Logical Phase Transitions: Understanding Collapse in LLM Logical Reasoning

TL;DR

The paper identifies a collapse in LLM logical reasoning at high logical depth, not a smooth degradation, and formalizes this with the Logical Complexity Metric (LoCM) and the discovery of Logical Phase Transitions (LPT). It then introduces a Neuro-Symbolic Curriculum Tuning framework that aligns NL and FOL representations and schedules training across complexity regimes, aiming to stabilize reasoning near phase-transition boundaries. A Neuro-Symbolic Alignment Dataset for Logical Reasoning (NSA-LR) provides paired NL/FOL data to support fine-grained analysis. Across five benchmarks and varying prompting strategies, the approach yields consistent robustness gains and better generalization to unseen logical structures, with code and data released for reproducibility.

Abstract

Symbolic logical reasoning is a critical yet underexplored capability of large language models (LLMs), providing reliable and verifiable decision-making in high-stakes domains such as mathematical reasoning and legal judgment. In this study, we present a systematic analysis of logical reasoning under controlled increases in logical complexity, and reveal a previously unrecognized phenomenon, which we term Logical Phase Transitions: rather than degrading smoothly, logical reasoning performance remains stable within a regime but collapses abruptly beyond a critical logical depth, mirroring physical phase transitions such as water freezing beyond a critical temperature threshold. Building on this insight, we propose Neuro-Symbolic Curriculum Tuning, a principled framework that adaptively aligns natural language with logical symbols to establish a shared representation, and reshapes training dynamics around phase-transition boundaries to progressively strengthen reasoning at increasing logical depths. Experiments on five benchmarks show that our approach effectively mitigates logical reasoning collapse at high complexity, yielding average accuracy gains of +1.26 in naive prompting and +3.95 in CoT, while improving generalization to unseen logical compositions. Code and data are available at https://github.com/AI4SS/Logical-Phase-Transitions.

Figures (12)

• Figure 1: Overview of our work. (Left) Physical phase transitions: as temperature increases, matter undergoes abrupt state changes (solid $\rightarrow$ liquid at $0^\circ$C, liquid $\rightarrow$ gas at $100^\circ$C). (Right) Logical phase transitions: as the Logical Complexity Metric (LoCM) increases, LLM reasoning accuracy drops sharply, revealing a phase-transition-like behavior in logical reasoning.
• Figure 2: Overview.Logical Complexity Measuring (left): We construct a Neuro-Symbolic alignment dataset (NSA-LR) and define the Logical Complexity Metric (LoCM), which quantifies reasoning difficulty via logical operators, nesting depth $d$, premise count $N_{\phi}$, and reasoning hops $H$. Logical Phase Transitions Discovery (center): LoCM reveals a phase transition where accuracy collapses to random guessing beyond a critical complexity threshold and partitions samples into Easy, Medium, and Hard pools. Neuro-Symbolic Curriculum Tuning (right): Stage 1 trains a mixed-semantics model $\theta_{\text{MIX}}$ and stage 2 applies curriculum optimization over the experience pool to obtain the final model $\theta^{*}$, mitigating reasoning collapse at high LoCM.
• Figure 3: Logical phase-transition curves across different models. Shaded regions denote the identified transition intervals, where accuracy drops sharply as LoCM increases. The dashed line marks the $1/3$ baseline corresponding to random guessing. Complete numerical results are provided in the Appendix \ref{['app:full_results']}.
• Figure 4: Ablation studies of the proposed multi-stage training framework. (Left: Q1) Comparison of curriculum learning and representation mixing. (Middle: Q2) Sensitivity analysis with respect to the mixing coefficient $\lambda$. (Right: Q3) Performance under different logical complexity regimes.
• Figure 5: Effect of prompting strategies on logical phase transition (LPT) curves. Left: Base model (Qwen2.5-7B, without fine-tuning) under Direct and CoT prompting. Right: Comparison between the base model and the finetuned model under Direct and CoT prompting.

Theorems & Definitions (1)

• Definition 1: LoCM