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]

References

1. ( Willsey et al. 2021 )
2. ( Willsey et al. 2021 )
3. ( Goharshady, Lam & Parreaux 2024 )
4. ( de Moura & Bjørner 2007 )
5. Moskal, Michał; Łopuszański, Jakub; Kiniry, Joseph R. (2008-05-06). "E-matching for Fun and Profit". Electronic Notes in Theoretical Computer Science. Proceedings of the 5th International Workshop on Satisfiability Modulo Theories (SMT 2007). 198 (2): 19–35. doi: 10.1016/j.entcs.2008.04.078 . ISSN   1571-0661.
6. Zhang, Yihong; Wang, Yisu Remy; Willsey, Max; Tatlock, Zachary (2022-01-12). "Relational e-matching". Proceedings of the ACM on Programming Languages. 6 (POPL): 35:1–35:22. doi: 10.1145/3498696 . S2CID   236924583.
7. Stepp, Michael Benjamin (2011). Equality saturation: engineering challenges and applications (PhD thesis). USA: University of California at San Diego. ISBN   978-1-267-03827-2.
8. ( Goharshady, Lam & Parreaux 2024 )
9. ( Flatt et al. 2022 , p. 2)
10. ( Tate et al. 2009 )
11. de Moura, Leonardo; Bjørner, Nikolaj (2008). "Z3: An Efficient SMT Solver". In Ramakrishnan, C. R.; Rehof, Jakob (eds.). Tools and Algorithms for the Construction and Analysis of Systems. Lecture Notes in Computer Science. Vol. 4963. Berlin, Heidelberg: Springer. pp. 337–340. doi: 10.1007/978-3-540-78800-3_24 . ISBN   978-3-540-78800-3.
12. Rümmer, Philipp (2012). "E-Matching with Free Variables". In Bjørner, Nikolaj; Voronkov, Andrei (eds.). Logic for Programming, Artificial Intelligence, and Reasoning. Proceedings. 18th International Conference, LPAR-18, Merida, Venezuela, March 11–15, 2012. Lecture Notes in Computer Science. Vol. 7180. Berlin, Heidelberg: Springer. pp. 359–374. doi:10.1007/978-3-642-28717-6_28. ISBN   978-3-642-28717-6.
13. ( Flatt et al. 2022 , p. 2)
14. Detlefs, David; Nelson, Greg; Saxe, James B. (May 2005). "Simplify: a theorem prover for program checking". Journal of the ACM. 52 (3): 365–473. doi:10.1145/1066100.1066102. ISSN   0004-5411. S2CID   9613854.
15. Joshi, Rajeev; Nelson, Greg; Randall, Keith (2002-05-17). "Denali: a goal-directed superoptimizer". ACM SIGPLAN Notices. 37 (5): 304–314. doi:10.1145/543552.512566. ISSN   0362-1340.
16. Yang, Yichen; Phothilimtha, Phitchaya Mangpo; Wang, Yisu Remy; Willsey, Max; Roy, Sudip; Pienaar, Jacques (2021-03-17). "Equality Saturation for Tensor Graph Superoptimization". arXiv: 2101.01332 [cs.AI].
17. Wang, Yisu Remy; Hutchison, Shana; Leang, Jonathan; Howe, Bill; Suciu, Dan (2020-12-22). "SPORES: Sum-Product Optimization via Relational Equality Saturation for Large Scale Linear Algebra". arXiv: 2002.07951 [cs.DB].
18. Stepp, Michael; Tate, Ross; Lerner, Sorin (2011). "Equality-Based Translation Validator for LLVM". In Gopalakrishnan, Ganesh; Qadeer, Shaz (eds.). Computer Aided Verification. Lecture Notes in Computer Science. Vol. 6806. Berlin, Heidelberg: Springer. pp. 737–742. doi: 10.1007/978-3-642-22110-1_59 . ISBN   978-3-642-22110-1.
19. "Wasm-mutate: Fuzzing WebAssembly Compilers with E-Graphs (EGRAPHS 2022) - PLDI 2022". pldi22.sigplan.org. Retrieved 2023-02-03.
20. Coward, Samuel; Constantinides, George A.; Drane, Theo (2022-03-17). "Abstract Interpretation on E-Graphs". arXiv: 2203.09191 [cs.LO].
    Coward, Samuel; Constantinides, George A.; Drane, Theo (2022-05-30). "Combining E-Graphs with Abstract Interpretation". arXiv: 2205.14989 [cs.DS].
21. Cao, David; Kunkel, Rose; Nandi, Chandrakana; Willsey, Max; Tatlock, Zachary; Polikarpova, Nadia (2023-01-09). "babble: Learning Better Abstractions with E-Graphs and Anti-Unification". Proceedings of the ACM on Programming Languages. 7 (POPL): 396–424. arXiv: 2212.04596 . doi:10.1145/3571207. ISSN   2475-1421. S2CID   254536022.

• 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.

External links

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