Qualitative Reachability for Open Interval Markov Chains

TL;DR

This paper investigates qualitative reachability in open interval Markov chains, providing polynomial-time algorithms that determine whether certain states can be reached with probability 0 or 1, even with open or half-open intervals.

Contribution

It introduces polynomial-time algorithms for qualitative reachability in open interval Markov chains without relying on interval closure, handling open and half-open intervals.

Findings

Algorithms work in polynomial time for both semantics.

Methods can approximate reachability probabilities arbitrarily closely.

New approach extends previous work by not requiring interval closure.

Abstract

Interval Markov chains extend classical Markov chains with the possibility to describe transition probabilities using intervals, rather than exact values. While the standard formulation of interval Markov chains features closed intervals, previous work has considered also open interval Markov chains, in which the intervals can also be open or half-open. In this paper we focus on qualitative reachability problems for open interval Markov chains, which consider whether the optimal (maximum or minimum) probability with which a certain set of states can be reached is equal to 0 or 1. We present polynomial-time algorithms for these problems for both of the standard semantics of interval Markov chains. Our methods do not rely on the closure of open intervals, in contrast to previous approaches for open interval Markov chains, and can characterise situations in which probability 0 or 1 can be attained not exactly but arbitrarily closely.