Metatheory.jl: Fast and Elegant Algebraic Computation in Julia with Extensible Equality Saturation

Abstract

We introduce Metatheory.jl: a lightweight and performant general purpose symbolics and metaprogramming framework meant to simplify the act of writing complex Julia metaprograms and to significantly enhance Julia with a native term rewriting system, based on state-of-the-art equality saturation techniques, and a dynamic first class AST pattern matching system that is dynamically composable in an algebraic fashion, taking full advantage of the language's powerful reflection capabilities. Our contribution allows to perform general purpose symbolic mathematics, manipulation, optimization, synthesis or analysis of syntactically valid Julia expressions with a clean and concise programming interface, both during compilation or execution of programs.

Symbolic-Numeric Integration of Univariate Expressions based on Sparse Regression

“…On the lowest level, TermInterface.jl and Metatheory.jl packages [17,10,16] provide an efficient expression rewriting engine and rule definition eDSL. Rules are written as regular Julia expressions, thus parsed into S-expressions and then efficiently compiled to callable functions that perform pattern matching and support advanced conditional selection [18,10,16].…”

Automated Code Optimization with E-Graphs

“…• chapter 2 introduces the package Metatheory.jl [8], designed to provide generic term rewriting utilities for Julia code and symbolic expressions, and we are going to talk about its role in a redesign of the Julia symbolic mathematics package ecosystem. We then describe how Metatheory.jl provides a pure-Julia implementation of a novel technique called e-graph rewriting [58] that generalizes an algorithm called equality saturation [53], in order to provide a new abstraction over term rewriting that supports equational rewrite rules and is thus particularly suitable for high-level domain specific compiler optimizations and symbolic mathematics.…”

“…The key focus of this work resides in the innovative expression rewriting features offered by the Metatheory.jl [8] package, discussed in section 2.3, specifically term rewriting and code analysis through equality graphs, detailed in subsection 2.3.2.…”

“…The main software contribution to this work consists in the Julia package Metatheory.jl 3 [8]. Metatheory.jl is a general purpose term rewriting, metaprogramming and algebraic computation library for the Julia programming language, designed to take advantage of the powerful reflection capabilities to bridge the gap between symbolic mathematics, abstract interpretation, equational reasoning, optimization, composable compiler transforms, and advanced homoiconic pattern matching features.…”

“…This novel technique allows users to define efficient term-rewriting systems with equational rules without having to worry about termination or the ordering of rules, resulting in algebraically compositional rewrite systems and efficient algorithms for minimizing cost functions on chains of expression rewrites. The Metatheory.jl [8] Julia package provides a generic rewriting backend relying on the equality saturation algorithm and data structures called e-graphs [58]. For defining e-graphs, associated concepts and procedures we adopt the naming conventions and syntax introduced in [58] and [62].…”