Reasoning: From Reflection to Solution

What is reasoning? This question has driven centuries of philosophical inquiry, from Aristotle's syllogisms to modern computational complexity theory. In the age of large language models achieving superhuman performance on benchmarks like GSM8K (95\% accuracy) and HumanEval (90\% pass@1), we must ask: have these systems learned to \emph{reason}, or have they learned to \emph{pattern-match over reasoning traces}? This paper argues for a specific answer: \textbf{reasoning is iterative operator application in state spaces, converging to fixed points}. This definition is not merely philosophical -- it has concrete architectural implications that explain both the failures of current systems and the path to genuine reasoning capabilities. Our investigation begins with a puzzle (OpenXOR), progresses through theory (OpenOperator), and culminates in a working solution (OpenLM) that achieves 76\% accuracy where state-of-the-art LLMs achieve 0\%. This is not about criticizing existing systems, but about \emph{understanding what reasoning requires} and \emph{building architectures that provide it}.

Multi-Preference Lambda-weighted Listwise DPO for Dynamic Preference Alignment

While large-scale unsupervised language models (LMs) capture broad world knowledge and reasoning capabilities, steering their behavior toward desired objectives remains challenging due to the lack of explicit supervision. Existing alignment techniques, such as reinforcement learning from human feedback (RLHF), rely on training a reward model and performing reinforcement learning to align with human preferences. However, RLHF is often computationally intensive, unstable, and sensitive to hyperparameters. To address these limitations, Direct Preference Optimization (DPO) was introduced as a lightweight and stable alternative, enabling direct alignment of language models with pairwise preference data via classification loss. However, DPO and its extensions generally assume a single static preference distribution, limiting flexibility in multi-objective or dynamic alignment settings. In this paper, we propose a novel framework: Multi-Preference Lambda-weighted Listwise DPO, which extends DPO to incorporate multiple human preference dimensions (e.g., helpfulness, harmlessness, informativeness) and enables dynamic interpolation through a controllable simplex-weighted formulation. Our method supports both listwise preference feedback and flexible alignment across varying user intents without re-training. Empirical and theoretical analysis demonstrates that our method is as effective as traditional DPO on static objectives while offering greater generality and adaptability for real-world deployment.