Approximate reasoning for real-time probabilistic processes

TL;DR

This paper introduces a pseudo-metric for generalized semi-Markov processes that extends bisimulation to continuous-time probabilistic models, enabling approximate reasoning about quantitative properties.

Contribution

It develops a new pseudo-metric framework for continuous-time probabilistic processes, with a fixed point characterization and logical semantics, broadening bisimulation applicability.

Findings

Pseudo-metric extends bisimulation to broader classes of distributions.

The approach is insensitive to arbitrary distance articulations, relying on an intrinsic uniformity.

Quantitative properties are continuous with respect to the pseudo-metric.

Abstract

We develop a pseudo-metric analogue of bisimulation for generalized semi-Markov processes. The kernel of this pseudo-metric corresponds to bisimulation; thus we have extended bisimulation for continuous-time probabilistic processes to a much broader class of distributions than exponential distributions. This pseudo-metric gives a useful handle on approximate reasoning in the presence of numerical information -- such as probabilities and time -- in the model. We give a fixed point characterization of the pseudo-metric. This makes available coinductive reasoning principles for reasoning about distances. We demonstrate that our approach is insensitive to potentially ad hoc articulations of distance by showing that it is intrinsic to an underlying uniformity. We provide a logical characterization of this uniformity using a real-valued modal logic. We show that several quantitative properties of interest are continuous with respect to the pseudo-metric. Thus, if two processes are metrically close, then observable quantitative properties of interest are indeed close.