Complete Dynamic Logic of Communicating Hybrid Programs

2024-08-09

arXiv.org

AI Summary

This paper addresses the formal verification of parallel hybrid systems—comprising discrete dynamics, continuous dynamics, and synchronous evolution. We propose dynamic logic for communicating hybrid programs (dLCHP), equipped with a relatively complete proof calculus. Our key innovation is the first modal formulation of assumption-commitment (AC) reasoning for hybrid systems: we decouple traditional parallel composition rules into modular axioms, achieving a balance among expressiveness, flexibility, and relative completeness. The foundation integrates differential dynamic logic (dL), AC reasoning, and communicating trajectory logic. Theoretical contributions include: (1) relative completeness of dLCHP with respect to communicating trajectories and differential equation properties; and (2) proof-theoretic equivalence between dLCHP and dL, establishing that verifying parallel hybrid systems incurs no greater logical complexity than verifying a single hybrid system.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

This article presents a relatively complete proof calculus for the dynamic logic of communicating hybrid programs dLCHP. Beyond traditional hybrid systems mixing discrete and continuous dynamics, communicating hybrid programs feature parallel interactions of hybrid systems. This not only compounds the subtleties of hybrid and parallel systems but adds the truly simultaneous synchronized evolution of parallel hybrid dynamics as a new challenge. To enable compositional reasoning about communicating hybrid programs nevertheless, dLCHP combines differential dynamic logic dL and assumption-commitment reasoning. To maintain the logical essence of dynamic logic axiomatizations, dLCHP's proof calculus presents a new modal logic view onto ac-reasoning. This modal view drives a decomposition of classical monolithic proof rules for parallel systems reasoning into new modular axioms, which yields better flexibility and simplifies soundness arguments. Adequacy of the proof calculus is shown by two completeness results: First, dLCHP is complete relative to the logic of communication traces and differential equation properties. This result proves the new modular modal view sufficient for reasoning about parallel hybrid systems, and captures modular strategies for reasoning about concrete parallel hybrid systems. The second result proof-theoretically aligns dLCHP and dL by proving that reasoning about parallel hybrid systems is exactly as hard as reasoning about hybrid systems, continuous systems, or discrete systems. This completeness result reveals the possibility of representational succinctness in parallel hybrid systems proofs.

Problem

• Develops proof calculus for communicating hybrid programs logic
• Addresses parallel hybrid systems with synchronized continuous dynamics
• Enables compositional reasoning via assumption-commitment and trace mechanisms

Innovation

• Combines differential dynamic logic with mutual abstraction
• Uses assumption-commitment reasoning for parallel subsystems
• Enables equitranslation between Ω-FOD and FOD