A Uniform Substitution Calculus for Differential Dynamic Logic

TL;DR

This paper presents a new proof calculus for differential dynamic logic based solely on uniform substitution, simplifying implementation and clarifying the logic's semantics.

Contribution

It introduces a uniform substitution-based calculus for dL, replacing schema variables with a finite set of axioms and internalizing differential invariants as first-class axioms.

Findings

Simplifies proof calculus for differential dynamic logic

Enables implementation based on axioms rather than schemata

Internalizes differential invariants and derivations as axioms

Abstract

This paper introduces a new 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 rely on axioms rather than axiom schemata, substantially simplifying implementations. Instead of nontrivial schema variables and soundness-critical side conditions on the occurrence patterns of variables, the resulting calculus adopts only a finite number of ordinary dL formulas as axioms. The static semantics of differential dynamic logic is captured exclusively in uniform substitutions and bound variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this paper introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivations as first-class axioms in dL.