Near-Linear Time Algorithms for Streett Objectives in Graphs and MDPs

TL;DR

This paper introduces randomized near-linear time algorithms for verifying Streett objectives in graphs and MDPs, significantly improving efficiency over previous methods and enabling faster model checking for reactive systems.

Contribution

The paper presents the first randomized near-linear time algorithms for Streett objectives in graphs and MDPs, achieving optimal expected running time up to poly-log factors.

Findings

Expected running time is $ ilde{O}(m + b)$ for the algorithms.

Algorithms are near-linear in input size, improving over previous quadratic or sub-quadratic methods.

Applicable to verification of omega-regular objectives in reactive systems.

Abstract

The fundamental model-checking problem, given as input a model and a specification, asks for the algorithmic verification of whether the model satisfies the specification. Two classical models for reactive systems are graphs and Markov decision processes (MDPs). A basic specification formalism in the verification of reactive systems is the strong fairness (aka Streett) objective, where given different types of requests and corresponding grants, the requirement is that for each type, if the request event happens infinitely often, then the corresponding grant event must also happen infinitely often. All omega-regular objectives can be expressed as Streett objectives and hence they are canonical in verification. Consider graphs/MDPs with n vertices, m edges, and a Streett objectives with k pairs, and let b denote the size of the description of the Streett objective for the sets of requests and grants. The current best-known algorithm for the problem requires time $O(min(n^2, m \sqrt{m \log n}) + b \log n)$. In this work, we present randomized near-linear time algorithms, with expected running time $\widetilde{O}(m + b)$, where the $\widetilde{O}$ notation hides poly-log factors. Our randomized algorithms are near-linear in the size of the input, and hence optimal up to poly-log factors.