  State-space models (SSMs) have emerged as a potential alternative
architecture for building large language models (LLMs) compared to the
previously ubiquitous transformer architecture. One theoretical weakness of
transformers is that they cannot express certain kinds of sequential
computation and state tracking (Merrill & Sabharwal, 2023), which SSMs are
explicitly designed to address via their close architectural similarity to
recurrent neural networks (RNNs). But do SSMs truly have an advantage (over
transformers) in expressive power for state tracking? Surprisingly, the answer
is no. Our analysis reveals that the expressive power of SSMs is limited very
similarly to transformers: SSMs cannot express computation outside the
complexity class $\mathsf{TC}^0$. In particular, this means they cannot solve
simple state-tracking problems like permutation composition. It follows that
SSMs are provably unable to accurately track chess moves with certain notation,
evaluate code, or track entities in a long narrative. To supplement our formal
analysis, we report experiments showing that Mamba-style SSMs indeed struggle
with state tracking. Thus, despite its recurrent formulation, the "state" in an
SSM is an illusion: SSMs have similar expressiveness limitations to
non-recurrent models like transformers, which may fundamentally limit their
ability to solve real-world state-tracking problems.
