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Temporal Logic Guided Universal Task Representations for Reinforcement Learning

Hao Zhang, Zhangli Zhou, Zhen Kan

arXiv:2608.15509v1cs.ROcs.FLcs.LG

TL;DR

Existing task representations struggle with efficiency, generalization, and interpretability across single- and multi-task settings. LOTUS addresses this with an LTL-based universal representation framework, whose experiments show consistent improvements over existing methods.

  • Problem

    Existing task representation methods often struggle with training efficiency, generalization performance, and interpretability across single-task and multi-task scenarios.

  • Method

    LOTUS uses an RGTN-based LTL encoder, reward-driven direct updates, and bisimulation-metric optimization for task representations.

  • Results

    LOTUS consistently outperforms existing methods in learning efficiency, generalization, and representation quality across diverse task scenarios.

  • Takeaways & Limitations

    LOTUS provides a task representation framework that integrates with existing RL algorithms across single-task and multi-task settings.

  • Takeaways & Limitations

    Because LOTUS relies on sc-LTL, it is limited to finite-horizon tasks and cannot learn infinite-horizon tasks such as patrol tasks.

Abstract

from arXiv · show

Task guided agents demonstrate strong performance in a wide range of complex tasks. However, most existing task representation algorithms are tailored to specific contexts and struggle to generalize across diverse scenarios. Moreover, they typically depend on gradient signals from reinforcement learning controllers to update their weights, which can degrade both representation quality and learning efficiency. To overcome these limitations, we propose LOTUS, a temporal logic inspired universal task representation framework that can be seamlessly integrated into any RL algorithm to enhance agent performance across diverse task settings. Specifically, we design a novel task representation architecture capable of modeling relationships and extracting task semantics from LTL formulas. We further introduce a more effective update mechanism that treats the LTL encoder as a policy, thereby improving representation capacity. To enhance stability and robustness, LOTUS leverages the bisimulation metric, which provides theoretical guarantees for LTL representation, including behavioral equivalence, optimality fidelity, and trajectory robustness. Experimental results show that LOTUS outperforms most existing methods in learning efficiency, generalization capability, and representation quality. Specifically, LOTUS accelerates convergence over 20% in single-task scenarios, achieves a 15%-45% higher success rate in unseen manipulation tasks, and improves generalization performance over 25% in complex multi-task environments with increased sub-goal depth or conjunctions. The corresponding code, videos, and appendix are available at: https://lotus-website.github.io/.

I. INTRODUCTION · II. PRELIMINARIES · A. sc-LTL and LTL Progression

The paper introduces LOTUS, a universal LTL-based task representation framework addressing limited generalization, efficiency, stability, and interpretability in reinforcement learning. It focuses on sc-LTL, whose progression updates remaining instructions from observed truth assignments.

  • I. INTRODUCTION: Existing task-guided methods often struggle with training efficiency, generalization performance, and interpretability in both single-task and multi-task scenarios.These limitations persist despite task relationship modeling and temporal abstraction approaches.
  • I. INTRODUCTION: Existing LTL encoders are specialized either for multi-task generalization or single-task learning, leaving no universal framework for diverse task scenarios.This motivates a representation approach that remains robust across different task settings.
  • I. INTRODUCTION: Indirect RL-based encoder updates can destabilize task representations and degrade representation quality and learning efficiency.LOTUS instead directly updates the LTL encoder through a tailored loss function and treats it as a policy using rewards.
  • I. INTRODUCTION: LOTUS targets complex robotic tasks by providing a universal task representation that integrates with any RL framework across single-task and multi-task settings.The framework is designed to improve sampling efficiency, performance, and generalization.
  • I. INTRODUCTION: LOTUS introduces the RGTN module to encode LTL instructions and extract task semantics through specialized attention mechanisms.The framework also includes direct updating, bisimulation-metric optimization, and theoretical guarantees.
  • I. INTRODUCTION: LOTUS experiments report improvements over baseline methods in learning efficiency, generalization, and representation quality.The reported evaluation covers multiple task scenarios, generalization tests, ablations, and real-world performance.
  • A. sc-LTL and LTL Progression: Co-safe LTL is a finite-horizon subclass of LTL suited to robotic instructions, using atomic propositions, Boolean operators, and temporal operators.This work focuses on sc-LTL to represent instructions such as approaching, grasping, and lifting objects.
  • A. sc-LTL and LTL Progression: LTL progression takes a formula and the current truth-assignment label as input, producing a formula that specifies the remaining instructions.For an atomic proposition p, progression returns True when p belongs to the current label and False otherwise.

