Symbolic Algorithms for Graphs and Markov Decision Processes with Fairness Objectives

Krishnendu Chatterjee,Monika Henzinger,Simin Oraee,Viktor Toman,Veronika Loitzenbauer

doi:10.1007/978-3-319-96142-2_13

Abstract

Given a model and a specification, the fundamental model-checking problem asks for algorithmic verification of whether the model satisfies the specification. We consider graphs and Markov decision processes (MDPs), which are fundamental models for reactive systems. One of the very basic specifications that arise in verification of reactive systems is the strong fairness (aka Streett) objective. Given different types of requests and corresponding grants, the objective requires 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. To handle the state-space explosion, symbolic algorithms are required that operate on a succinct implicit representation of the system rather than explicitly accessing the system. While explicit algorithms for graphs and MDPs with Streett objectives have been widely studied, there has been no improvement of the basic symbolic algorithms. The worst-case numbers of symbolic steps required for the basic symbolic algorithms are as follows: quadratic for graphs and cubic for MDPs. In this work we present the first sub-quadratic symbolic algorithm for graphs with Streett objectives, and our algorithm is sub-quadratic even for MDPs. Based on our algorithmic insights we present an implementation of the new symbolic approach and show that it improves the existing approach on several academic benchmark examples.

Highlights

In this work we present faster symbolic algorithms for graphs and Markov decision processes (MDPs) with strong fairness objectives
The most basic algorithm for the problem for graphs is based on repeated strongly connected component (SCC) computation, and informally can be described as follows: for a given SCC, (a) if for every request type that is present in the SCC the corresponding grant type is present in the SCC, the SCC is identified as “good”, (b) else vertices of each request type that has no corresponding grant type in the SCC are removed, and the algorithm recursively proceeds on the remaining graph
For MDPs, the SCC computation has to be replaced by MEC computation, and the current best-known symbolic algorithm for MEC computation requires O(n2) symbolic steps

Summary

Introduction

In this work we present faster symbolic algorithms for graphs and Markov decision processes (MDPs) with strong fairness objectives. For the fundamental model-checking problem, the input consists of a model and a specification, and the algorithmic verification problem is to check whether the model satisfies the specification. We first describe the specific model-checking problem we consider and our contributions. Models: Graphs and MDPs. Two standard models for reactive systems are graphs and Markov decision processes (MDPs). Vertices of a graph represent states of a reactive system, edges represent transitions of the system, and infinite paths of the graph represent non-terminating trajectories of the reactive system. MDPs extend graphs with probabilistic transitions that represent reactive systems with uncertainty. Graphs and MDPs are the de-facto model of reactive systems with nondeterminism, and nondeterminism with stochastic aspects, respectively [3,19]

Objectives

Methods

Results

Conclusion