Grammars of Formal Uncertainty: When to Trust LLMs in Automated Reasoning Tasks (2505.20047v1)

Abstract: LLMs show remarkable promise for democratizing automated reasoning by generating formal specifications. However, a fundamental tension exists: LLMs are probabilistic, while formal verification demands deterministic guarantees. This paper addresses this epistemological gap by comprehensively investigating failure modes and uncertainty quantification (UQ) in LLM-generated formal artifacts. Our systematic evaluation of five frontier LLMs reveals Satisfiability Modulo Theories (SMT) based autoformalization's domain-specific impact on accuracy (from +34.8% on logical tasks to -44.5% on factual ones), with known UQ techniques like the entropy of token probabilities failing to identify these errors. We introduce a probabilistic context-free grammar (PCFG) framework to model LLM outputs, yielding a refined uncertainty taxonomy. We find uncertainty signals are task-dependent (e.g., grammar entropy for logic, AUROC>0.93). Finally, a lightweight fusion of these signals enables selective verification, drastically reducing errors (14-100%) with minimal abstention, transforming LLM-driven formalization into a reliable engineering discipline.

Evaluating LLMs: Addressing Uncertainty in Automated Formal Reasoning

The paper "Grammars of Formal Uncertainty: When to Trust LLMs in Automated Reasoning Tasks" undertakes a rigorous examination of the epistemological gap inherent in combining LLMs with formal verification systems. As the field of automated reasoning seeks to benefit from LLMs' capabilities in generating formal artifacts like code and proofs, this research systematically explores the uncertainty that arises when probabilistic LLMs are applied to deterministic formal methods.

Key Insights and Contributions

The paper systematically evaluates five frontier LLMs across four formal reasoning datasets, assessing the performance of LLM-generated Satisfiability Modulo Theories (SMT) programs in tasks ranging from logical reasoning to factual verification. The analysis highlights varied impacts of SMT-based autoformalization, improving accuracy by up to 34.8% on logic-centric tasks like ProofWriter, while decreasing it by 44.5% on datasets demanding factual computation. Results indicate that traditional uncertainty quantification methods, such as token probability entropy, fail to adequately predict formalization errors.

Leveraging probabilistic context-free grammars (PCFGs), the authors establish a refined framework for evaluating uncertainty in LLM outputs. This approach moves beyond simple probabilistic predictions, examining ensembles of SMT programs generated by LLMs and employing PCFGs to model their distributional characteristics. PCFG-derived metrics reveal task-dependent uncertainty signals—significantly strong predictors emerge for logical tasks (AUROC>0.93), underscoring the metric’s utility in guiding verification processes.

The paper's core contributions lie in:

1. Systematic Evaluation: Providing empirical evidence of LLM performance across diverse reasoning tasks, quantifying the inherent uncertainty and failure modes within formal verification artifacts.
2. PCFG Framework: Introducing a probabilistic method that bridges neural outputs with formal logic, facilitating a deeper understanding of classical epistemic and aleatoric distinctions.
3. Uncertainty Metrics: Developing 25 dedicated metrics, outlining a nuanced taxonomy indicative of neural-symbolic cognitive structures, and deploying a fusion approach that substantially reduces error rates in automated formalizations.

Implications and Future Directions

The findings in this paper have significant implications for both theoretical and practical aspects of AI and formal verification. Practically, the demonstrated reduction in errors through selective verification marks a path towards reliable, LLM-driven engineering processes. Theoretically, the insights into task-specific uncertainty offer avenues to refine neurosymbolic structures and improve formal reasoning capabilities. These highlight the need for architectures sensitive to modality distinctions and task-dependent reliability.

Future research might further explore the alignment of formal and textual reasoning pathways, potentially through joint model training. This reflects the ongoing challenge of reconciling LLMs’ generative strengths with formal methods’ deterministic requirements, a theme underscored by this work's emphasis on bridging probabilistic and deterministic paradigms through structured model approaches like PCFGs.

In conclusion, this paper advances the discourse on leveraging LLMs for automated reasoning, providing a clearer understanding of their capabilities and limitations in formal environments. The proposed framework and metrics represent a substantial contribution towards integrating probabilistic AI systems with deterministic verification procedures, serving as a roadmap for future research and application in the domain of AI-driven reasoning and verification.