Reasoning-trained language models often spend more tokens on harder problems, but longer chains of thought do not show whether a model is merely computing for more steps or following a different internal trajectory. We study this distinction through hidden-state trajectories during chain-of-thought generation across competitive programming, mathematics, and Boolean satisfiability. Raw trajectory geometry is strongly shaped by generation length: longer generations mechanically alter path statistics, so difficulty-dependent comparisons are misleading without adjustment. After residualizing trajectory statistics on length, difficulty remains systematically coupled to corrected trajectory geometry across all domains studied. The clearest reasoning-specific separation appears in the code domain, where harder problems show more direct corrected trajectories and less heterogeneous local curvature in reasoning-trained models than in matched instruction-tuned baselines. Corrected difficulty-geometry coupling is weaker, but still present, in mathematics and Boolean satisfiability. Prompt-stage linear probes do not mirror the code-domain separation, and behavioral annotations show that stronger corrected coupling co-occurs with strategy shifts and uncertainty monitoring. Together, these findings establish length correction as a prerequisite for generation-time trajectory analysis and show that reasoning training can be associated with distinct corrected trajectory geometry, with the strength of the effect depending on the domain.

Numerical reasoning is vital for natural language processing models to understand and process numerical information in real-world scenarios. Most current methods first generate the Intermediate Meaning Representations (IMRs) of questions and then generate answers. Current SOTA methods generate programs as IMRs with large language models (LLMs). Intuitively, equations have fewer restrictions and closer semantics to the question than programs, leading to higher generation accuracy. However, current LLMs generate equations worse than programs, where we assume that the equation data is rare in pre-training data compared to programs. So in this paper, we try to use equations as IMRs to solve the numerical reasoning task by addressing two problems: (1) Theoretically, how to prove that the equation is an IMR with higher generation accuracy than programs; (2) Empirically, how to improve the generation accuracy of equations with LLMs. For the first problem, we propose and prove a proposition to theoretically compare the generation accuracy of different IMRs. For the second problem, we present a method called Boosting Numerical Reason\textbfing by Decomposing the Generation of Equations (Bridge), which can improve the accuracy of LLMs in generating equations as IMRs by reducing the tendency of generating constant expressions and programs. Our method improves the performance by 2.2%, 0.9%, and 1.7% on GSM8K, SVAMP, and Algebra datasets compared to the previous state-of-the-art methods under the single reasoning path setting. Our codes and prompts are released in https://github.com/zirui-HIT/Bridge_for_Numerical_Reasoning.

The ability of a system to meet its requirements is a strong determinant of success. Thus effective requirements specification is crucial. Explicit Requirements are well-defined needs for a system to execute. IMplicit Requirements (IMRs) are assumed needs that a system is expected to fulfill though not elicited during requirements gathering. Studies have shown that a major factor in the failure of software systems is the presence of unhandled IMRs. Since relevance of IMRs is important for efficient system functionality, there are methods developed to aid the identification and management of IMRs. In this paper, we emphasize that Common Sense Knowledge, in the field of Knowledge Representation in AI, would be useful to automatically identify and manage IMRs. This paper is aimed at identifying the sources of IMRs and also proposing an automated support tool for managing IMRs within an organizational context. Since this is found to be a present gap in practice, our work makes a contribution here. We propose a novel approach for identifying and managing IMRs based on combining three core technologies: common sense knowledge, text mining and ontology. We claim that discovery and handling of unknown and non-elicited requirements would reduce risks and costs in software development.