The B Book

In this chapter we will present the basic concepts of term rewriting that are needed in this book. More details on term rewriting, its applications, and related subjects can be found in the textbook of Baader and Nipkow [BN98]. Readers versed in German are also referred to the textbooks of Avenhaus [Ave95], Bundgen [Bun98], and Drosten [Dro89]. Moreover, there are several survey articles [HO80, DJ90, Klo92, Pla93] that can also be consulted.

Term Rewriting Systems

We implement a large Isabelle/HOL repository of algebras for application in modelling computing systems. They subsume computational logics such as dynamic and Hoare logics and form a basis for various software development methods. Isabelle has recently been extended by automated theorem provers and SMT solvers. We use these integrated tools for automatically proving several rather intricate refinement and termination theorems. We also automate a modal correspondence result and soundness and relative completeness proofs of propositional Hoare logic. These results show, for the first time, that Isabelle's tool integration makes automated algebraic reasoning particularly simple. This is a step towards increasing the automation of formal methods.

/pdf/automating-algebraic-methods-in-isabelle-26qle0gvdk.pdf

Automating algebraic methods in isabelle

http://www.csse.canterbury.ac.nz/walter.guttmann/publications/0024.pdf

Normal design algebra

Modal Kleene algebras are relatives of dynamic logics that support program construction and verification by equational reasoning. We describe their application in implementing versatile program correctness components in interactive theorem provers such as Isabelle/HOL. Starting from a weakest precondition based component with a simple relational store model, we show how variants for Hoare logic, strongest postconditions and program refinement can be built in a principled way. Modularity of the approach is demonstrated by variants that capture program termination and recursion, memory models for programs with pointers, and program trace semantics.

/pdf/modal-kleene-algebra-applied-to-program-correctness-1xpwmfrhnr.pdf

Modal Kleene Algebra Applied to Program Correctness

Five algebraic notions of termination are formalised, analysed and compared: wellfoundedness or Noetherity, L\"ob's formula, absence of infinite iteration, absence of divergence and normalisation. The study is based on modal semirings, which are additively idempotent semirings with forward and backward modal operators. To model infinite behaviours, idempotent semirings are extended to divergence semirings, divergence Kleene algebras and omega algebras. The resulting notions and techniques are used in calculational proofs of classical theorems of rewriting theory. These applications show that modal semirings are powerful tools for reasoning algebraically about the finite and infinite dynamics of programs and transition systems.

Algebraic Notions of Termination

Five algebraic notions of termination are formalised, analysed and compared:
wellfoundedness or Noetherity, L\"ob's formula, absence of infinite iteration,
absence of divergence and normalisation. The study is based on modal semirings,
which are additively idempotent semirings with forward and backward modal
operators. To model infinite behaviours, idempotent semirings are extended to
divergence semirings, divergence Kleene algebras and omega algebras. The
resulting notions and techniques are used in calculational proofs of classical
theorems of rewriting theory. These applications show that modal semirings are
powerful tools for reasoning algebraically about the finite and infinite
dynamics of programs and transition systems.

https://www.informatik.uni-augsburg.de/lehrstuehle/dbis/pmi/publications/all_pmi_tech-reports/tr-2006-23_des_moe_str/tr-2006-23_des_moe_str.pdf

Bernhard Moeller

Papers

Algebraic Notions of Termination

Algebraic Notions of Termination