# On Skolem-hardness and saturation points in Markov decision processes

@article{Piribauer2020OnSA, title={On Skolem-hardness and saturation points in Markov decision processes}, author={Jakob Piribauer and Christel Baier}, journal={ArXiv}, year={2020}, volume={abs/2004.11441} }

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…

## 9 Citations

### On the Skolem Problem and the Skolem Conjecture

- MathematicsLICS
- 2022

It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences (LRS) over the integers, namely whether a given such sequence has a zero…

### Skolem Meets Schanuel

- MathematicsMFCS
- 2022

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding…

### The variance-penalized stochastic shortest path problem

- Computer ScienceICALP
- 2022

It is shown that the optimal VPE in MDPs with non-negative weights as well as an optimal deterministic finite-memory scheduler can be computed in exponential space and the threshold problem whether the maximal VPE exceeds a given rational is shown to be EXPTIME-hard and to lie in NEXPTIME.

### A Universal Skolem Set of Positive Lower Density

- Mathematics, Computer ScienceMFCS
- 2022

In this work, a Skolem set of positive lower density is constructed for the class of simple LRS, a set ofpositive integers relative to which theSkolem Problem is decidable.

### On Decidability of Time-bounded Reachability in CTMDPs

- MathematicsICALP
- 2020

It is shown that any unconditional decidability result would imply unconditionalDecidability of the bounded continuous Skolem problem, or equivalently, the problem of checking if an exponential polynomial has a non-tangential zero in a bounded interval.

### Universal Skolem Sets

- Mathematics, Computer Science2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021

The notion of a Universal Skolem Set is introduced: an infinite subset of the positive integers such that there is an effective procedure that inputs a linear recurrence sequence u and decides whether u(n) = 0 for some $n \in \mathcal{S}$.

### The Orbit Problem for Parametric Linear Dynamical Systems

- MathematicsCONCUR
- 2021

A parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems is studied, showing decidability in the case of one parameter and Skolem-hardness with two or more parameters, and a reduction from the well-knownSkolem problem for linear recurrence sequences is exhibited.

### Erratum to "Frequency Linear-time Temporal Logic"

- MathematicsArXiv
- 2020

This work corrects the proof of a theorem stating that satisfiability of frequency linear-time temporal logic is undecidable and states that it is correct.

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