Easter Egg: Equality Reasoning Based on E-Graphs with Multiple Assumptions

E-graphs are a prominent data structure that has been increasing in popularity in recent years due to their ex-panding range of applications in various formal reasoning tasks. E-graphs allow systematic and efficient treatment of equality, which is pervasive in automated reasoning based on proofs. E-graphs handle equality well, but are severely limited in their handling of case splitting and other aspects of propositional reasoning, such as resolution, which introduce branching in provers and solvers. As a consequence, most tools resort to using e-graphs locally, recreating them ad-hoc when they are needed, and then discarding them. In exploratory scenarios, where it is necessary to retain multiple branches simultaneously, this limitation proves to be prohibitive. In particular, in theory exploration-a process where lemmas are discovered and then proven-this poses a significant challenge. Theory exploration must enumerate a space of possible assumptions, and must retain all of them to make progress. This poses a severe limitation on the ability to harness e-graphs for the task. Our key observation is that in exploratory reasoning tasks, branching represents versions of the same e-graph each with an added assumption, such as “x > y” or “is_sorted l”. Essentially, each e-graph represents an equality relation, and each branch corresponds to a matching coarsened equality relation. Based on this observation, we present an extension to e-graphs, called Colored E-Graphs, as a way to efficiently represent all of the coarsened equality relations in a single structure. A colored e-graph is a memory-efficient equivalent of multiple copies of an e-graph, with a much lower overhead. This is attained by sharing as much as possible between different cases, while carefully tracking which conclusion is true under which assumption. It can be viewed as adding multiple “color-coded” layers on top of the original e-graph structure, representing different assumptions. We run experiments and demonstrate that our colored e-graphs can support large numbers of assumptions and terms with space requirements that are about 10x lower, and with slightly improved performance.