# Disjunctive Reachability Queries in MDPs (Model-Checking Problem)

## Description

Given a model of a system and an objective, the model-checking problem asks whether the model satisfies the objective.

In this case, the model is a Markov Decision Process (MDP), and the objective is reachability: given a set of target vertices $T\subseteq V$, determine whether there is an infinite path that visits a vertex in $T$ at least once (i.e. you want to reach some vertex in $T$).

Furthermore, given $k$ reachability objectives, the disjunctive reachability query question asks whether there is a strategy for player 1 to ensure that one of the reachability objectives is satisfied with probability 1.

Disjunctive queries do not coincide with disjunctive objectives on MDPs.

## Related Problems

Generalizations: Reachability in MDPs

Related: Conjunctive Reachability Queries in MDPs, Safety in MDPs, Disjunctive Safety Queries in MDPs, Conjunctive Safety Queries in MDPs, Safety in Graphs, Disjunctive Queries of Safety in Graphs, Disjunctive coBüchi Objectives, Generalized Büchi Games

## Parameters

$n$: number of vertices

$m$: number of edges

$MEC$: O(\min(n^2, m^{1.5}))

## Table of Algorithms

Currently no algorithms in our database for the given problem.

## Reductions FROM Problem

Problem | Implication | Year | Citation | Reduction |
---|---|---|---|---|

Triangle Detection | assume: Strong Triangle then: there is no combinatorial $O(n^{3-\epsilon})$ or $O((k \cdot n^{2})^{1-\epsilon})$ algorithm for any $\epsilon > {0}$ for target. The bounds holf for dense MDPs with $m=\Theta(n^{2})$ |
2016 | https://dl.acm.org/doi/pdf/10.1145/2933575.2935304 | link |

OV | assume: Strong Triangle then: there is no $O(m^{2-\epsilon})$ or $O((k \cdot m)^{1-\epsilon})$ algorithm, for any $\epsilon > {0}$ for target. |
2016 | https://dl.acm.org/doi/pdf/10.1145/2933575.2935304 | link |