E-graph

Let ${\displaystyle \Sigma }$ be a set of uninterpreted functions, where ${\displaystyle \Sigma _{n}}$ is the subset of ${\displaystyle \Sigma }$ consisting of functions of arity ${\displaystyle n}$. Let ${\displaystyle \mathbb {id} }$ be a countable set of opaque identifiers that may be compared for equality, called e-class IDs. The application of ${\displaystyle f\in \Sigma _{n}}$ to e-class IDs ${\displaystyle i_{1},i_{2},\ldots ,i_{n}\in \mathbb {id} }$ is denoted ${\displaystyle f(i_{1},i_{2},\ldots ,i_{n})}$ and called an e-node.

The e-graph then represents equivalence classes of e-nodes, using the following data structures:[1]

• A union-find structure ${\displaystyle U}$ representing equivalence classes of e-class IDs, with the usual operations ${\displaystyle \mathrm {find} }$, ${\displaystyle \mathrm {add} }$ and ${\displaystyle \mathrm {merge} }$. An e-class ID ${\displaystyle e}$ is canonical if ${\displaystyle \mathrm {find} (U,e)=e}$; an e-node ${\displaystyle f(i_{1},\ldots ,i_{n})}$ is canonical if each ${\displaystyle i_{j}}$ is canonical (${\displaystyle j}$ in ${\displaystyle 1,\ldots ,n}$).
• An association of e-class IDs with sets of e-nodes, called e-classes. This consists of
  • a hashcons ${\displaystyle H}$ (i.e. a mapping) from canonical e-nodes to e-class IDs, and
  • an e-class map ${\displaystyle M}$ that maps e-class IDs to e-classes, such that ${\displaystyle M}$ maps equivalent IDs to the same set of e-nodes: ${\displaystyle \forall i,j\in \mathbb {id} ,M[i]=M[j]\Leftrightarrow \mathrm {find} (U,i)=\mathrm {find} (U,j)}$

Let ${\displaystyle V}$ be a set of variables and let ${\displaystyle \mathrm {Term} (\Sigma ,V)}$ be the smallest set that includes the 0-arity function symbols (also called constants), includes the variables, and is closed under application of the function symbols. In other words, ${\displaystyle \mathrm {Term} (\Sigma ,V)}$ is the smallest set such that ${\displaystyle V\subset \mathrm {Term} (V,\Sigma )}$, ${\displaystyle \Sigma _{0}\subset \mathrm {Term} (\Sigma ,V)}$, and when ${\displaystyle x_{1},\ldots ,x_{n}\in \mathrm {Term} (\Sigma ,V)}$ and ${\displaystyle f\in \Sigma _{n}}$, then ${\displaystyle f(x_{1},\ldots ,x_{n})\in \mathrm {Term} (\Sigma ,V)}$. A term containing variables is called a pattern, a term without variables is called ground.

An e-graph ${\displaystyle E}$ represents a ground term ${\displaystyle t\in \mathrm {Term} (\Sigma ,\emptyset )}$ if one of its e-classes represents ${\displaystyle t}$. An e-class ${\displaystyle C}$ represents ${\displaystyle t}$ if some e-node ${\displaystyle f(i_{1},\ldots ,i_{n})\in C}$ does. An e-node ${\displaystyle f(i_{1},\ldots ,i_{n})\in C}$ represents a term ${\displaystyle g(j_{1},\ldots ,j_{n})}$ if ${\displaystyle f=g}$ and each e-class ${\displaystyle M[i_{k}]}$ represents the term ${\displaystyle j_{k}}$ (${\displaystyle k}$ in ${\displaystyle 1,\ldots ,n}$).

e-matching is an operation that takes a pattern ${\displaystyle p\in \mathrm {Term} (\Sigma ,V)}$ and an e-graph ${\displaystyle E}$, and yields all pairs ${\displaystyle (\sigma ,C)}$ where ${\displaystyle \sigma \subset V\times \mathbb {id} }$ is a substitution mapping the variables in ${\displaystyle p}$ to e-class IDs and ${\displaystyle C\in \mathbb {id} }$ is an e-class ID such that each term ${\displaystyle \sigma (p)}$ is represented by ${\displaystyle C}$. There are several known algorithms for e-matching,[3][4] the relational e-matching algorithm is based on worst-case optimal joins and is worst-case optimal.[5]

Equality saturation is a technique for building optimizing compilers using e-graphs.[6] It operates by applying a set of rewrites using e-matching until the e-graph is saturated, a timeout is reached, an e-graph size limit is reached, a fixed number of iterations is exceeded, or some other halting condition is reached. After rewriting, an optimal term is extracted from the e-graph according to some cost function, usually related to AST size or performance considerations.

E-graphs are used in automated theorem proving. They are a crucial part of modern SMT solvers such as Z3[7] and CVC4. They are also used in the Simplify theorem prover of ESC/Java.[8]

Equality saturation is used in specialized optimizing compilers,[9] e.g. for deep learning[10] and linear algebra.[11] Equality saturation has also been used for translation validation applied to the LLVM toolchain.[12]

E-graphs have been applied to several problems in program analysis, including fuzzing,[13] abstract interpretation,[14][15] and library learning.[16]

• de Moura, Leonardo; Bjørner, Nikolaj (2007). "Efficient E-Matching for SMT Solvers". In Pfenning, Frank (ed.). Automated Deduction – CADE-21. Lecture Notes in Computer Science. Vol. 4603. Berlin, Heidelberg: Springer. pp. 183–198. doi:10.1007/978-3-540-73595-3_13. ISBN 978-3-540-73595-3.
• Willsey, Max; Nandi, Chandrakana; Wang, Yisu Remy; Flatt, Oliver; Tatlock, Zachary; Panchekha, Pavel (2021-01-04). "egg: Fast and extensible equality saturation". Proceedings of the ACM on Programming Languages. 5 (POPL): 23:1–23:29. arXiv:2004.03082. doi:10.1145/3434304. S2CID 226282597.
• Tate, Ross; Stepp, Michael; Tatlock, Zachary; Lerner, Sorin (2009-01-21). "Equality saturation". Proceedings of the 36th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages. POPL '09. Savannah, GA, USA: Association for Computing Machinery. pp. 264–276. doi:10.1145/1480881.1480915. ISBN 978-1-60558-379-2. S2CID 2138086.

• The Egg Project
• A Colab notebook explaining e-graphs