Lipschitz Robustness of Timed I/O Systems

TL;DR

This paper investigates the robustness of timed I/O systems using Lipschitz continuity, providing decision procedures for certain classes of systems and analyzing their computational complexity.

Contribution

It formalizes Lipschitz robustness for timed I/O systems, introduces decision procedures for specific models, and analyzes complexity results.

Findings

K-robustness is undecidable for discrete transducers.

A class of timed transducers admits a polynomial space decision procedure.

K-robustness for asynchronous circuits is PSPACE-complete.

Abstract

We present the first study of robustness of systems that are both timed as well as reactive (I/O). We study the behavior of such timed I/O systems in the presence of "uncertain inputs" and formalize their robustness using the analytic notion of Lipschitz continuity. Thus, a timed I/O system is K-(Lipschitz) robust if the perturbation in its output is at most K times the perturbation in its input. We quantify input and output perturbation using "similarity functions" over timed words such as the timed version of the Manhattan distance and the Skorokhod distance. We consider two models of timed I/O systems --- timed transducers and asynchronous sequential circuits. While K-robustness is undecidable even for discrete transducers, we identify a class of timed transducers which admits a polynomial space decision procedure for K-robustness. For asynchronous sequential circuits, we reduce K-robustness w.r.t. timed Manhattan distances to K-robusness of discrete letter-to-letter transducers and show PSPACE-compeleteness of the problem.