A Complete Uniform Substitution Calculus for Differential Dynamic Logic

Abstract

This article introduces a relatively complete proof calculus for differential dynamic logic (dL) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform substitutions make it possible to use axioms instead of axiom schemata, thereby substantially simplifying implementations. Instead of subtle schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many axiom schema instances to sound ones, the resulting calculus adopts only a finite number of ordinary dL formulas as axioms, which uniform substitutions instantiate soundly. The static semantics of differential dynamic logic and the soundness-critical restrictions it imposes on proof steps is captured exclusively in uniform substitutions and variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this article introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivatives as first-class axioms to reason about differential equations axiomatically. The resulting axiomatization of differential dynamic logic is proved to be sound and relatively complete.

Tactical contract composition for hybrid system component verification

“…For specifying and verifying correctness statements about hybrid systems, we use differential dynamic logic (dL) [28,32,35], which supports hybrid programs as a pro- ?Q Aborts run if formula Q is not true 1 A continuous evolution along the differential equation system x = θ for an arbitrary real duration within the region described by formula Q gram notation for hybrid systems. The syntax and informal semantics of hybrid programs is summarized in Table 1.…”

“…The semantics of dL [28,32,35] is a Kripke semantics in which the states of the Kripke model are the states of the hybrid system. Let R denote the set of real numbers and V denote the set of variables.…”

“…A state is a map ν : V → R assigning a real value ν(x) to each variable x ∈ V. We write ν | φ if formula φ is true at state ν. [28,32,35] and Appendix A.…”

“…As a basis for our approach, we use differential dynamic logic dL [28,32,35], which is a hybrid systems specification and verification logic that is already compositional for each of its operators. Reasoning in dL splits models along the dL operators into smaller pieces.…”

Decoupling Abstractions of Non-linear Ordinary Differential Equations

“…This property is crucial to soundness of the abstraction. A rather different, but in a certain sense more general, concept was explored by Platzer, who introduced differential ghosts [19], where the original dynamics is augmented by introducing some fresh variables whose rate of change may feature the newly introduced variables themselves, but is not restricted in the same way as in Def. 2.…”