This thesis puts forward a flexible and principled approach to the development of
construction and verification tools for imperative programs, in which the
control flow and the data level are cleanly separated. The approach is inspired
by algebraic principles and benefits from an algebraic semantics layer.
It is programmed in the Isabelle/HOL interactive theorem prover and yields
simple lightweight mathematical components as well as program construction and
verification tools that are themselves correct by construction.
First, a simple tool is implemented using Kleeene algebra with tests (KAT)
for the control flow of while-programs, which is the most compact verification
formalism for imperative programs, and their standard relational semantics for
the data level. A reference formalisation of KAT in Isabelle/HOL is then
presented, providing three different formalisations of tests. The structured
comprehensive libraries for these algebras include an algebraic account of
Hoare logic for partial correctness. Verification condition generation and
program construction rules are based on equational reasoning and supported by
powerful Isabelle tactics and automated theorem proving.
Second, the tool is expanded to support different programming features and
verification methods. A basic program construction tool is developed by adding
an operation for the specification statement and one single axiom. To include
recursive procedures, KATs are expanded further to quantales with tests,
where iteration and the specification statement can be defined explicitly.
Additionally, a nondeterministic extension supports the verification of simple
concurrent programs.
Finally, the approach is also applied to separation logic, where the
control-flow is modelled by power series with convolution as separating
conjunction. A generic construction lifts resource monoids to assertion and
predicate transformer quantales. The data level is captured by concrete
store-heap models. These are linked to the algebra by soundness proofs.
A number of examples shows the tools at work.
