Automatic pre- and postconditions for partial differential equations

Abstract

Based on an automata-theoretic and algebraic framework, we study equational reasoning for Initial Value Problems (ivps) of polynomial Partial Differential Equations (pdes). We first characterize the solutions of a pde system Σ in terms of the final morphism from a coalgebra induced by Σ to the coalgebra of formal power series (fps). fps solutions conservatively extend the classical analytic ones. To express ivps in their general form, we then introduce stratified systems, where the specification of a function can be decomposed into distinct subsystems of pdes. We lift the existence and uniqueness result of fps solutions to stratified systems. We then give a relatively complete algorithm to compute weakest preconditions and strongest postconditions for such systems. To some extent, this result reduces equational reasoning on pde initial value problems to algebraic reasoning. We illustrate some experiments conducted with a proof-of-concept implementation of the method.