Equality Saturation: A New Approach to Optimization

Abstract

Abstract. Optimizations in a traditional compiler are applied sequentially, with each optimization destructively modifying the program to produce a transformed program that is then passed to the next optimization. We present a new approach for structuring the optimization phase of a compiler. In our approach, optimizations take the form of equality analyses that add equality information to a common intermediate representation. The optimizer works by repeatedly applying these analyses to infer equivalences between program fragments, thus saturating the intermediate representation with equalities. Once saturated, the intermediate representation encodes multiple optimized versions of the input program. At this point, a profitability heuristic picks the final optimized program from the various programs represented in the saturated representation. Our proposed way of structuring optimizers has a variety of benefits over previous approaches: our approach obviates the need to worry about optimization ordering, enables the use of a global optimization heuristic that selects among fully optimized programs, and can be used to perform translation validation, even on compilers other than our own. We present our approach, formalize it, and describe our choice of intermediate representation. We also present experimental results showing that our approach is practical in terms of time and space overhead, is effective at discovering intricate optimization opportunities, and is effective at performing translation validation for a realistic optimizer.

Translation Validation for Synchronous Data-Flow Specification in the SIGNAL Compiler

“…On the other hand, Tate et al [15] proposed a framework for translation validation. Given a function in the input program and the corresponding optimized version of the function in the output program, they compute two value-graphs to represent the computations of the variables.…”

“…The value-graph of the computation of x is φ (= (a, b), d, 0). Replacing the definition of a, b and d, and normalizing this graph, we get: by (7) φ(c, 0, 0) by (16) 0 by (15) Optimization-specific Rules Based on the optimizations of the Signal compiler, we have a number of optimization-specific rules in a way that reflexes the effects of specific optimizations of the compiler. These rules do not always reduce the graph or make it simpler.…”

A Formally Verified SSA-Based Middle-End

“…Stepp et al [21] report on a translation validator for LLVM. Their validator uses Equality Saturation [22], which views optimizations as equality analyses. Their tool does not validate GVN.…”

Language support for dynamic, hierarchical data partitioning

“…Existing techniques for proving equivalence can be classified into three categories: sound algorithms for loop-free code [1,6,10,11,23]; algorithms that analyze finite unwindings of loops or finite spaces of inputs [17,20,28,30]; algorithms that require knowledge of the particular transformations used for turning one program into another [24,36] and the order in which the transformations have been applied [14,25,29]. In contrast, our approach to handling equivalence checking of loops does not assume any knowledge about the optimizations performed.…”

“…In particular, DDEC restricts the expressiveness of invariants required for a proof to be conjunctions of linear or nonlinear equalities. However, for most interesting intra-procedural optimizations, simple equalities appear to be sufficiently expressive [25,32,36]. DDEC is also currently unable to reason about floating point computations, simply because the current generation of offthe-shelf SMT solvers do not support floating point reasoning.…”