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原文摘要

Recurrent and state-space language models compress the past into a state of fixed size. The state must answer questions about the future now, and it must remain useful as new tokens arrive, including tokens the model samples itself. We show that the second requirement can cost far more memory than the first. For a known hidden Markov source with positive transition probabilities, a state of $O(\log(1/\varepsilon))$ bits answers every question about the future to accuracy $\varepsilon$, however far ahead it looks, because the Bayesian filter forgets its rounding errors. Generating the future is different. When a model feeds each sampled token back into its state, the small errors in its next-token distributions add up, and $T$ tokens can require a state of order $\log(T/\varepsilon)$ bits. The lower bound comes from the KL chain rule: at every step, a model with $K$ states pays for approximating the true next-token distribution by one of only $K$ distributions. The argument applies to any source, including a language model continuing a prompt, and a small-ball estimate makes it tight for two-state sources. The extra $\log T$ disappears if the model keeps private randomness, or if error is budgeted per token, as in perplexity. Statistics of the recent past show the same gap between describing and maintaining. The number of ones among the last $n$ bits takes $\log n$ bits to write down. But if each bit is a noisy copy of the one $n$ steps earlier, tracking that count to a mean-square error $E^2$ well below its variance costs order $n/(1+E^2)+\log n$ bits. Information about each bit must be kept until the bit leaves the window, and a queue holding the counts of recent blocks keeps just enough of it.