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

Computer Science > Logic in Computer Science [Submitted on 3 Sep 2026] Title:Target Discounted Sum Problem on Markov Chains with Applications to Markov Decision Processes View PDF HTML (experimental)Abstract:The discounted sum is a way to aggregate a sequence of weights from a finite alphabet $\Sigma$, i.e., for a discount factor $\lambda$, the discounted sum of a sequence $w_0 w_1 w_2 \cdots$ over $\Sigma$ is $\sum_{i \in \mathbb{N}} w_i \lambda^i$. 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 $t$. We study and solve a probabilistic variant of this problem, i.e., the target discounted-sum problem on Markov chains. To do this, we prove that the event consisting of paths whose discounted sum is equal to the target and has infinitely many distinct suffix sums has probability zero. This structural property allows us to solve the target discounted-sum problem on Markov chains using an automata-theoretic technique. We apply our technical results to Markov decision processes with target discounted-sum objectives: we show that the infimum value and the finite-memory supremum value are computable in pseudo-polynomial time and are attained by deterministic finite-memory strategies. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.