B. Labeled MDP and Reinforcement Learning · C. Bisimulation Metrics in RL · D. Challenges and Problem Formulation

The paper formulates temporal-logic tasks as labeled MDPs, uses bisimulation metrics to compare state behavior, and identifies challenges in learning universal, effectively updated task representations. It addresses these challenges by augmenting the MDP with LTL progression and task-dependent rewards.

  • B. Labeled MDP and Reinforcement Learning: An sc-LTL task is modeled by a labeled MDP containing states, terminal states, actions, transitions, atomic propositions, labeling, rewards, discounting, and an initial-state distribution.The labeling function maps states to proposition sets and detects events relevant to determining whether an LTL specification is satisfied.
  • B. Labeled MDP and Reinforcement Learning: The robot follows a deterministic policy that maps each state to an action, receives the corresponding reward, and seeks an optimal policy maximizing expected discounted return.
  • C. Bisimulation Metrics in RL: Bisimulation abstracts high-dimensional state spaces by grouping states with identical behavior, while the bisimulation metric quantifies similarity through a p-th Wasserstein distance between transition distributions.
  • D. Challenges and Problem Formulation: The running example considers diverse task scenarios where one task representation module must guide the robot through different complex tasks specified by varying proposition sets.Possible solutions are shown as differently colored trajectories.
  • D. Challenges and Problem Formulation: The central challenges are designing an LTL encoder effective in single- and multi-task settings, updating representations without indirect RL-controller gradients, and improving learning efficiency.
  • D. Challenges and Problem Formulation: LOTUS aims to learn a universal task representation module that supports diverse scenarios while updating representations more effectively and stably through an augmented task-driven labeled MDP.The augmentation incorporates an LTL instruction and exploits LTL progression.
  • D. Challenges and Problem Formulation: The task-driven labeled MDP augments states with the progression closure of the LTL task and combines environmental and task-associated rewards to address non-Markovian rewards.The task-associated reward depends on whether the progression satisfies or falsifies the LTL task.

III. ALGORITHM DESIGN · A. Overview of LOTUS

LOTUS is a universal LTL task-representation framework designed to support efficient learning across single-task and multi-task scenarios while improving update stability. Its overview combines graph-based task-semantic modeling, relational and attention mechanisms, and direct encoder updates within exploration and training phases.

  • III. ALGORITHM DESIGN: LOTUS is introduced as a novel task-representation framework for LTL formulas that facilitates learning across diverse task scenarios and updates representations more effectively and stably.The paper presents the framework in Section III and develops its technical components in Sections III-B through III-F.
  • A. Overview of LOTUS: The task-representation module guides agents efficiently in both single-task and multi-task scenarios while improving learning efficiency through direct updates and stable representation performance.This design is intended to improve representation stability without relying solely on indirect learning effects.
  • A. Overview of LOTUS: LOTUS represents each LTL formula as a directed graph to capture dependencies among subtasks and expose task semantics across single-task and multi-task settings.The graph structure provides the basis for subsequent task-representation processing.
  • A. Overview of LOTUS: During training, LOTUS optimizes the policy using its own update method and updates the LTL encoder through a separate procedure.Algorithm 1 divides execution into exploration and training phases, with encoder updating occurring during training.
  • A. Overview of LOTUS: The RGTN task-representation module converts LTL formulas into directed graphs, models different edge-type relationships with relational GNNs, and integrates global information through multi-view attention.This architecture generates LTL representations intended to guide agents across diverse scenarios and differs from prior indirect gradient-based updates.
  • A. Overview of LOTUS: During exploration, LOTUS progresses the current LTL formula, converts it into a directed graph, encodes the graph, augments observations, guides interaction, and stores resulting transitions.These operations correspond to the exploration phase in Algorithm 1.
  • A. Overview of LOTUS: The subsequent algorithm sections describe graph construction from LTL instructions, RGTN design, direct encoder updating from environmental feedback, and bisimulation-metric use.These components provide the stated development path for extracting semantics, generating representations, updating the encoder, and addressing representation properties.

