A Formalization of Unique Solutions of Equations in Process Algebra

TL;DR

This thesis formalizes key properties and theorems of Milner's CCS in HOL4, extending the theory with recent developments like contraction relations and introducing a new variant called observational contraction.

Contribution

It provides a comprehensive formalization of CCS, including classical and recent theories, and introduces a novel contraction variant with improved properties.

Findings

Formalized classical properties of bisimulation and observation congruence

Extended the theory with contraction and expansion relations

Discovered and proved properties of a new observational contraction

Abstract

In this thesis, a comprehensive formalization of Milner's Calculus of Communicating Systems (also known as CCS) has been done in HOL theorem prover (HOL4), based on an old work in HOL88. This includes all classical properties of strong/weak bisimulation equivalences and observation congruence, a theory of congruence for CCS, various versions of "bisimulation up to" techniques, and several deep theorems, namely the "coarsest congruence contained in weak equivalence", and three versions of the "unique solution of equations" theorem in Milner's book. This work is further extended to support recent developments in Concurrency Theory, namely the "contraction" relation and the related "unique solutions of contractions" theorem found by Prof. Davide Sangiorgi, University of Bologna. As a result, a rather complete theory of "contraction" (and a similar relation called "expansion") for CCS is also formalized in this thesis. Further more, a new variant of contraction called "observational contraction" was found by the author during this work, based on existing contraction relation. It's formally proved that, this new relation is preserved by direct sums of CCS processes, and has a more elegant form of the "unique solutions of contractions" theorem without any restriction on the CCS grammar.