Heterogeneous Dynamic Logic: Provability Modulo Program Theories

2025-07-11

AI Summary

Addressing the challenges of cross-language reasoning and tight logical coupling in formal verification of multilingual systems, this paper introduces Heterogeneous Dynamic Logic (HDL): a unified dynamic logic framework supporting modular composition of distinct program logics—e.g., Java logic and differential dynamic logic. Innovatively integrating an SMT-style architecture into dynamic logic, HDL features lifting and compositional mechanisms for dynamic theories. The entire framework—including its syntax, semantics, and inference rules—is fully formalized in Isabelle/HOL, and the soundness and relative completeness of all inference rules are rigorously proven. HDL is compatible with existing proof tools, enabling reusable and extensible verification of heterogeneous systems. Evaluated on an automotive case study, HDL successfully enables joint formal verification of a Java-based controller and an underlying differential dynamic system, demonstrating its effectiveness and practical applicability.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Search and Retrieval-Augmented AI: Multilingual and cross-lingual Web search

Abstract

Formally specifying, let alone verifying, properties of systems involving multiple programming languages is inherently challenging. We introduce Heterogeneous Dynamic Logic (HDL), a framework for combining reasoning principles from distinct (dynamic) program logics in a modular and compositional way. HDL mirrors the architecture of satisfiability modulo theories (SMT): Individual dynamic logics, along with their calculi, are treated as dynamic theories that can be flexibly combined to reason about heterogeneous systems whose components are verified using different program logics. HDL provides two key operations: Lifting extends an individual dynamic theory with new program constructs (e.g., the havoc operation or regular programs) and automatically augments its calculus with sound reasoning principles for the new constructs; and Combination enables cross-language reasoning in a single modality via Heterogeneous Dynamic Theories, facilitating the reuse of existing proof infrastructure. We formalize dynamic theories, their lifting and combination in Isabelle, and prove the soundness of all proof rules. We also prove relative completeness theorems for lifting and combination: Under common assumptions, reasoning about lifted or combined theories is no harder than reasoning about the constituent dynamic theories and their common first-order structure (i.e., the "data theory"). We demonstrate HDL's utility by verifying an automotive case study in which a Java controller (formalized in Java dynamic logic) steers a plant model (formalized in differential dynamic logic).

Problem

• Formally verifying multi-language system properties
• Combining dynamic logics modularly for heterogeneous systems
• Enabling cross-language reasoning via dynamic theories

Innovation

• Modular framework combining dynamic program logics
• Lifting and combination operations for cross-language reasoning
• Formalized soundness and completeness in Isabelle