E-graph

In computer science, an e-graph is a data structure that stores an equivalence relation over terms of some language.

Definition and operations

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

Invariants

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.

Operations

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.

Equivalent formulations

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]

E-matching

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]

Extraction

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]

Complexity

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

Equality saturation

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.

Applications

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]