Combining E-Graphs with Abstract Interpretation

E-graphs are a data structure that compactly represents equivalent expressions. They are constructed via the repeated application of rewrite rules. Often in practical applications, conditional rewrite rules are crucial, but their application requires the detection -- at the time the e-graph is being built -- that a condition is valid in the domain of application. Detecting condition validity amounts to proving a property of the program. Abstract interpretation is a general method to learn such properties, traditionally used in static analysis tools. We demonstrate that abstract interpretation and e-graph analysis naturally reinforce each other through a tight integration because (i) the e-graph clustering of equivalent expressions induces natural precision refinement of abstractions and (ii) precise abstractions allow the application of deeper rewrite rules (and hence potentially even greater precision). We develop the theory behind this intuition and present an exemplar interval arithmetic implementation, which we apply to the FPBench suite.