A lattice-theoretic framework for circular assume-guarantee reasoning

We develop an abstract lattice-theoretic framework within which we study soundness and other properties of circular assume-guarantee (A-G) rules constrained by side conditions. We identify a particular side condition, non-blockingness, which admits an intelligible inductive proof of the soundness of circular A-G reasoning. Besides, conditional circular rules based on non-blockingness turn out to be complete in various senses and stronger than a large class of sound conditional A-G rules. In this respect, our framework enlightens the foundations of circular A-G reasoning. Due to its abstractness, the framework can be instantiated to many concrete settings. We show several known circular A-G rules for compositional verification to be instances of our generic rules. Thus, we do the circularity-breaking inductive argument once to establish soundness of our generic rules, which then implies soundness of all the instances without resorting to technically complicated circularity-breaking arguments for each single rule. In this respect, our framework unifies many approaches to circular A-G reasoning and provides a starting point for the systematic development of new circular A-G rules.