B. Convert LTL Specification to Directed Graph · C. Encode LTL Representation by RGTN

The method converts each LTL specification into a directed graph whose nodes and typed edges expose task structure, then encodes it with RGTN to capture relational and global task semantics. RGTN combines multi-relation and multi-head attention with normalization, residual connections, and feature fusion, while its update still depends on reinforcement-learning signals.

  • B. Convert LTL Specification to Directed Graph: LTL tasks are represented as learnable sequence-vector embeddings rather than sorted automaton-state indices, improving compatibility with reinforcement learning.The representation is intended to provide stronger inductive biases for generalization and more expressive capabilities.
  • B. Convert LTL Specification to Directed Graph: Each LTL formula φ becomes a directed graph Gφ = (V, E, R) with source and target nodes, typed edges, parent-operator links, and self-loops across time steps.Progressed subtasks connect to their parent operators through directed edges.
  • C. Encode LTL Representation by RGTN: RGTN is proposed as a parameterized LTL encoder to guide agents across single-task, multi-task, and sparse-reward settings.The section contrasts prior approaches: LTL2Action uses R-GCN for task-conditional multi-task policies but performs poorly in single-task settings, while T2TL uses a Transformer.
  • C. Encode LTL Representation by RGTN: RGTN stacks Relational Graph Transformer layers containing relational-attention, head-attention, and feedforward sub-layers, with layer normalization and residual connections.Layer normalization is applied before each sub-layer, and residual connections follow each block to stabilize gradient propagation.
  • C. Encode LTL Representation by RGTN: Multi-Relation Attention groups edge features by relation type and integrates them so the agent captures diverse task semantics from heterogeneous graph structure.The formulation derives relation-specific queries, keys, and values from the directed graph representation.
  • C. Encode LTL Representation by RGTN: Multi-head attention fuses features across heads to capture global context, while an additional projection integrates relation and head attention outputs into a higher-dimensional space.The added attention-fusion projection is denoted Wcat.
  • C. Encode LTL Representation by RGTN: RGTN’s multi-head attention and residual connections capture node-level subtasks for single-task performance, whereas multi-relation attention and directed graphs exploit edge structure for multi-task training efficiency.Despite these design benefits, the RGTN update mechanism still depends on reinforcement-learning signals.

D. The Direct Update of LTL Encoder · E. Optimizing LTL Encoder by Bisimulation Metric

LOTUS directly updates the LTL encoder by modeling it as a stochastic policy in a taskable MDP and optimizing it with policy gradients. It then uses a bisimulation metric and latent-space constraints to improve representation stability and robustness.

