How Do Transformers Learn Variable Binding in Symbolic Programs?

Yiwei Wu; Atticus Geiger; Raphaël Millière

Home/Archive/arXiv

arXiv · 2505.20896

How Do Transformers Learn Variable Binding in Symbolic Programs?

Yiwei Wu·Atticus Geiger·Raphaël Millière

View original source ↗Read source document ↗Export metadata ↓

Abstract

Variable binding -- the ability to associate variables with values -- is fundamental to symbolic computation and cognition. Although classical architectures typically implement variable binding via addressable memory, it is not well understood how modern neural networks lacking built-in binding operations may acquire this capacity. We investigate this by training a Transformer to dereference queried variables in symbolic programs where variables are assigned either numerical constants or other variables. Each program requires following chains of variable assignments up to four steps deep to find the queried value, and also contains irrelevant chains of assignments acting as distractors. Our analysis reveals a developmental trajectory with three distinct phases during training: (1) random prediction of numerical constants, (2) a shallow heuristic prioritizing early variable assignments, and (3) the emergence of a systematic mechanism for dereferencing assignment chains. Using causal interventions, we find that the model learns to exploit the residual stream as an addressable memory space, with specialized attention heads routing information across token positions. This mechanism allows the model to dynamically track variable bindings across layers, resulting in accurate dereferencing. Our results show how Transformer models can learn to implement systematic variable binding without explicit architectural support, bridging connectionist and symbolic approaches. To facilitate reproducible research, we developed Variable Scope, an interactive web platform for exploring our findings at https://variablescope.org

Explore related subjects

cs.LG cs.AI cs.CL

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yiwei Wu, Atticus Geiger, Raphaël Millière. 2025-05-30. How Do Transformers Learn Variable Binding in Symbolic Programs?. https://arxiv.org/abs/2505.20896

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rollout Total Correlation for Deep Reinforcement Learning

Learning task-relevant representations is crucial for reinforcement learning. Recent approaches aim to learn such representations by improving the temporal consistency in the observed transitions. However, they only consider individual transitions and can fail to achieve long-term consistency. Instead, we argue that capturing aspects of the state that correlate with other states and actions of the trajectory---even more distant in the future---could further help in extracting task-relevant information. Hence, in this paper we investigate how to learn representations by maximizing the rollout total correlation, the correlation among all learned representations and actions within the trajectories produced by the agent. For improving rollout total correlation, we propose to combine two complementary lower bounds based on a generative and a discriminative model, combined with a simple and effective technique of chunk-wise mini-batching. Furthermore, we propose an intrinsic reward based on the learned representation for better exploration. Experimental evaluations on a set of challenging image-based simulated control tasks show that our method achieves better sample efficiency, and robustness to both white noise and natural video backgrounds compared to leading baselines.

cs.LG↗

arXiv2026

Reward Shaping to Mitigate Reward Hacking in RLHF

Reinforcement learning from human feedback (RLHF) is widely used to align large language models (LLMs) with human preferences. However, RLHF remains vulnerable to \emph{reward hacking}, whereby a policy exploits imperfections in the reward function instead of learning the intended behavior, thereby undermining alignment. Although reward shaping can stabilize RLHF training and partially mitigate reward hacking, shaping methods and their underlying design principles have not been systematically investigated. To address this gap, we conduct a comprehensive study of prevalent reward-shaping techniques. Our analysis identifies two key design principles: (1) the reinforcement-learning reward should be bounded, and (2) it should grow rapidly at first and then gradually saturate. Motivated by these principles, we propose Preference as Reward (PAR), a novel method that uses the latent preferences encoded in the reward model as the reinforcement-learning signal. We further show that PAR possesses two variance-reduction properties that stabilize RLHF training and substantially widen the practical window for early stopping. Our evaluation consists of two parts. First, we compare PAR with several other reward-shaping strategies using Proximal Policy Optimization (PPO) as the reinforcement-learning algorithm and Gemma2-2B as the base model. Second, we compare PAR with the vanilla baseline (i.e., unshaped reward) across four base models and four reinforcement-learning algorithms. In the first set of experiments, PAR consistently outperforms other reward-shaping methods and also reflects high data efficiency and robustness. The second set of experiments shows that PAR is particularly effective for actor-critic RL algorithms when value estimates become unstable and demonstrates its effectiveness across different base models. The code is available at https://github.com/PorUna-byte/PAR.

cs.LG↗

arXiv2026

Trajectory Entropy Reinforcement Learning for Robust Robot Motor Skill Learning

Simplicity is a critical inductive bias for designing data-driven controllers, especially when robustness is important. Despite the impressive results of deep reinforcement learning in complex control tasks, it is prone to capturing intricate and spurious correlations between observations and actions, leading to failure under slight perturbations to the environment. To tackle this problem, in this work we introduce a novel inductive bias towards simple policies in reinforcement learning. The simplicity inductive bias is introduced by minimizing the entropy of entire action trajectories, corresponding to the number of bits required to describe information in action trajectories after the agent observes state trajectories. Our reinforcement learning agent, Trajectory Entropy Reinforcement Learning, is optimized to minimize the trajectory entropy while maximizing rewards. We show that the trajectory entropy can be effectively estimated by learning a variational parameterized action prediction model, and use the prediction model to construct an information-regularized reward function. Furthermore, we construct a practical algorithm that enables the joint optimization of models, including the policy and the prediction model. Experimental evaluations on several high-dimensional locomotion tasks show that our learned policies produce more cyclical and consistent action trajectories, and achieve superior performance, and robustness to noise and dynamic changes than the state-of-the-art.

cs.LG↗