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Register a new useregg: Fast and Extensible Equality Saturation
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Added by
Max Willsey
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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An e-graph efficiently represents a congruence relation over many expressions. Although they were originally developed in the late 1970s for use in automated theorem provers, a more recent technique known as equality saturation repurposes e-graphs to implement state-of-the-art, rewrite-driven compiler optimizations and program synthesizers. However, e-graphs remain unspecialized for this newer use case. Equality saturation workloads exhibit distinct characteristics and often require ad-hoc e-graph extensions to incorporate transformations beyond purely syntactic rewrites. This work contributes two techniques that make e-graphs fast and extensible, specializing them to equality saturation. A new amortized invariant restoration technique called rebuilding takes advantage of equality saturations distinct workload, providing asymptotic speedups over current techniques in practice. A general mechanism called e-class analyses integrates domain-specific analyses into the e-graph, reducing the need for ad hoc manipulation. We implemented these techniques in a new open-source library called egg. Our case studies on three previously published applications of equality saturation highlight how eggs performance and flexibility enable state-of-the-art results across diverse domains.
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Many compilers, synthesizers, and theorem provers rely on rewrite rules to simplify expressions or prove equivalences. Developing rewrite rules can be difficult: rules may be subtly incorrect, profitable rules are easy to miss, and rulesets must be rechecked or extended whenever semantics are tweaked. Large rulesets can also be challenging to apply: redundant rules slow down rule-based search and frustrate debugging. This paper explores how equality saturation, a promising technique that uses e-graphs to apply rewrite rules, can also be used to infer rewrite rules. E-graphs can compactly represent the exponentially large sets of enumerated terms and potential rewrite rules. We show that equality saturation efficiently shrinks both sets, leading to faster synthesis of smaller, more general rulesets. We prototyped these strategies in a tool dubbed ruler. Compared to a similar tool built on CVC4, ruler synthesizes 5.8X smaller rulesets 25X faster without compromising on proving power. In an end-to-end case study, we show ruler-synthesized rules which perform as well as those crafted by domain experts, and addressed a longstanding issue in a popular open source tool.
Recent program synthesis techniques help users customize CAD models(e.g., for 3D printing) by decompiling low-level triangle meshes to Constructive Solid Geometry (CSG) expressions. Without loops or functions, editing CSG can require many coordinated changes, and existing mesh decompilers use heuristics that can obfuscate high-level structure. This paper proposes a second decompilation stage to robustly shrink unstructured CSG expressions into more editable programs with map and fold operators. We present Szalinski, a tool that uses Equality Saturation with semantics-preserving CAD rewrites to efficiently search for smaller equivalent programs. Szalinski relies on inverse transformations, a novel way for solvers to speculatively add equivalences to an E-graph. We qualitatively evaluate Szalinski in case studies, show how it composes with an existing mesh decompiler, and demonstrate that Szalinski can shrink large models in seconds.
One of the major optimizations employed in deep learning frameworks is graph rewriting. Production frameworks rely on heuristics to decide if rewrite rules should be applied and in which order. Prior research has shown that one can discover more optimal tensor computation graphs if we search for a better sequence of substitutions instead of relying on heuristics. However, we observe that existing approaches for tensor graph superoptimization both in production and research frameworks apply substitutions in a sequential manner. Such sequential search methods are sensitive to the order in which the substitutions are applied and often only explore a small fragment of the exponential space of equivalent graphs. This paper presents a novel technique for tensor graph superoptimization that employs equality saturation to apply all possible substitutions at once. We show that our approach can find optimized graphs with up to 16% speedup over state-of-the-art, while spending on average 48x less time optimizing.
We investigate the termination problem of a family of multi-path polynomial programs (MPPs), in which all assignments to program variables are polynomials, and test conditions of loops and conditional statements are polynomial equalities. We show that the set of non-terminating inputs (NTI) of such a program is algorithmically computable, thus leading to the decidability of its termination. To the best of our knowledge, the considered family of MPPs is hitherto the largest one for which termination is decidable. We present an explicit recursive function which is essentially Ackermannian, to compute the maximal length of ascending chains of polynomial ideals under a control function, and thereby obtain a complete answer to the questions raised by Seidenberg. This maximal length facilitates a precise complexity analysis of our algorithms for computing the NTI and deciding termination of MPPs. We extend our method to programs with polynomial guarded commands and show how an incomplete procedure for MPPs with inequality guards can be obtained. An application of our techniques to invariant generation of polynomial programs is further presented.
At fast rotation rates the coronal activity of G- and K-type stars has been observed to saturate and then decline again at even faster rotation rates -- a phenomenon dubbed super-saturation. In this paper we investigate coronal activity in fast-rotating M-dwarfs using deep XMM-Newton observations of 97 low-mass stars of known rotation period in the young open cluster NGC 2547, and combine these with published X-ray surveys of low-mass field and cluster stars of known rotation period. Like G- and K-dwarfs, we find that M-dwarfs exhibit increasing coronal activity with decreasing Rossby number N_R, the ratio of period to convective turnover time, and that activity saturates at L_x/L_bol ~ 10^-3 for log N_R < -0.8. However, super-saturation is not convincingly displayed by M-dwarfs, despite the presence of many objects in our sample with log N_R < -1.8, where super-saturation is observed to occur in higher mass stars. Instead, it appears that a short rotation period is the primary predictor of super-saturation; P <=0.3d for K-dwarfs and perhaps P <=0.2d for M-dwarfs. These observations favour the centrifugal stripping model for super-saturation, where coronal structures are forced open or become radiatively unstable as the Keplerian co-rotation radius moves inside the X-ray emitting coronal volume.
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