Decisive Markov Chains

TL;DR

This paper investigates the properties of decisive infinite-state Markov chains, establishing conditions for decidability of safety and liveness problems, and analyzing the limitations of expressing certain probabilities within Tarski-algebra.

Contribution

It introduces the concept of decisiveness for infinite Markov chains, characterizes classes where decisiveness holds, and analyzes the decidability and computational aspects of safety and liveness problems.

Findings

Decisiveness holds for Markov chains with finite attractors and globally coarse chains.

Path enumeration algorithms terminate for decisive chains, enabling approximate safety and liveness analysis.

Exact probabilities of repeated reachability cannot be uniformly expressed in Tarski-algebra for studied classes.

Abstract

We consider qualitative and quantitative verification problems for infinite-state Markov chains. We call a Markov chain decisive w.r.t. a given set of target states F if it almost certainly eventually reaches either F or a state from which F can no longer be reached. While all finite Markov chains are trivially decisive (for every set F), this also holds for many classes of infinite Markov chains. Infinite Markov chains which contain a finite attractor are decisive w.r.t. every set F. In particular, this holds for probabilistic lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains are decisive. This class includes probabilistic vector addition systems (PVASS) and probabilistic noisy Turing machines (PNTM). We consider both safety and liveness problems for decisive Markov chains, i.e., the probabilities that a given set of states F is eventually reached or reached infinitely often, respectively. 1. We express the qualitative problems in abstract terms for decisive Markov chains, and show an almost complete picture of its decidability for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm of Iyer and Narasimha terminates for decisive Markov chains and can thus be used to solve the approximate quantitative safety problem. A modified variant of this algorithm solves the approximate quantitative liveness problem. 3. Finally, we show that the exact probability of (repeatedly) reaching F cannot be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS, PVASS or (P)NTM.