Reasoning about distributive laws in a concurrent refinement algebra

Distributive laws are important for algebraic reasoning in arithmetic and logic. They are equally important for algebraic reasoning about concurrent programs. In existing theories such as Concurrent Kleene Algebra, only partial correctness is handled, and many of its distributive laws are weak, in the sense that they are only refinements in one direction, rather than equalities. The focus of this paper is on strengthening our theory to support the proof of strong distributive laws that are equalities, and in doing so come up with laws that are quite general. Our concurrent refinement algebra supports total correctness by allowing both finite and infinite behaviours. It supports the rely/guarantee approach of Jones by encoding rely and guarantee conditions as rely and guarantee commands. The strong distributive laws may then be used to distribute rely and guarantee commands over sequential compositions and into (and out of) iterations. For handling data refinement of concurrent programs, strong distributive laws are essential.