Quantitative Timed Simulation Functions and Refinement Metrics for Timed Systems (Full Version)

TL;DR

This paper introduces quantitative timed refinement and simulation metrics for timed systems, providing algorithms to compute timing mismatches using game-theoretic approaches, which help quantify and analyze timing differences in system behaviors.

Contribution

It proposes new quantitative metrics for timed system refinement, incorporating zenoness checks, and develops algorithms based on game theory to compute these metrics over timed automata.

Findings

Algorithms for computing three types of timing mismatches.

Introduction of game objectives for timed simulation analysis.

Quantitative metrics that incorporate zenoness checks.

Abstract

We introduce quantatitive timed refinement and timed simulation (directed) metrics, incorporating zenoness check s, for timed systems. These metrics assign positive real numbers between zero and infinity which quantify the \emph{timing mismatches} between two timed systems, amongst non-zeno runs. We quantify timing mismatches in three ways: (1) the maximal timing mismatch that can arise, (2) the "steady-state" maximal timing mismatches, where initial transient timing mismatches are ignored; and (3) the (long-run) average timing mismatches amongst two systems. These three kinds of mismatches constitute three important types of timing differences. Our event times are the \emph{global times}, measured from the start of the system execution, not just the time durations of individual steps. We present algorithms over timed automata for computing the three quantitative simulation distances to within any desired degree of accuracy. In order to compute the values of the quantitative simulation distances, we use a game theoretic formulation. We introduce two new kinds of objectives for two player games on finite-state game graphs: (1) eventual debit-sum level objectives, and (2) average debit-sum level objectives. We present algorithms for computing the optimal values for these objectives in graph games, and then use these algorithms to compute the values of the timed simulation distances over timed automata.