Target discounted-sum problem for arbitrary rational discount factors

Determine, for a finite integer alphabet, an arbitrary rational discount factor, and a rational target, whether there exists an infinite sequence whose discounted sum equals the target, thereby resolving the general target discounted-sum problem beyond the case in which the reciprocal of the discount factor is an integer.

Background

The target discounted-sum problem asks whether a sequence over a finite integer alphabet can have a prescribed discounted sum. The paper notes that the problem is connected to questions in automata theory, dynamical systems, and number representations. Existing results cited in the paper solve the case where the reciprocal of the discount factor is an integer, while the general rational-discount-factor case remains unresolved.

The paper solves a stochastic variant on finite Markov chains, but does not resolve the original non-stochastic existence problem for arbitrary rational discount factors. The unresolved problem is also the reason that unrestricted supremum optimization for the corresponding MDP objective remains open.

References

The target discounted-sum problem, which is currently open, asks, given $\lambda,\Sigma$ and a target $t$, whether there exists an infinite sequence over $\Sigma$ whose discounted sum is equal to $t$.

— Target Discounted Sum Problem on Markov Chains with Applications to Markov Decision Processes  (2609.03670 - Bertrand et al., 3 Sep 2026) in Abstract; Section 1, Introduction; Section 1, Related work

However, they leave open the punctual case, where the objective is a single target value.

— Target Discounted Sum Problem on Markov Chains with Applications to Markov Decision Processes  (2609.03670 - Bertrand et al., 3 Sep 2026) in Section 1, Related work