Existing work on value alignment typically characterizes value relations statically, ignoring how alignment interventions—such as prompting, fine-tuning, or preference optimization—reshape the broader value system. In practice, aligning a target value can implicitly shift other values, creating value trade-offs that remain largely unmeasured.We introduce the VAT, a framework that quantifies value trade-offs by measuring how alignment-induced changes propagate across interconnected values relative to achieved on-target gain. VAT captures the system-level dynamics of value expression under alignment intervention, enabling evaluation of both intended improvements and unintended side effects.Using a controlled scenario–action dataset grounded in Schwartz value theory, we collect paired pre–post normative judgments and analyze alignment effects across models, values, and interventions. Results show that alignment often produces uneven and structured co-movement among values, revealing systematic trade-offs between target and non-target values. These effects are largely invisible under conventional target-only evaluation, but become evident via VAT, highlighting process-level alignment risks and offering new insights into the dynamic nature of value alignment in LLMs.Dataset and code are open-sourced.

We present Prolog-MATH, a curated corpus designed to support mathematical reasoning in large language models (LLMs) through logic programming. Each verbal math problem in the dataset is paired with a chain-of-thought explanation to generate Prolog program via a two-stage automated pipeline. In the first stage, an LLM (e.g., Deepseek-V3) predicts a set of relevant mathematical predicates that could be useful in solving the problem. In the second stage, the LLM uses these suggested predicates along with the expected answer type to gen- erate a complete Prolog program. To improve coverage, we fine-tune an open-source LLM us- ing supervised fine-tuning, followed by GRPO (Group Relative Policy Optimization) training to address problems that Deepseek-V3 fails to solve. To support this training, we propose a predicate-aware reward function that evaluates how well the generated solution incorporates the suggested predicates, complementing the standard binary reward. Experimental results show that: 1) Our two-stage pipeline achieves 81.3% solution coverage on the MATH training set; 2) GRPO training with the predicate-aware reward function enables a series of base models to correctly solve additional problems missed by Deepseek-V3, further increasing solution coverage to 97.4%. Data and source code can be obtained at the Github repository.