Building program construction and verification tools from algebraic principles

Abstract

We present a principled modular approach to the development of construction and verification tools for imperative programs, in which the control flow and the data flow are cleanly separated. Our simplest verification tool uses Kleene algebra with tests for the control flow of while-programs and their standard relational semantics for the data flow. It is expanded to a basic program construction tool by adding an operation for the specification statement and one single axiom. To include recursive procedures, Kleene algebras with tests are expanded further to quantales with tests. In this more expressive setting, iteration and the specification statement can be defined explicitly and stronger program transformation rules can be derived. Programming our approach in the Isabelle/HOL interactive theorem prover yields simple lightweight mathematical components as well as program construction and verification tools that are correct by construction themselves. Verification condition generation and program construction rules are based on equational reasoning and supported by powerful Isabelle tactics and automated theorem proving. A number of examples shows our tools at work.

Towards Verification of Cyber-Physical Systems with UTP and Isabelle/HOL

“…Altogether we have proven over 200 hundred laws of predicate and relational calculus, many of which can be imported either from HOL or by Armstrong's algebraic hierarchy [1]. This then gives us the foundation on which to build UTP theories for Cyber-Physical Systems.…”

“…Isabelle is a powerful proof assistant that can be used both for the mechanisation of mathematics, and for the application of such mechanisations to program verification, which is famously illustrated by the seL4 microkernel verification project [26]. Another excellent example is the use of Kleene algebras to build program verification tools [1], from which Hoare logics, weakest-precondition calculi, rely-guarantee calculi, and separation logics have been created. Specifically of interest for CPS, there has also been a lot of recent work on formalising calculus, analysis, and ordinary differential equations (ODEs) in Isabelle [25], which can then be applied to verification of hybrid systems.…”

“…The sledgehammer tool [4], for example, integrates a host of firstorder automated theorem provers and SMT solvers, which often shoulder the burden of proof effort. Sledgehammer is used, for example, by [1], both at the theory engineering level, for constructing an algebraic hierarchy of Kleene algebras, and also at the verification level, where it is used to discharge first-order proof obligations. For verification of hybrid systems, it will also be necessary to integrate Isabelle with Computer Algebra Systems (CAS) like Mathematica, MATLAB, or SageMath, to provide solutions to differential equations, an approach that has been previously well used by the KeYmaera tool [30,31].…”

Unifying Heterogeneous State-Spaces with Lenses

“…Thus one can apply predicate calculus to reason about programs, as well as prove the algebraic laws of programming themselves [19]. These laws can then be applied to construct semantic presentations for the purpose of verification, such as operational semantics, Hoare calculi, separation logic, and refinement calculi, to name a few [2,9]. This further enables the application of automated theorem provers to build program verification tools, an approach which has seen multiple successes [1,22].…”

A Discrete Geometric Model of Concurrent Program Execution

“…We have already built mathematical components for variants of Kleene algebras, regular algebras and relation algebras in Isabelle [6,17,4,13,2], integrated some of them into verification components for sequential programs [16,5,18], local reasoning with separation logic [12] and the rely-guarantee calculus [3]. In all of them, an abstract algebraic layer has been linked via formal soundness proofs with concrete computational models, e.g.…”