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 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:
  • A union-find structure representing equivalence classes of e-class IDs, with the usual operations, and. An e-class ID is canonical if ; an e-node is canonical if each is canonical.
  • An association of e-class IDs with sets of e-nodes, called e-classes. This consists of
  • * a hashcons from e-nodes to e-class IDs, and
  • * an e-class map that maps e-class IDs to e-classes, such that maps equivalent IDs to the same set of e-nodes:

    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 are congruent when. The hashcons invariant states that the hashcons maps canonical e-nodes to their e-class ID.

Operations

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.

Equivalent formulations

An e-graph can also be formulated as a bipartite graph where
  • is the set of e-class IDs,
  • is the set of e-nodes, and
  • 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 be a set of variables and let be the smallest set that includes the 0-arity function symbols, 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 .
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.

Extraction

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.

Complexity

  • An e-graph with n equalities can be constructed in O 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-based solvers that use conflict-driven clause learning, 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.