Language Models Need Inductive Biases to Count Inductively

Counting constitutes a core skill underlying a wide range of tasks, such as formal language recognition, multi-hop reasoning and simulating algorithms. Generaliz- ing counting inductively is central to task success on out-of-distribution (OOD) instances where testing inputs are longer than those seen in training. While there is a large body of literature reporting poor length generalization in language models, few papers have tried to distill the “reasoning” failure to the simplest case of count- ing failure. We aim to provide a broader picture on whether various language model architectures can a) learn to count, and b) generalize counting inductively. This work provides extensive empirical results on architectures ranging from RNNs, Transformers, State-Space Models and RWKV. We present carefully-designed task formats, auxiliary tasks and positional embeddings to avoid limitations in general- ization with OOD-position and OOD-vocabulary. We find that while traditional RNNs trivially achieve inductive counting, Transformers have to rely on positional embeddings (PEs) to count OOD. Further analyses on interpreting the learned solution reveal that different PEs encode different inductive biases that facilitate counting in different task formats. As counting is the basis for many arguments concerning the expressivity of Transformers, our finding calls for the community to reexamine the application scope of primitive functions defined in formal charac- terizations. Finally, modern RNNs also largely underperform traditional RNNs in generalizing counting inductively, hinting at the tradeoff modern RNNs struggle to balance between parallelized training and maintaining their recurrent nature.