  • D. The Direct Update of LTL Encoder: LOTUS models the LTL encoder as a stochastic policy in a taskable MDP, allowing task representations to be sampled directly and optimized with policy gradients.This approach is called TLPG, whose first step models LTL Progression interaction at the task level.
  • D. The Direct Update of LTL Encoder: The task-level policy produces LTL representations that guide agent-environment interaction, while task-level rewards are obtained as the LTL specification progresses.The representation induces propositions through σ = L(s), linking environmental states to task-level progression.
  • D. The Direct Update of LTL Encoder: TLPG directly optimizes the LTL encoder with an on-policy loss after representing its interaction with LTL Progression as a task-level MDP.The encoder’s policy weights are updated rather than relying only on indirect gradients from the downstream RL controller.
  • D. The Direct Update of LTL Encoder: The encoder update combines task rewards with environmental rewards, supporting both sparse- and dense-reward settings and potentially accelerating optimization.Dense rewards from DexArt are described as enabling higher success rates.
  • E. Optimizing LTL Encoder by Bisimulation Metric: TLPG can suffer from sampling bias and reward fluctuations, motivating LOTUS to use a bisimulation metric for more stable and robust representation updates.Sampling noise may impair performance, while dense-reward environments can make sampled representations unstable.
  • E. Optimizing LTL Encoder by Bisimulation Metric: The bisimulation metric evaluates LTL-formula similarity through reward differences and transition-distribution differences, with smaller distances indicating greater similarity.A weighting coefficient balances the reward and transition contributions, while the max operator is difficult to compute in high-dimensional spaces.
  • E. Optimizing LTL Encoder by Bisimulation Metric: LOTUS simplifies bisimulation computation using the encoder policy and further constrains latent representations toward the desired formula-distance structure.The method enforces comparable latent-space scale and uses a Gaussian latent dynamics model with the 2-Wasserstein distance.
  • E. Optimizing LTL Encoder by Bisimulation Metric: During execution, LOTUS converts each LTL task into a directed graph, extracts task semantics with RGTN, and progressively shortens the task through atomic propositions and LTL Progression.The resulting representation is concatenated with observations for downstream RL control and reward collection.

F. Theoretical Guarantees of LTL Encoder

This section establishes three theoretical guarantees for LOTUS’s LTL representations under the bisimulation metric: behavioral equivalence, optimality fidelity, and trajectory robustness. Together, the results show that sufficiently accurate clustering and encoding preserve value functions and cumulative rewards.

  • Overview: LOTUS’s bisimulation metric provides guarantees of behavioral equivalence, optimality fidelity, and trajectory robustness for LTL representations.These are the three guarantees stated for the LTL encoder.
  • Behavioral Equivalence: If LOTUS’s policy continuously improves and converges to the optimal policy, the metric between LTL formulas has a least fixed point that is a π∗-bisimulation metric.The metric is defined through couplings of the corresponding transition distributions.
  • Optimality Fidelity: Clustering task states within an ϵ-neighborhood yields latent and original optimal value functions that are bounded under the same convergence assumptions.The encoder’s learning error L measures the difference between encoded distances and the bisimulation metric.
  • Trajectory Robustness: For an original trajectory and its encoded counterpart, LOTUS bounds the difference between their expected cumulative rewards.The bound depends on the trajectory horizon, discount factor, cluster radius ϵ, encoder error L, and the metric’s contraction parameter.
  • Trajectory Robustness: When the cluster radius ϵ and encoder error L are sufficiently small, the learned latent space does not change cumulative rewards for the same trajectory.This indicates that the representation retains essential information from the original state space.

IV. EXPERIMENTS

The experiments evaluate LOTUS against state-of-the-art algorithms in both simulation and real-world settings. The section considers multiple evaluation aspects.

  • LOTUS is evaluated against state-of-the-art algorithms.
  • The evaluation includes simulation settings.
  • The evaluation includes real-world settings.

1) Performance: … B. Main Performance in Diverse Task Scenarios

LOTUS is evaluated across diverse single-task and multi-task settings, with experiments examining task guidance, encoding capability, generalization, and efficiency. Results indicate that LTL-guided methods improve performance, while LOTUS further offers faster convergence and more stable training through its update mechanism and bisimulation metric.

  • 1) Performance:: The evaluation examines task guidance in single-task and multi-task scenarios, encoding capability, task generalization, and the efficiency of LOTUS’s roles in policy improvement.
  • 1) Environments:: The benchmark includes four dense-reward Robosuite tasks—Stack, Nut Assembly, Cleanup, and Peg Insertion—and sparse-reward LetterWorld and ZoneEnv environments.
  • 2) Baselines:: LOTUS is empirically compared with DFA, LSTM, and T2TL, representing product-MDP, sequence-vector, and Transformer-based LTL encoders, respectively.
  • B. Main Performance in Diverse Task Scenarios: Figure 6 reports discounted-return curves for four multi-task learning algorithms across LetterWorld and ZoneEnv.
  • B. Main Performance in Diverse Task Scenarios: Figure 7 evaluates normalized-reward curves in four manipulation tasks, where Transformer, HGT, RGTN, and LOTUS improve the No LTL baseline’s training efficiency.
  • B. Main Performance in Diverse Task Scenarios: Most LTL-guided methods outperform the algorithm without LTL representation, supporting the value of extracting task semantics from LTL.
  • B. Main Performance in Diverse Task Scenarios: LOTUS achieves faster convergence and more stable training across environments by combining TLPG with the bisimulation metric.
  • B. Main Performance in Diverse Task Scenarios: LOTUS accelerates task convergence over indirect encoder-update methods such as LTL2Action by removing dependence on RL-controller gradients, while the bisimulation metric mitigates representation instability.

