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On Skolem-Hardness and Saturation Points in Markov Decision Processes

Authors Jakob Piribauer , Christel Baier

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Author Details

Jakob Piribauer
  • Technische Universität Dresden, Germany
Christel Baier
  • Technische Universität Dresden, Germany

Funding

The authors are supported by the DFG through the DFG grant 389792660 as part of TRR 248, the Cluster of Excellence EXC 2050/1 (CeTI, project ID 390696704, as part of Germany’s Excellence Strategy), the Research Training Group QuantLA (GRK 1763), the Collaborative Research Centers CRC 912 (HAEC), the DFG-project BA-1679/11-1, and the DFG-project BA-1679/12-1.

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Jakob Piribauer and Christel Baier. On Skolem-Hardness and Saturation Points in Markov Decision Processes. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 138:1-138:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.138

Abstract

The Skolem problem and the related Positivity problem for linear recurrence sequences are outstanding number-theoretic problems whose decidability has been open for many decades. In this paper, the inherent mathematical difficulty of a series of optimization problems on Markov decision processes (MDPs) is shown by a reduction from the Positivity problem to the associated decision problems which establishes that the problems are also at least as hard as the Skolem problem as an immediate consequence. The optimization problems under consideration are two non-classical variants of the stochastic shortest path problem (SSPP) in terms of expected partial or conditional accumulated weights, the optimization of the conditional value-at-risk for accumulated weights, and two problems addressing the long-run satisfaction of path properties, namely the optimization of long-run probabilities of regular co-safety properties and the model-checking problem of the logic frequency-LTL. To prove the Positivity- and hence Skolem-hardness for the latter two problems, a new auxiliary path measure, called weighted long-run frequency, is introduced and the Positivity-hardness of the corresponding decision problem is shown as an intermediate step. For the partial and conditional SSPP on MDPs with non-negative weights and for the optimization of long-run probabilities of constrained reachability properties (aU b), solutions are known that rely on the identification of a bound on the accumulated weight or the number of consecutive visits to certain sates, called a saturation point, from which on optimal schedulers behave memorylessly. In this paper, it is shown that also the optimization of the conditional value-at-risk for the classical SSPP and of weighted long-run frequencies on MDPs with non-negative weights can be solved in pseudo-polynomial time exploiting the existence of a saturation point. As a consequence, one obtains the decidability of the qualitative model-checking problem of a frequency-LTL formula that is not included in the fragments with known solutions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Logic and verification
Keywords
  • Markov decision process
  • Skolem problem
  • stochastic shortest path
  • conditional expectation
  • conditional value-at-risk
  • model checking
  • frequency-LTL

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