Robust Probabilistic Model Checking with Continuous Reward Domains

TL;DR

This paper introduces a new method for probabilistic model checking that accurately handles continuous reward distributions using moment matching with Erlang mixtures, enabling more comprehensive quality property analysis.

Contribution

It presents a novel approach combining moment generating functions and Erlang mixtures to approximate continuous reward distributions with bounded error in probabilistic model checking.

Findings

The method accurately approximates reward distributions in continuous spaces.

It provides bounded error guarantees for distributional approximations.

Experimental results show improved scalability and accuracy over existing approaches.

Abstract

Probabilistic model checking traditionally verifies properties on the expected value of a measure of interest. This restriction may fail to capture the quality of service of a significant proportion of a system's runs, especially when the probability distribution of the measure of interest is poorly represented by its expected value due to heavy-tail behaviors or multiple modalities. Recent works inspired by distributional reinforcement learning use discrete histograms to approximate integer reward distribution, but they struggle with continuous reward space and present challenges in balancing accuracy and scalability. We propose a novel method for handling both continuous and discrete reward distributions in Discrete Time Markov Chains using moment matching with Erlang mixtures. By analytically deriving higher-order moments through Moment Generating Functions, our method approximates the reward distribution with theoretically bounded error while preserving the statistical properties of the true distribution. This detailed distributional insight enables the formulation and robust model checking of quality properties based on the entire reward distribution function, rather than restricting to its expected value. We include a theoretical foundation ensuring bounded approximation errors, along with an experimental evaluation demonstrating our method's accuracy and scalability in practical model-checking problems.