Our goal is to learn about CSP refinement by implementing a refinement checker. So a good first step is to make sure we’re all on the same page about what refinement is, and then to step through the refinement algorithm that we mean to implement. (If nothing else, that will help make sure I don’t go off on too many tangents while implementing it!)
I’ve mentioned refinement elsewhere on this blog a few times (for instance, here). The basic idea is that in CSP, you use the same process language to describe the system you’re designing or investigating, as well as the properties that you would like that system to have. (This is unlike most other formal methods, where you have separate languages for the system and the properties.) In CSP, the system’s process is typically called \(Impl\) (for implementation), and the property description process is typically called \(Spec\) (for specification).
CSP then defines several semantic models that provide rigorous mathematical definitions of what a process’s behavior is. You perform a refinement check within the context of a particular semantic model. A successful refinement check tells you that the property defined by \(Spec\) “holds” — specifically, that all of the behaviors of \(Impl\) are also allowed behaviors of \(Spec\). A failed refinement check gives you a counterexample — that is, a specific behavior of \(Impl\) that was disallowed by \(Spec\).
The three most common semantic models are traces, failures, and failures-divergences. We’ll go into more detail about the mathematics behind these semantic models in later posts; for now, the 10,000-foot overview is that:
In my post about the Read Atomic concurreny model, I use a traces refinement check to verify that Read Atomic doesn’t allow “unrepeatable reads”. In this example, the \(Spec\) process is a description of the Read Atomic concurrency model, while the \(Impl\) process is a “fake” implementation that immediately tries to perform an unrepeatable read. Because that unrepeatable read isn’t allowed by the Read Atomic process, the traces refinement check fails.
Given all of that, how do we write a program that can perform refinement checks for us? The answer is strewn throughout Bill Roscoe’s textbook, The theory and practice of concurrency. The bulk of FDR’s refinement algorithm is described in Appendix C (p. 541). At a high level, we need to:
Load in a description of the \(Spec\) and \(Impl\) processes, transforming them each into a labeled transition system (LTS).
Normalize the \(Spec\) process, resulting in a normalized LTS.
Perform a simultaneous breadth-first search through the \(Spec\)’s normalized LTS and \(Impl\)’s (non-normalized) LTS, looking for a counterexample to the refinement.
If we find any counterexample, the refinement check fails. If we don’t, the refinement check succeeds.
This is enough to get started for now; in later posts I’ll drill down into each of these steps in (much) more detail, and show how to implement them in HST.