Target Discounted Sum Problem on Markov Chains with Applications to Markov Decision Processes

The discounted sum is a way to aggregate a sequence of weights from a finite alphabet $Σ$, i.e., for a discount factor $λ$, the discounted sum of a sequence $w_0 w_1 w_2 \cdots$ over $Σ$ is $\sum_{i \in \mathbb{N}} w_i λ^i$. The target discounted-sum problem, which is currently open, asks, given $λ,