C. Generalization Evaluation · D. Quantitative Evaluation · E. Visualization of Trajectories in ZoneEnv

LOTUS generalizes across increasingly difficult motion-planning and manipulation tasks, maintaining strong returns, stability, and success rates. Statistical tests and trajectory visualizations further show where LOTUS is advantageous, including significantly better manipulation performance and fewer steps across ZoneEnv scenarios.

  • C. Generalization Evaluation: LOTUS is evaluated on LetterEnv and ZoneEnv under i.i.d. tasks and harder settings with increased sequence depth and conjunctions.The evaluation covers partially ordered and avoidance tasks.
  • C. Generalization Evaluation: Across initial task distributions, LOTUS achieves higher returns and lower variance across all reported task scenarios.RGTN further supports the effectiveness of the task-representation architecture.
  • C. Generalization Evaluation: Graph-modeling methods show more robust performance as task depth and conjunctions increase, while cDFA declines more sharply than other algorithms.The passage attributes this robustness to graph neural networks for multi-task motion planning.
  • C. Generalization Evaluation: In most multi-task settings, LOTUS achieves the best returns with greater stability, while Transformer, DeepLTL, and other methods show scenario-dependent strengths.DeepLTL performs strongly across most scenarios, whereas Transformer generalizes more strongly in manipulation environments with fewer sub-goals and clearer regularity.
  • D. Quantitative Evaluation: Using p<0.05, LOTUS’s advantages are not significant in most LetterWorld and ZoneEnv scenarios, while RGTN has higher p-values in most cases.The passage links the limited significance to the greater challenge created by increased task depth and conjunctions.
  • D. Quantitative Evaluation: Using p<0.05, LOTUS significantly outperforms other algorithms in most scenarios across the four manipulation tasks.DeepLTL nonetheless achieves a p-value of 0.409 relative to LOTUS on the unseen Trashcan task.
  • E. Visualization of Trajectories in ZoneEnv: As sub-goal counts increase, LSTM-guided agents become ineffective at planning efficient solutions.The observation comes from trajectory visualizations for increasingly complex partially ordered tasks in ZoneEnv.
  • E. Visualization of Trajectories in ZoneEnv: LOTUS completes every visualized ZoneEnv task in fewer steps than all other methods, while HGT avoids failures and R-GCN often exhausts interaction steps in disjunctive scenarios.R-GCN failures occur in cases such as (Y ∨R) and (J ∨R) within φ3.

F. Ablation Study

The ablations show that LTL semantics, policy-based representation sampling, and bisimulation-based stabilization improve performance across scenarios. Architectural studies further identify Multi-Relation Attention, Multi-Head Attention, edge-specific designs, and concatenative fusion as important for representation quality and generalization.

