Unique Solutions of Contractions, CCS, and their HOL Formalisation

The unique solution of contractions is a proof technique for bisimilarity that overcomes certain syntactic constraints of Milner's "unique solution of equations" technique. The paper presents an overview of a rather comprehensive formalisation of the core of the theory of CCS in the HOL theorem prover (HOL4), with a focus towards the theory of unique solutions of contractions. (The formalisation consists of about 20,000 lines of proof scripts in Standard ML.) Some refinements of the theory itself are obtained. In particular we remove the constraints on summation, which must be weakly-guarded, by moving to rooted contraction, that is, the coarsest precongruence contained in the contraction preorder.Comment: In Proceedings EXPRESS/SOS 2018, arXiv:1808.0807

Formalising the pi-calculus using nominal logic

We formalise the pi-calculus using the nominal datatype package, based on ideas from the nominal logic by Pitts et al., and demonstrate an implementation in Isabelle/HOL. The purpose is to derive powerful induction rules for the semantics in order to conduct machine checkable proofs, closely following the intuitive arguments found in manual proofs. In this way we have covered many of the standard theorems of bisimulation equivalence and congruence, both late and early, and both strong and weak in a uniform manner. We thus provide one of the most extensive formalisations of a process calculus ever done inside a theorem prover. A significant gain in our formulation is that agents are identified up to alpha-equivalence, thereby greatly reducing the arguments about bound names. This is a normal strategy for manual proofs about the pi-calculus, but that kind of hand waving has previously been difficult to incorporate smoothly in an interactive theorem prover. We show how the nominal logic formalism and its support in Isabelle accomplishes this and thus significantly reduces the tedium of conducting completely formal proofs. This improves on previous work using weak higher order abstract syntax since we do not need extra assumptions to filter out exotic terms and can keep all arguments within a familiar first-order logic.Comment: 36 pages, 3 figure

Unique solutions of contractions, CCS, and their HOL formalisation

International audienceThe unique solution of contractions is a proof technique for (weak) bisimilarity that overcomes certainsyntactic limitations of Milner’s “unique solution of equations” theorem. This paper presents an overview ofa comprehensive formalisation of Milner’s Calculus of Communicating Systems (CCS) in the HOL theoremprover (HOL4), with a focus towards the theory of unique solutions of equations and contractions. Theformalisation consists of about 24,000 lines (1MB) of code in total. Some refinements of the “unique solutionof contractions” theory itself are obtained. In particular we remove the constraints on summation, whichmust be guarded, by moving from contraction to rooted contraction. We prove the “unique solution ofrooted contractions” theorem and show that rooted contraction is the coarsest precongruence contained inthe contraction preorder