Algebraic Notions of Termination

TL;DR

This paper introduces and compares five algebraic notions of termination within modal semirings, extending to divergence structures, and demonstrates their utility in classical rewriting theory proofs.

Contribution

It formalizes and analyzes five algebraic termination notions using modal semirings and extends these to infinite behaviors with divergence structures.

Findings

Modal semirings effectively model finite and infinite program behaviors.

Divergence semirings enable algebraic reasoning about infinite computations.

Classical rewriting theorems are proved using these algebraic frameworks.

Abstract

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.