Composition: A Way to Make Proofs Harder

Compositionality: The Significant Difference (Proceedings of the COMPOS'97 Symposium), Willem-Paul de Roever, Hans Langmaack, and Amir Pnueli editors. Lecture Notes in Computer Science. |

Systems are complicated. We master their complexity by building them from simpler components. This suggests that to master the complexity of reasoning about systems, we should prove properties of the separate components and then combine those properties to deduce properties of the entire system. In concurrent systems, the obvious choice of component is the process. So, compositional reasoning has come to mean deducing properties of a system from properties of its processes.

I have long felt that this whole approach is rather silly. You don’t design a mutual exclusion algorithm by first designing the individual processes and then hoping that putting them together guarantees mutual exclusion. Similarly, anyone who has tried to deduce mutual exclusion from properties proved by considering the processes in isolation knows that it’s the wrong way to approach the problem. You prove mutual exclusion by finding a global invariant, and then showing that each process’s actions maintains the invariant. TLA makes the entire reasoning process completely mathematical–the specifications about which one reasons are mathematical formulas, and proving correctness means proving a single mathematical formula. A mathematical proof is essentially decompositional: you apply a deduction rule to reduce the problem of proving a formula to that of proving one or more simpler formulas.

This paper explains why traditional compositional reasoning is just one particular, highly constrained way of decomposing the proof. In most cases, it’s not a very natural way and results in extra work. This extra work is justified if it can be done by a computer. In particular, decomposition along processes makes sense if the individual processes are simple enough to be verified by model checking. TLA is particularly good for doing this because, as illustrated by [119], it allows a great deal of flexibility in choosing what constitutes a process.