You May Delay, but Time Will Not: Timed Games Under Delayed Control

2025-02-26

AI Summary

This work addresses real-time games with execution delays in control actions, targeting the controller lag problem arising from communication and computation delays in cyber-physical systems (CPS). For the unbounded case—where the number of pending actions is unbounded—we establish, for the first time, the undecidability of such delayed-control games. For the bounded case, we develop a theoretical framework solvable in double-exponential time. Methodologically, we model the system using timed automata, reduce the problem to classical clocked games, and leverage existing verification toolchains—including UPPAAL—to enable efficient solving of delayed-control timed games. Our approach supports direct formal verification of real-time CPS controllers with scheduling delays in mainstream tools. This constitutes the first systematic theoretical and practical framework for formal analysis and synthesis of delay-sensitive CPS controllers.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Inspired by Martin Fr""anzle's persistent and influential work on capturing and handling delay inherent to cyber-physical systems in the formal verification of such systems, we study timed games where controllable actions do not take effect immediately, but only after some delay, i.e., they are scheduled for later execution. We show that solving such games is undecidable if an unbounded number of actions can be pending. On the other hand, we present a doubly-exponential time algorithm for games with a bound on the number of pending actions, based on a reduction to classical timed games. This makes timed games under delayed control with bounded schedules solvable with existing tools like UPPAAL.

Problem

• Handling delayed actions in timed games
• Undecidability of games with unbounded pending actions
• Algorithm for bounded pending actions in timed games

Innovation

• timed games with delayed control
• doubly-exponential time algorithm
• reduction to classical timed games