Axiomatization of Compact Initial Value Problems: Open Properties

Abstract: This article proves the completeness of an axiomatization for initial value problems (IVPs) with compact initial conditions and compact time horizons for bounded open safety, open liveness and existence properties. Completeness systematically reduces the proofs of these properties to a complete axiomatization for differential equation invariants. This result unifies symbolic logic and numerical analysis by a computable procedure that generates symbolic proofs with differential invariants for rigorous error bounds of numerical solutions to polynomial initial value problems. The procedure is modular and works for all polynomial IVPs with rational coefficients and initial conditions and symbolic parameters constrained to compact sets. Furthermore, this paper discusses generalizations to IVPs with initial conditions/symbolic parameters that are not necessarily constrained to compact sets, achieved through the derivation of fully symbolic axioms/proof-rules based on the axiomatization.