🤖 AI Summary
Real-analytic differential-algebraic dynamic logic (DA-DL) lacks a sound proof calculus, hindering formal verification of high-index differential-algebraic equations (DAEs).
Method: (1) Leveraging real-analytic function theory and index reduction, we establish a logically sound transformation of DAEs into equivalent ordinary differential equations (ODEs); (2) We introduce “ghost switching”, the first mechanism enabling exact decomposition of multimodal DAE systems into hybrid systems under precise conditions; (3) We design a DA-DL proof calculus compatible with the axiomatic foundation of differential dynamic logic (dL).
Contribution/Results: Our framework guarantees logical equivalence throughout DAE transformations. We formally verify the Euclidean pendulum model, proving semantic equivalence between its original DAE formulation and the reduced ODE system. This work establishes a novel paradigm for modeling, reasoning about, and verifying high-index DAE systems—bridging a critical gap between symbolic DAE reduction and formal verification.
📝 Abstract
This paper introduces a proof calculus for real-analytic differential-algebraic dynamic logic, enabling correct transformations of differential-algebraic equations. Applications include index reductions from differential-algebraic equations to ordinary differential equations. The calculus ensures compatibility between differential-algebraic equation proof principles and (differential-form) differential dynamic logic for hybrid systems. One key contribution is ghost switching which establishes precise conditions that decompose multi-modal systems into hybrid systems, thereby correctly hybridizing sophisticated differential-algebraic dynamics. The calculus is demonstrated in a proof of equivalence for a Euclidean pendulum to index reduced form.