States Hidden in Hidden States: LLMs Emerge Discrete State Representations Implicitly

Large Language Models (LLMs) exhibit various emergent abilities. Among these abilities, some might reveal the internal working mechanisms of models. In this paper, we uncover a novel emergent capability in models: the intrinsic ability to perform extended sequences of calculations without relying on chain-of-thought step-by-step solutions. Remarkably, the most advanced models can directly output the results of two-digit number additions with lengths extending up to 15 addends. We hypothesize that the model emerges Implicit Discrete State Representations (IDSRs) within its hidden states and performs symbolic calculations internally. To test this hypothesis, we design a sequence of experiments that look into the hidden states. Specifically, we first confirm that IDSRs exist. Then, we provide interesting observations about the formation of IDSRs from layer, digit, and sequence perspectives. Finally, we confirm that models indeed use IDSRs to produce the final answers. However, we also discover that these state representations are far from lossless in current open-sourced models, leading to inaccuracies in their final performance. Our work presents a novel exploration of LLMs' symbolic calculation abilities and the underlying mechanisms.