Differential Equation Axiomatization

Abstract

We prove the completeness of an axiomatization for differential equation invariants. First, we show that the differential equation axioms in differential dynamic logic are complete for all algebraic invariants. Our proof exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. Cleverly chosen differential ghosts are the proof-theoretical counterpart of dark matter. They create new hypothetical state, whose relationship to the original state variables satisfies invariants that did not exist before. The reflection of these new invariants in the original system then enables its analysis.We then show that extending the axiomatization with existence and uniqueness axioms makes it complete for all local progress properties, and further extension with a real induction axiom makes it complete for all real arithmetic invariants. This yields a parsimonious axiomatization, which serves as the logical foundation for reasoning about invariants of differential equations. Moreover, our results are purely axiomatic, and so the axiomatization is suitable for sound implementation in foundational theorem provers.

Differential Equation Invariance Axiomatization

“…is article extends the authors' earlier conference version [34] beyond polynomial term languages and presents a di erential equation invariance axiomatization. For extended term languages meeting a set of extended term conditions, this article presents the following contributions:…”

“…e improvement (7) is signi cant for conceptual, implementation, and proof purposes. All algebraic (and analytic) invariants can now be proved using only a constant number of scalar di erential ghosts compared to the earlier result [34] which uses a quadratic number of vectorial di erential ghosts.…”

“…is presentation is omi ed as it is not the focus for this article. Readers are referred to the authors' earlier conference version [34] for details.…”

“…2a, this corresponds to the case where both blue lines lie exactly on the horizontal axis. An alternative derivation of rule vdbx is given in the authors' earlier conference version [34] based on Liouville's formula [46, §15.III]. at alternative derivation has an alternative geometric interpretation as a continuous change of basis that is expressed purely syntactically [34] but requires the use of vectorial di erential ghosts and a number of ghost variables that is quadratic in the dimension.…”

“…Any such extension automatically inherits all of the aforementioned completeness results. (6) e authors' earlier results and proofs [34] are generalized to extended term languages. (7) A stronger proof-theoretical result is shown for algebraic (and analytic) completeness compared to [34] using only scalar di erential ghosts.…”