Rewrite Rule Inference Using Equality Saturation

OOPSLA 2021, August 2021
Distinguished Paper
  author = {Nandi, Chandrakana 
        and Willsey, Max 
        and Zhu, Amy 
        and Wang, Yisu Remy 
        and Saiki, Brett 
        and Anderson, Adam 
        and Schulz, Adriana 
        and Grossman, Dan 
        and Tatlock, Zachary},
  title = {Rewrite Rule Inference Using Equality Saturation},
  year = {2021},
  issue_date = {October 2021},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {5},
  number = {OOPSLA},
  url = {},
  doi = {10.1145/3485496},
  journal = {Proc. ACM Program. Lang.},
  month = {oct},
  articleno = {119},
  numpages = {28},
  keywords = {Program Synthesis, Rewrite Rules, Equality Saturation}
Given a grammar and interpreter for a target domain, Ruler uses e-graphs and equality saturation to efficiently enumerate potential rewrite rules and iteratively select a small set of general, orthogonal rules. Ruler supports various validation strategies to ensure soundness and speed up synthesis, including constraint solving (e.g., SMT), model checking, and fuzzing.


Many compilers, synthesizers, and theorem provers rely on rewrite rules to simplify expressions or prove equivalences. Developing rewrite rules can be difficult: rules may be subtly incorrect, profitable rules are easy to miss, and rulesets must be rechecked or extended whenever semantics are tweaked. Large rulesets can also be challenging to apply: redundant rules slow down rule-based search and frustrate debugging. This paper explores how equality saturation, a promising technique that uses e-graphs to apply rewrite rules, can also be used to infer rewrite rules. E-graphs can compactly represent the exponentially large sets of enumerated terms and potential rewrite rules. We show that equality saturation efficiently shrinks both sets, leading to faster synthesis of smaller, more general rulesets.

We prototyped these strategies in a tool dubbed Ruler. Compared to a similar tool built on CVC4, ruler synthesizes 5.8x smaller rulesets 25x faster without compromising on proving power. In an end-to-end case study, we show ruler-synthesized rules which perform as well as those crafted by domain experts, and addressed a longstanding issue in a popular open source tool.