In computer science, an e-graph is a data structure that stores an equivalence relation over terms of some language.
Let be a set of uninterpreted functions, where is the subset of consisting of functions of arity . Let be a countable set of opaque identifiers that may be compared for equality, called e-class IDs. The application of to e-class IDs is denoted and called an e-node.
The e-graph then represents equivalence classes of e-nodes, using the following data structures:
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 are congruent when . The hashcons invariant states that the hashcons maps canonical e-nodes to their e-class ID.
E-graphs expose wrappers around the , , and 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 where
There is a directed edge from each e-class to each of its members, and from each e-node to each of its children.
Let be a set of variables and let 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, is the smallest set such that , , and when and , then . A term containing variables is called a pattern, a term without variables is called ground.
An e-graph represents a ground term if one of its e-classes represents . An e-class represents if some e-node does. An e-node represents a term if and each e-class represents the term ( in ).
e-matching is an operation that takes a pattern and an e-graph , and yields all pairs where is a substitution mapping the variables in to e-class IDs and is an e-class ID such that the term is represented by . 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.
Given an e-class and a cost function that maps each function symbol in 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.
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.
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 and linear algebra. 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.