Feedback Does Not Increase the Capacity of Approximately Memoryless Surjective POST Channels

We study a class of finite-state channels, known as POST channels, in which the previous channel output serves as the current state. A POST channel is deemed approximately memoryless when the state-dependent transition matrices are sufficiently close to one another. For this family of channels, under a surjectivity condition on the associated memoryless reference channel, we show that the feedback capacity coincides with the non-feedback capacity. This result extends Shannon’s classical theorem on discrete memoryless channels and demonstrates that the phenomenon holds well beyond the strictly memoryless case.