  • Component Ablation: No LTL performs worst across all scenarios, while TLPG achieves a higher Faucet success rate than RGTN by sampling representations through a policy-modeled LTL encoder.LOTUS further improves representation stability during training through the bisimulation metric.
  • Architecture Ablation: Removing Multi-Relation Attention weakens training efficiency and Seen-set performance, while its edge-type modeling captures logical and temporal dependencies and improves Unseen-set generalization.The ablation attributes this benefit to modeling four edge types in LTL directed graphs.
  • Architecture Ablation: Removing Multi-Head Attention causes poor generalization across Unseen sets, whereas removing residual connections has a smaller effect but lowers Toilet Unseen success by 7.11% versus full RGTN.The MHA ablation supports its importance for task representation.
  • MRA Key Designs: Fixed edge-type parameters underperform on challenging Faucet tasks, while sharing edge-type information weakens edge-type features and reduces Faucet learning efficiency and generalization relative to RGTN.The findings support independently transmitted and learnable edge-type information in MRA.
  • MHA Key Designs: Attention across all nodes per head reduces challenging-task success by 10.45% and 15.93% relative to RGTN, while edge-based attention on adjacent nodes is important for both tasks.The all-node design performs well on Toilet but declines on Faucet.
  • Fusion Projection Module: RGTNfuse_concat outperforms direct-addition and learnable-weight fusion because concatenation preserves multilevel semantic information, expands expressive capacity, and improves adaptability in generalized scenarios.Direct addition can homogenize distinct features, while learnable weighting remains inferior without precise manual tuning.

G. Visualization of LTL Task Embedding Space

The study visualizes LTL task embeddings using t-SNE and pairwise distance metrics across six LetterEnv task types. RGTN produces clear, tightly grouped clusters by task structure, unlike the scattered and intermixed embeddings of R-GCN.

  • Visualization methodology: Six LetterEnv task types are evaluated using 2D t-SNE projections, pairwise cosine similarity, and Euclidean distance.The tasks vary by avoidance or partial ordering and by i.i.d., increased depth, or increased conjuncts.
  • Visualization results: RGTN clearly separates Avoidance and Partially-Ordered tasks into two nonoverlapping clusters in the t-SNE embedding space.Embeddings from the same task type are tightly grouped, indicating strong task-type clustering.
  • Visualization results: R-GCN produces scattered and intermingled embeddings, contrasting with RGTN’s structure-sensitive clustering.The comparison concerns task representations visualized in the two-dimensional t-SNE space.

H. Computational Cost and Memory Usage … V. CONCLUSIONS

The paper reports that LOTUS maintains computational efficiency, extends task evaluation to longer formulas, and demonstrates real-world applicability, while acknowledging finite-horizon limitations and outlining future directions. Its conclusion emphasizes RGTN, direct updates, and bisimulation-based guarantees as foundations for broad RL integration.

  • H. Computational Cost and Memory Usage: LOTUS does not significantly increase runtime compared with other algorithms because computation and update processes were optimized using DGL.Algorithms without LTL representation guidance consume the least time but have degraded performance.
  • I. Longer Task Evaluation: With the same number of sub-goals, RGTN achieves higher returns than cDFAs and LTL2Action on longer, deeply nested LTL formulas.The result demonstrates the superiority of RGTN’s architecture for encoding task semantics.
  • I. Longer Task Evaluation: At 35 sub-tasks, all methods struggle to complete the task, indicating that RGTN has reached its encoding limit.The paper suggests integrating RGTN with knowledge graphs to improve reasoning and encoding in more complex scenarios.
  • J. Real–World Experimental Results: LOTUS was evaluated on real-world Stack and Cleanup tasks using a Realsense D455 camera, AprilTag-labeled object poses, and a Franka Emika Panda robot.Snapshots are provided in Fig. 10, with a full experiment video available on the project website.
  • K. Limitations: LOTUS is limited by sc-LTL’s expressiveness to finite-horizon tasks and lacks ω-regular completeness compared with full LTL.This limitation prevents learning infinite-horizon tasks constructed with operators such as □.
  • V. CONCLUSIONS: LOTUS is a universal task representation framework integrable with existing RL algorithms, while RGTN encodes LTL instructions and extracts task semantics.The framework uses direct updates and the bisimulation metric to improve stability.
  • V. CONCLUSIONS: Bisimulation-based LTL representations provide guarantees of behavioral equivalence, optimality fidelity, and trajectory robustness.Extensive experiments show that LOTUS consistently outperforms existing methods.
  • V. CONCLUSIONS: Future work will investigate automata-based representations, LLM-level LTL encoders, and improved learning efficiency across diverse task settings.The first direction targets more complex scenarios such as infinite horizon LTL tasks.
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