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We present an algorithm that canonicalizes the algebraic representations of the topological semantics of machine knitting programs. Machine knitting is a staple technology of modern textile production where hundreds of mechanical needles are manipulated to form yarn into interlocking loop structures. Our semantics are defined using a variant of a monoidal category, and they closely correspond to string diagrams. We formulate our canonicalization as an Abstract Rewriting System (ARS) over words in our category, and prove that our algorithm is correct and runs in polynomial time. 

Programming low-level controls for knitting machines is a meticulous, time-consuming task that demands specialized expertise. Recently, there has been a shift towards automatically generating low-level knitting machine programs from high-level knit representations that describe knit objects in a more intuitive, user-friendly way. Current high-level systems trade off expressivity for ease-of-use, requiring ad-hoc trapdoors to access the full space of machine capabilities, or eschewing completeness in the name of utility. Thus, advanced techniques either require ad-hoc extensions from domain experts, or are entirely unsupported. Furthermore, errors may emerge during the compilation from knit object representations to machine instructions. While the generated program may describe a valid machine control sequence, the fabricated object is topologically different from the specified input, with little recourse for understanding and fixing the issue. To address these limitations, we introduce instruction graphs, an intermediate representation capable of capturing the full range of machine knitting programs. We define a semantic mapping from instruction graphs to fenced tangles, which make them compatible with the established formal semantics for machine knitting instructions. We establish a semantics-preserving bijection between machine knittable instruction graphs and knit programs that proves three properties - upward, forward, and ordered (UFO) - are both necessary and sufficient to ensure the existence of a machine knitting program that can fabricate the fenced tangle denoted by the graph. As a proof-of-concept, we implement an instruction graph editor and compiler that allows a user to transform an instruction graph into UFO presentation and then compile it to a machine program, all while maintaining semantic equivalence. In addition, we use the UFO properties to more precisely characterize the limitations of existing compilers. This work lays the groundwork for more expressive and reliable automated knitting machine programming systems by providing a formal characterization of machine knittability. 

High-performance kernel libraries are critical to exploiting accelerators and specialized instructions in many applications. Because compilers are difficult to extend to support diverse and rapidly-evolving hardware targets, and automatic optimization is often insufficient to guarantee state-of-the-art performance, these libraries are commonly still coded and optimized by hand, at great expense, in low-level C and assembly. To better support development of high-performance libraries for specialized hardware, we propose a new programming language, Exo, based on the principle of exocompilation: externalizing target-specific code generation support and optimization policies to user-level code. Exo allows custom hardware instructions, specialized memories, and accelerator configuration state to be defined in user libraries. It builds on the idea of user scheduling to externalize hardware mapping and optimization decisions. Schedules are defined as composable rewrites within the language, and we develop a set of effect analyses which guarantee program equivalence and memory safety through these transformations. We show that Exo enables rapid development of state-of-the-art matrix-matrix multiply and convolutional neural network kernels, for both an embedded neural accelerator and x86 with AVX-512 extensions, in a few dozen lines of code each. 

We present a lightweight Coq framework for optimizing tensor kernels written in a pure, functional array language. Optimizations rely on user scheduling using series of verified, semantics-preserving rewrites. Unusually for compilation targeting imperative code with arrays and nested loops, all rewrites are source-to-source within a purely functional language. Our language comprises a set of core constructs for expressing high-level computation detail and a set of what we call reshape operators, which can be derived from core constructs but trigger low-level decisions about storage patterns and ordering. We demonstrate that not only is this system capable of deriving the optimizations of existing state-of-the-art languages like Halide and generating comparably performant code, it is also able to schedule a family of useful program transformations beyond what is reachable in Halide.