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: ${\displaystyle \forall i,j\in \mathbb {id} ,M[i]=M[j]\Leftrightarrow \mathrm {find} (U,i)=\mathrm {find} (U,j)}$ 

In addition to the above structure, a valid e-graph conforms to several data structure invariants.^[2] Two e-nodes are equivalent if they are in the same e-class. The congruence invariant states that an e-graph must ensure that equivalence is closed under congruence, where two e-nodes ${\displaystyle f(i_{1},\ldots ,i_{n}),f(j_{1},\ldots ,j_{n})}$  are congruent when ${\displaystyle \mathrm {find} (U,i_{k})=\mathrm {find} (U,j_{k}),k\in \{1,\ldots ,n\}}$ . The hashcons invariant states that the hashcons maps canonical e-nodes to their e-class ID.

E-graphs expose wrappers around the ${\displaystyle \mathrm {add} }$ , ${\displaystyle \mathrm {find} }$ , and ${\displaystyle \mathrm {merge} }$  operations from the union-find that preserve the e-graph invariants. The last operation, e-matching, is described below.

An e-graph can also be formulated as a bipartite graph ${\displaystyle G=(N\uplus \mathrm {id} ,E)}$  where

• ${\displaystyle \mathrm {id} }$  is the set of e-class IDs (as above),
• ${\displaystyle N}$  is the set of e-nodes, and
• ${\displaystyle E\subseteq (\mathrm {id} \times N)\cup (N\times \mathrm {id} )}$  is a set of directed edges.

There is a directed edge from each e-class to each of its members, and from each e-node to each of its children.^[3]

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 ${\displaystyle V\subset \mathrm {Term} (\Sigma ,V)}$ , ${\displaystyle \Sigma _{0}\subset \mathrm {Term} (\Sigma ,V)}$ , 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 the term ${\displaystyle \sigma (p)}$  is represented by ${\displaystyle C}$ . There are several known algorithms for e-matching,^[4][5] the relational e-matching algorithm is based on worst-case optimal joins and is worst-case optimal.^[6]

Given an e-class and a cost function that maps each function symbol in ${\displaystyle \Sigma }$  to a natural number, the extraction problem is to find a ground term with minimal total cost that is represented by the given e-class. This problem is NP-hard.^[7] There is also no constant-factor approximation algorithm for this problem, which can be shown by reduction from the set cover problem. However, for graphs with bounded treewidth, there is a linear-time, fixed-parameter tractable algorithm.^[8]

• An e-graph with n equalities can be constructed in O(n log n) time.^[9]

Equality saturation is a technique for building optimizing compilers using e-graphs.^[10] 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^[11] and CVC4, where they are used to decide the empty theory by computing the congruence closure of a set of equalities, and e-matching is used to instantiate quantifiers.^[12] In DPLL(T)-based solvers that use conflict-driven clause learning (also known as non-chronological backtracking), e-graphs are extended to produce proof certificates.^[13] E-graphs are also used in the Simplify theorem prover of ESC/Java.^[14]

Equality saturation is used in specialized optimizing compilers,^[15] e.g. for deep learning^[16] and linear algebra.^[17] Equality saturation has also been used for translation validation applied to the LLVM toolchain.^[18]

E-graphs have been applied to several problems in program analysis, including fuzzing,^[19] abstract interpretation,^[20] and library learning.^[21]

• 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.
• Flatt, Oliver; Coward, Samuel; Willsey, Max; Tatlock, Zachary; Panchekha, Pavel (October 2022). "Small Proofs from Congruence Closure". In A. Griggio; N. Rungta (eds.). Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022. TU Wien Academic Press. pp. 75–83. doi:10.34727/2022/isbn.978-3-85448-053-2_13. ISBN 978-3-85448-053-2. S2CID 252118847.
• Goharshady, Amir Kafshdar; Lam, Chun Kit; Parreaux, Lionel (2024-10-08). "Fast and Optimal Extraction for Sparse Equality Graphs". Proceedings of the ACM on Programming Languages. 8 (OOPSLA2): 361:2551–361:2577. doi:10.1145/3689801.

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