We study planning for finite-horizon partially observable Markov decision processes (POMDPs) with finite state, action, and observation spaces. Both exact and approximate planning are known to be computationally hard in this setting, so prior work has explored natural assumptions that make planning easier. Recent work [Golowich et al., 2022] gave quasipolynomial-time approximate planning algorithms for observable POMDPs. These are POMDPs whose observation channels satisfy an $\ell_1$-observability condition, but notably, no assumptions are required on the transition dynamics. In a similar vein, for every fixed $ρ>0$, we give quasipolynomial-time approximate planning algorithms for $ρ$-decodable POMDPs. We show that decodability is implied by both $\ell_1$- and $\ell_2$-observability and is strictly weaker than observability. Moreover, we show that the optimal decodability constant can be computed in polynomial time, whereas computing the optimal $\ell_1$-observability constant is NP-hard in the worst case. We also show that the fast filter-forgetting guarantee underlying the approach of [Golowich et al., 2022] does not extend directly from $\ell_1$-observability to constant $\ell_2$-observability. Thus, decodability is an easy-to-check, unifying assumption under which quasipolynomial-time approximate planning for POMDPs is possible.

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