CriticNL2LTL Module for LTL Constraint Refinement

• CriticNL2LTL is a module that translates, validates, and refines natural language into LTL constraints for both embodied agent planning and cognitive modeling.
• It integrates LLM reasoning with formal verification methods to generate safe, efficient symbolic policies that reduce failures and enhance task completion.
• The module employs iterative actor–critic loops and unsupervised refinement to ensure high accuracy in constraint synthesis and dynamic system integration.

The CriticNL2LTL module is a formal component in language-model-driven systems designed to translate, validate, and refine Linear Temporal Logic (LTL) constraints derived from natural language. It appears in multiple recent frameworks—for LLM-based embodied agents as well as for interpretable cognitive agent construction. At its core, CriticNL2LTL synthesizes the reasoning capabilities of LLMs with the rigor of temporal logic to yield actionable, safe, and efficient symbolic policies over discrete decision domains such as high-level task planning and cognitive modeling (Gokhale et al., 4 Jul 2025, Deng et al., 20 Dec 2025).

1. Concept and Role within LLM-to-LTL Architectures

CriticNL2LTL is the critic component within actor–critic and unsupervised pipeline architectures that operate over sequences of (state, action) pairs and natural language rules. Its primary role is to induct, verify, and adapt LTL constraints that shape agent behavior according to specified safety, efficiency, or domain-alignment objectives.

• In embodied agent planning (Gokhale et al., 4 Jul 2025), CriticNL2LTL processes full trajectories generated by an LLM-based actor. It analyzes these trajectories, identifies failures or inefficiencies, and proposes new or refined LTL constraints. These constraints are immediately integrated into a symbolic verifier (e.g., a Büchi automaton), closing the loop for safe and efficient future planning.
• In cognitive agent auto-formalization (Deng et al., 20 Dec 2025), CriticNL2LTL appears as an unsupervised logic revisor within a tree-of-LLMs structure. Given LTL formulas output by a base LLM from rich natural-language cognitive rules, the module validates and revises the logical form until all criteria for correctness and domain relevance are met, entirely without human-in-the-loop.

2. Formal Definitions and Theoretical Foundation

Let $S$ denote the set of abstract state descriptions (encoded as Boolean atoms), $A$ as the action set (one-hot encoded), and $\tau = (s_0, a_0, \ldots, s_T)$ a trajectory. The CriticNL2LTL module operates over:

• LTL Formulas: $AP = \{p_1, \ldots, p_n\}$; formulas constructed with $$\psi ::= \top \mid p_i \mid \neg \psi \mid \psi \land \psi \mid X\psi \mid F\psi \mid G\psi \mid \psi_1 U \psi_2$$.
• Target Formula Pattern: Only rules of the form $G(\varphi_s \Rightarrow X(\varphi_a))$ are induced: $\varphi_s$ is a Boolean over state atoms, $\varphi_a$ over actions.
• Correction Criteria: In cognitive domains, the module ensures logical fidelity (semantic match to input), syntactic well-formedness, and that atomic propositions align with a domain knowledge base.

CriticNL2LTL supports incremental, model-agnostic integration, allowing any LLM to play actor/reviser/critic roles subject to external enforcement via dynamic LTL constraint sets.

3. Algorithmic Workflow and Core Procedures

3.1 Actor–Critic Planning Loop (Gokhale et al., 4 Jul 2025)

• Online actor loop: At each state, an actor LLM observes $s_t$ (expressed in NL), proposes $a_t$, and receives instant feedback from an LTL verifier regarding constraint satisfaction.
• Offline CriticNL2LTL loop: Periodically, entire trajectories are parsed with
  • Trajectory parsing: Map $(s_t, a_t, \text{success})$ tuples to boolean feature and action encodings.
  • NL summarization: Encode the trajectory as natural-language bullets, augment with current LTL rule text.
  • Constraint proposal: Prompt the LLM critic with fixed templates and few-shot examples for new constraints on witnessed failures or inefficiencies.
  • Filtering: Accept only candidate rules that are traceable (witnessed antecedents in the data) and non-blocking (do not eliminate all valid actions in any observed state).
  • Verifier update: Add approved rules, regenerate Büchi automaton.

3.2 Unsupervised Critic Tree Refinement (Deng et al., 20 Dec 2025)

• Initialization: Produce $A_\text{init} = f_\theta(T)$, with $T$ as the NL rule and $f_\theta$ a fine-tuned LLM translator.
• Tree-based refinement: Using a revisor LLM and $\delta$ critics per depth, tree nodes $(A_v, C_v)$ (formula, context) are iteratively refined.
  • Each critic $k$ produces either "approve" or feedback string $r_k$.
  • If all critics approve a candidate, it is final; otherwise, feedback is appended to the context, and the revisor LLM generates a new candidate.
• Approval rule: $$Score(T, A) = (1/\delta) \sum_{k=1}^\delta s_k(T, A)$$, where $s_k(T, A) = 1$ for approval, $0$ otherwise.

