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 $\mathrm{\Sigma }$ be a set of uninterpreted functions, where ${\mathrm{\Sigma }}_n$ is the subset of $\mathrm{\Sigma }$ consisting of functions of arity $n$. Let $\mathbb{𝕚}\mathbb{𝕕}$ be a countable set of opaque identifiers that may be compared for equality, called e-class IDs. The application of $f\in {\mathrm{\Sigma }}_n$ to e-class IDs $i_1,i_2,\dots ,i_n\in \mathbb{𝕚}\mathbb{𝕕}$ is denoted $f(i_1,i_2,\dots ,i_n)$ and called an e-node.

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

• A union-find structure $U$ representing equivalence classes of e-class IDs, with the usual operations $\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}$, $\mathrm{a}\mathrm{d}\mathrm{d}$ and $\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}$. An e-class ID $e$ is canonical if $\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}(U,e)=e$; an e-node $f(i_1,\dots ,i_n)$ is canonical if each $i_j$ is canonical ($j$ in $1,\dots ,n$).
• An association of e-class IDs with sets of e-nodes, called e-classes. This consists of
  • a hashcons $H$ (i.e. a mapping) from e-nodes to e-class IDs, and
  • an e-class map $M$ that maps e-class IDs to e-classes, such that $M$ maps equivalent IDs to the same set of e-nodes: $\mathrm{\forall }i,j\in \mathbb{𝕚}\mathbb{𝕕},M[i]=M[j]\iff \mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}(U,i)=\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}(U,j)$

Invariants

In addition to the above structure, a valid e-graph conforms to several data structure invariants. 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 $f(i_1,\dots ,i_n),f(j_1,\dots ,j_n)$ are congruent when $\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}(U,i_k)=\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}(U,j_k),k\in \{1,\dots ,n\}$. The hashcons invariant states that the hashcons maps canonical e-nodes to their e-class ID.

Operations

E-graphs expose wrappers around the $\mathrm{a}\mathrm{d}\mathrm{d}$, $\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{d}$, and $\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}$ 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 $G=(N\uplus \mathrm{i}\mathrm{d},E)$ where

• $\mathrm{i}\mathrm{d}$ is the set of e-class IDs (as above),
• $N$ is the set of e-nodes, and
• $E\subseteq (\mathrm{i}\mathrm{d}\times N)\cup (N\times \mathrm{i}\mathrm{d})$ 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.

E-matching

Let $V$ be a set of variables and let $\mathrm{T}\mathrm{e}\mathrm{r}\mathrm{m}(\mathrm{\Sigma },V)$ 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, $\mathrm{T}\mathrm{e}\mathrm{r}\mathrm{m}(\mathrm{\Sigma },V)$ is the smallest set such that $V\subset \mathrm{T}\mathrm{e}\mathrm{r}\mathrm{m}(\mathrm{\Sigma },V)$, ${\mathrm{\Sigma }}_0\subset \mathrm{T}\mathrm{e}\mathrm{r}\mathrm{m}(\mathrm{\Sigma },V)$, and when $x_1,\dots ,x_n\in \mathrm{T}\mathrm{e}\mathrm{r}\mathrm{m}(\mathrm{\Sigma },V)$ and $f\in {\mathrm{\Sigma }}_n$, then $f(x_1,\dots ,x_n)\in \mathrm{T}\mathrm{e}\mathrm{r}\mathrm{m}(\mathrm{\Sigma },V)$. A term containing variables is called a pattern, a term without variables is called ground.

An e-graph $E$ represents a ground term $t\in \mathrm{T}\mathrm{e}\mathrm{r}\mathrm{m}(\mathrm{\Sigma },\mathrm{\varnothing })$ if one of its e-classes represents $t$. An e-class $C$ represents $t$ if some e-node $f(i_1,\dots ,i_n)\in C$ does. An e-node $f(i_1,\dots ,i_n)\in C$ represents a term $g(j_1,\dots ,j_n)$ if $f=g$ and each e-class $M[i_k]$ represents the term $j_k$ ($k$ in $1,\dots ,n$).

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

Extraction

Given an e-class and a cost function that maps each function symbol in $\mathrm{\Sigma }$ 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. 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.

Complexity

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

Equality saturation

Equality saturation is a technique for building optimizing compilers using e-graphs. 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 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. 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. E-graphs are also used in the Simplify theorem prover of ESC/Java.

Equality saturation is used in specialized optimizing compilers, e.g. for deep learning, linear algebra and auto-vectorization for digital signal processors. Equality saturation has also been used for translation validation applied to the LLVM toolchain.

E-graphs have been applied to several problems in program analysis, including fuzzing, abstract interpretation, and library learning.

References

• de Moura, Leonardo; Bjørner, Nikolaj (2007). "Efficient E-Matching for SMT Solvers". in Pfenning, Frank (in en). Automated Deduction – CADE-21. Lecture Notes in Computer Science. 4603. Berlin, Heidelberg: Springer. pp. 183–198. doi:10.1007/978-3-540-73595-3_13. ISBN 978-3-540-73595-3. https://link.springer.com/chapter/10.1007/978-3-540-73595-3_13. 
• 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. doi:10.1145/3434304. https://doi.org/10.1145/3434304. 
• 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. https://doi.org/10.1145/1480881.1480915. 
• Flatt, Oliver; Coward, Samuel; Willsey, Max; Tatlock, Zachary; Panchekha, Pavel (October 2022). "Small Proofs from Congruence Closure" (in en). 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. https://repositum.tuwien.at/handle/20.500.12708/81325. 
• 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. https://dl.acm.org/doi/abs/10.1145/3689801. 

External links

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