E-graph explained

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

Definition and operations

Let

\Sigma be a set of uninterpreted functions, where \Sigma _{n} is the subset of \Sigma consisting of functions of arity n . Let id be a countable set of opaque identifiers that may be compared for equality, called e-class IDs. The application of f\in\Sigma _{n} to e-class IDs i _{1,} i _{2,} \ldots, i _{n\inid} is denoted 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:

• A union-find structure

U representing equivalence classes of e-class IDs, with the usual operations find , add and merge . An e-class ID e is canonical if find (U, e) = e ; an e-node f(i _{1,\ldots,i} _{n)} is canonical if each i _{j} is canonical (j in 1 ,\ldots,n ).

• An association of e-class IDs with sets of e-nodes, called e-classes. This consists of

H (i.e. a mapping) from canonical 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: \forall i, j\in id ,M[i] = M[j]\Leftrightarrow find (U,i) = find (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,\ldots,i} _{n),f(j} _{1,\ldots,j} _{n)} are congruent when find (U, i _{k)=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

add , find , and merge operations from the union-find that preserve the e-graph invariants. The last operation, e-matching, is described below.

E-matching

Let

V be a set of variables and let 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, Term (\Sigma, V) is the smallest set such that V\subset Term ( \Sigma, V) , \Sigma _{0\subsetTerm(\Sigma,} V) , and when x _{1,} \ldots, x _{n\in} Term (\Sigma, V) and f\in\Sigma _{n} , then f(x _{1,\ldots,x} _{n)\inTerm(\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 Term (\Sigma, \emptyset) if one of its e-classes represents t . An e-class C represents t if some e-node f(i _{1,\ldots,i} _{n)\in} C does. An e-node f(i _{1,\ldots,i} _{n)\in} C represents a term g(j _{1,\ldots,j} _{n)} if f = g and each e-class M[i _{k]} represents the term j _{k} (k in 1 ,\ldots,n ).

e-matching is an operation that takes a pattern

p\in Term (\Sigma, V) and an e-graph E , and yields all pairs (\sigma, C) where \sigma\subset V x id is a substitution mapping the variables in p to e-class IDs and C\in id is an e-class ID such that each term \sigma(p) is represented by C . There are several known algorithms for e-matching,^[1] the relational e-matching algorithm is based on worst-case optimal joins and is worst-case optimal.^[2]

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^[3] 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.^[4]

Equality saturation is used in specialized optimizing compilers,^[5] e.g. for deep learning^[6] and linear algebra.^[7] Equality saturation has also been used for translation validation applied to the LLVM toolchain.^[8]

E-graphs have been applied to several problems in program analysis, including fuzzing,^[9] abstract interpretation,^[10] and library learning.^[11]

References

• Book: de Moura. Leonardo. Bjørner. Nikolaj. Automated Deduction – CADE-21 . Efficient E-Matching for SMT Solvers . 2007. Pfenning. Frank. https://link.springer.com/chapter/10.1007/978-3-540-73595-3_13. Lecture Notes in Computer Science. 4603. en. Berlin, Heidelberg. Springer. 183–198. 10.1007/978-3-540-73595-3_13. 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. 10.1145/3434304. 2004.03082. 226282597.
• Book: Tate. Ross. Stepp. Michael. Tatlock. Zachary. Lerner. Sorin. Proceedings of the 36th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages . Equality saturation . 2009-01-21. https://doi.org/10.1145/1480881.1480915. POPL '09. Savannah, GA, USA. Association for Computing Machinery. 264–276. 10.1145/1480881.1480915. 978-1-60558-379-2. 2138086.
• Book: Flatt . Oliver . Coward . Samuel . Willsey . Max . Tatlock . Zachary . Panchekha . Pavel . October 2022 . Small Proofs from Congruence Closure . A. Griggio . N. Rungta . Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022 . https://repositum.tuwien.at/handle/20.500.12708/81325 . en . TU Wien Academic Press . 75–83 . 10.34727/2022/isbn.978-3-85448-053-2_13 . 978-3-85448-053-2. 252118847 .

External links

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

Notes and References

1. 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) . en . 198 . 2 . 19–35 . 10.1016/j.entcs.2008.04.078 . 1571-0661. free .
2. 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 . 10.1145/3498696. 236924583 . free .
3. Book: de Moura. Leonardo. Bjørner. Nikolaj. Tools and Algorithms for the Construction and Analysis of Systems . Z3: An Efficient SMT Solver . 2008. Ramakrishnan. C. R.. Rehof. Jakob. Lecture Notes in Computer Science. 4963. en. Berlin, Heidelberg. Springer. 337–340. 10.1007/978-3-540-78800-3_24. 978-3-540-78800-3. free.
4. Detlefs . David . Nelson . Greg . Saxe . James B. . May 2005 . Simplify: a theorem prover for program checking . Journal of the ACM . 52 . 3 . 365–473 . 10.1145/1066100.1066102 . 9613854 . 0004-5411.
5. Joshi. Rajeev. Nelson. Greg. Randall. Keith. 2002-05-17. Denali: a goal-directed superoptimizer. ACM SIGPLAN Notices. 37. 5. 304–314. 10.1145/543552.512566. 0362-1340.
6. Yang. Yichen. Phothilimtha. Phitchaya Mangpo. Wang. Yisu Remy. Willsey. Max. Roy. Sudip. Pienaar. Jacques. 2021-03-17. Equality Saturation for Tensor Graph Superoptimization. cs.AI. 2101.01332.
7. 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. cs.DB. 2002.07951.
8. Book: Stepp. Michael. Tate. Ross. Lerner. Sorin. Computer Aided Verification . Equality-Based Translation Validator for LLVM . 2011. Gopalakrishnan. Ganesh. Qadeer. Shaz. Lecture Notes in Computer Science. 6806. en. Berlin, Heidelberg. Springer. 737–742. 10.1007/978-3-642-22110-1_59. 978-3-642-22110-1. free.
9. Web site: Wasm-mutate: Fuzzing WebAssembly Compilers with E-Graphs (EGRAPHS 2022) - PLDI 2022 . 2023-02-03 . pldi22.sigplan.org.
10. Coward . Samuel . Constantinides . George A. . Drane . Theo . 2022-03-17 . Abstract Interpretation on E-Graphs . cs.LO . 2203.09191 .
    Coward . Samuel . Constantinides . George A. . Drane . Theo . 2022-05-30 . Combining E-Graphs with Abstract Interpretation . cs.DS . 2205.14989 .
11. 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 . 10.1145/3571207 . 2212.04596 . 254536022 . 2475-1421.