4. Input Representation and Prompt Engineering

CriticNL2LTL's effectiveness relies on task-specific or domain-agnostic input representations and tailored LLM prompting:

• Trajectory representation: Chronological, atomized NL bullet lists (e.g., observed states, actions, and results).
• Constraint list: Human-annotated or previously auto-induced LTL rules, each with a plain-language rationale.
• LLM prompts: Fixed templates instructing the critic to detect state-action missteps or inefficiencies and propose precisely formatted $G(\varphi_s \Rightarrow X(\neg \mathrm{action}))$ rules; must preserve traceability and non-blockingness.

5. Integration and Iterative Refinement Logic

The CriticNL2LTL module is engineered for seamless integration and continual looped refinement:

• Constraint synthesis: Induced constraints are both permanent (safety) and adaptive (efficiency). Hand-specified and learned rules are consolidated and used to rebuild the symbolic verifier at each iteration.
• Non-blocking checks: For every unique observed state, ensure at least one action remains permitted after updating constraints to prevent deadlock.
• Termination: Iterative refinement continues until no new constraints are generated and empirical agent performance converges.

6. Empirical Outcomes and Evaluation

6.1 LTLCrit (Embodied Agents) (Gokhale et al., 4 Jul 2025)

• Safety and efficiency: Adding CriticNL2LTL to standard LLM planners (e.g., SayCan, InnerMonologue) yields 100% completion on the Minecraft diamond-mining benchmark, reduces average primitive steps per episode (e.g., from 12 to 9 for wooden tools), and sharply lowers failed unsafe actions (from $\sim23\%$ to $4.5\%$).

<table> <tr> <th>Method</th> <th>Completion (Diamond)</th> <th>Failed Actions (%)</th> </tr> <tr> <td>SayCan (no LTL)</td> <td\>0/5</td> <td>—</td> </tr> <tr> <td>SayCan + LTLCrit</td> <td\>5/5</td> <td>—</td> </tr> <tr> <td>InnerMonologue (no LTL)</td> <td\>4/5</td> <td\>23</td> </tr> <tr> <td>InnerMonologue + LTLCrit</td> <td\>5/5</td> <td\>4.5</td> </tr> </table>

• Ablation: Without learned efficiency constraints, completion rates drop slightly but primitive action usage rises by 10–15%. Excluding hand-authored safety rules reduces safety to $\sim$70%.

6.2 NL2CA (Cognitive Modeling) (Deng et al., 20 Dec 2025)

• Translation accuracy: On expert benchmarks, CriticNL2LTL achieves 80.6% accuracy and BLEU 0.517, substantially outperforming end-to-end monolithic LLMs and self-refinement methods.
• Scalability: Maintains high accuracy (90.2%) on synthetic benchmarks with zero human-in-the-loop.

Benchmark	Fine-tuned LLM	Self-Refine	CriticNL2LTL	Human Interactive
Dataset 1 ACC*	22.2%	63.9%	80.6%	94.4%
Dataset 1 BLEU	0.246	0.490–0.546	0.517	0.911
Dataset 2 ACC	97.5%	—	90.2%	78.3%
Dataset 2 BLEU	0.993	—	0.945	0.978

• Computation: For $\delta=2$, $D=2$, the worst-case cost is seven LLM calls per rule. Module is model-agnostic and parameterized to tune quality/cost.

7. Practical Considerations and Limitations

• Modularity: CriticNL2LTL is agnostic to the choice of LLMs for both actor and critic roles, as well as downstream integration points (Büchi automata for planning; symbolic production rule engines for cognition).
• Zero human-in-the-loop: In NL2CA, CriticNL2LTL operates unsupervised, contrasting with interactive prompt strategies.
• Operator support: Rules outside supported fragments (such as $F$, $U$ in some system instantiations) may trigger maskable inference errors during downstream translation or enforcement.
• Coverage: The synthesized rules are constrained by observable state distributions; rare or latent hazards may require explicit domain knowledge injection or enhanced exploration.
• Extensibility: Adjusting internal schemas or prompt libraries allows extension to new domains and operator vocabularies.

• "LTLCrit: A Temporal Logic-based LLM Critic for Safe and Efficient Embodied Agents" (Gokhale et al., 4 Jul 2025)
• "NL2CA: Auto-formalizing Cognitive Decision-Making from Natural Language Using an Unsupervised CriticNL2LTL Framework" (Deng et al., 20 Dec 2025)