • Relational evidence and task response are separate maps; interaction adequacy evaluates the resulting response separately as success or failure.The interaction-relevant map can represent exact response equivalence or the weaker adequacy equivalence.Source: §3.3 · pp. 13–14
  • Sufficiency holds when every configuration left indistinguishable after relational comparison and anchoring is equivalent under the chosen interaction-relevant map.Residual ambiguity need not collapse to a unique configuration; if relational evidence leaves task-distinguishable configurations, admissible anchors supply locally obtainable interaction-derived information to separate them.Source: §3.3 · pp. 14–15

4. Relational Sufficiency and Minimal Anchoring

The paper asks when relational evidence preserves enough information for an embodied interaction despite unresolved representational ambiguity. It formalizes sufficiency and minimal sensorimotor anchoring in terms of whether that ambiguity can change the task-relevant outcome.

4.1. Relational sufficiency and interaction invariance

  • Relational evidence is sufficient exactly when equal evidence implies equal interaction-relevant outcomes, or equivalently when each ambiguity fiber lies within one interaction-equivalence class.Agents need not identify a unique configuration if they can determine its interaction-equivalence class.Source: §4.1 · p. 15
  • Common-shift ambiguity is harmless for relative synchronization but changes absolute hand-off timing, so the same relational evidence is sufficient only for the relative task.The relative response is invariant to common shifts; the absolute response is not.Source: §4.1 · p. 17

4.2. Minimal sensorimotor anchoring

  • An anchor set restores sufficiency exactly when it separates every pair that relational evidence cannot distinguish but the interaction can.Anchors need only resolve task-conflicting ambiguity, not identify the world state or reconstruct a world model.Source: §4.2 · p. 18
  • The local first-order lower bound counts independent directions invisible to relational evidence but visible to the interaction-relevant map.It is the dimension of the epistemically invisible tangent space after removing directions that are also interaction-invariant.Source: §4.2 · p. 19
  • The lower bound does not guarantee that suitable physical anchors exist; attaining it requires an admissible anchor family rich enough to resolve the mismatch.The stopping condition is task-relative: anchoring is complete once every ambiguity that can change the interaction-relevant quantity is resolved.Source: §4.2 · pp. 19–20
  • For the hand-off example, the lower bound is zero anchors for relative synchronization and at least one for absolute interaction.One independent piece of absolute temporal information can separate the common-shift ambiguity without reconstructing the complete physical state.Source: §4.2 · pp. 19–20

5. Two Variants of the Asynchronous Hand-off Scenario

The two hand-off variants share the observer model and relational evidence but differ in what the interaction treats as relevant. That difference determines whether common-mode ambiguity can remain unresolved or requires an anchor.

5.1. Common observer model and unresolved common-mode ambiguity

  • The asynchronous observers use heterogeneous native representations transported into the same hand-off-time comparison space, and transport accuracy depends on observation timing and prediction horizon.No particular noise model is required for the argument.Source: §5.1 · p. 21

5.2. S1-R: relative synchronization

  • In S1-R, common-mode shifts are invisible to pairwise evidence but also interaction-invariant, making that evidence response- and adequacy-sufficient with no additional anchor.Resolving the absolute temporal offset would refine the representation without changing the interaction-relevant quantity.Source: §5.2 · pp. 22–23

5.3. S1-A: absolute hand-off timing and minimal anchoring

  • In S1-A, a common-mode shift preserves pairwise agreement yet can violate absolute timing tolerances; one scalar sensorimotor anchor separates that ambiguity and is locally minimal.The variants keep the observer model and relational evidence fixed; only interaction invariance changes.Source: §5 · p. 20, §5.3 · p. 24
  • A Boolean timing-window cue can establish adequacy over the specified tolerance regime without reconstructing the response, so adequacy can require less information.The cue is not response-sufficient, but it is adequate over the specified tolerance regime.Source: §5.3 · p. 25

6. Discussion

Relational sufficiency asks whether unresolved representational ambiguity can change the response to a specified interaction, rather than requiring globally identical representations. The discussion frames task-relevant alignment as a stopping condition while outlining boundaries on the theory’s scope.

6. Discussion

  • The required compatibility is determined by whether relational evidence leaves ambiguities that cross interaction-equivalence classes.A sufficient relational quotient need not reconstruct a unique configuration or globally accessible world representation.Source: §6 · p. 26

6.1. Alignment, anchoring, and interaction-relevant representation

  • Alignment can be task-over-refined when it distinguishes configurations equivalent for the chosen interaction, though those distinctions may help other tasks or future interactions.Disagreement matters for the current interaction only when it crosses interaction-equivalence classes.Source: §6.1 · p. 26
  • Relative synchronization leaves the common-mode degree of freedom irrelevant, while absolute hand-off timing makes that same degree of freedom task-relevant and requires an anchor.The landmark in S1-A supplies one additional relation, sufficient because the unresolved ambiguity has one task-relevant dimension.Source: §6.1 · p. 27
  • Successful action may require less information than identifying the exact physical response, which may itself require less than complete representational identification.The Boolean-anchor example permits successful interaction with bounded relational evidence and a one-bit environmental constraint even when the exact response remains unidentified.Source: §6.1 · p. 27
  • For learning systems, reward and representational-complexity costs may favor quotienting joint relational configurations by task relevance, but this implication is left for future empirical investigation.The discussion presents this as a narrower multi-observer implication, not a new reinforcement-learning result.Source: §6.1 · p. 28

6.2. What changes with one or more than two observers?

  • The analysis focuses on two distinct observers; a single-observer extension is treated as degenerate, while pairwise sufficiency may not ensure sufficiency across three or more observers.A multi-observer theory would need to address cycle inconsistency, non-transitive congruity, higher-order constraints, and anchor distribution.Source: §6.2 · pp. 28–29

6.3. Scope, assumptions, and limitations

  • The results are conditional on a fixed interaction and interaction-relevant map; the paper also assumes transport is defined and uses deterministic responses and local first-order bounds.The scenario is analytically solvable rather than an embodied benchmark; equality between the lower bound and required anchors depends on anchor richness.Source: §6.3 · pp. 29–31
  • The analysis does not establish external validity for physical or simulated robots, nor does it address active, state-changing information acquisition.Transport construction is a separate problem, and the current static anchoring results assume admissible anchors are available before the task action.Source: §6.3 · pp. 30–31

7. Conclusions

The paper formalizes when heterogeneous observer representations can support coherent interaction without resolving every disagreement. Its hand-off examples show that needed anchoring depends on interaction invariances, yielding a task-relative stopping condition for alignment.

7. Conclusions

  • Relational evidence is sufficient when its residual ambiguity lies entirely within an interaction-induced equivalence class; otherwise, anchoring is needed only for evidence-invisible distinctions relevant to the interaction.The analysis establishes this principle for two embodied observer agents and analytically controlled tasks.Source: §7 · pp. 31–32
  • Relative synchronization requires no additional scalar anchor, whereas absolute timing requires one for exact response sufficiency.The two variants hold the observer agents, representations, transports, and relational evidence fixed; the difference arises from interaction invariances.Source: §7 · p. 31
  • Alignment or grounding need only continue until the remaining ambiguity is behaviorally irrelevant to the interaction, not until representations fully agree.Active state-changing anchoring, stochastic interactions, learned transport, and multiple-observer networks remain directions for extending the analysis.Source: §7 · pp. 31–32
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