An algebraic array shape inference system for MATLAB®

Abstract

The problem of inferring array shapes ahead of time in languages that exhibit both implicit and dynamic typing is a critical one because the ramifications of its solution are the better organization of array storage through compaction and reuse, and the generation of high-performance code through specialization by shape. This article addresses the problem in a prototypical implicitly and dynamically typed array language called MATLAB. The approach involves modeling the language's shape semantics using an algebraic system, and applying term rewriting techniques to evaluate expressions under this algebra. Unlike prior efforts at array shape determination, this enables the deduction of valuable shape information even when array extents are compile-time unknowns. Furthermore, unlike some previous methods, our approach doesn't impose monotonicity requirements on an operator's shape semantics. The work also describes an inference methodology and reports measurements from a type inference engine called . In a benchmark suite of 17 programs, the shape inference subsystem in detected the equivalence of over 61% of the symbolic shapes in six programs, and over 57% and 37% of the symbolic shapes in two others. In the remaining nine programs, all array shapes were inferred to be compile-time constants.

Just-in-time shape inference for array-based languages

“…The MAGICA system [8] was a static compiler that statically inferred shapes through a symbolic shape inference algorithm. Our approach is quite different from MAGICA because we perform shape inference and specialization at runtime, and take into account program information obtained at runtime.…”

A Dimension Abstraction Approach to Vectorization in Matlab

“…McCosh finds sets of legal type configurations using propositional logic constructs and infers the size of the variables by formulating this problem as a special version of the clique problem which she solves in polynomial time [15]. Joisha et al present a framework to infer array shape and even allow certain static inferences to be made when complete static knowledge is lacking, by establishing what useful inferences can be made regarding the shapes of statically indeterminable arrays [11,10]. These techniques could be integrated with our vectorizer to deliver an end-to-end solution to the vectorization of Matlab programs.…”

Library Function Selection in Compiling Octave

“…One way to address the performance problem that continues to hinder large scale application development in these languages is to compile these languages to a relatively lower-level language for which there are known good compilers [7,11,3]. Usually, the first step in this translation process is inferring types of the variables from their definitions and uses so that the individual high-level operations in the original program * The work in this paper was supported in part by the National Science could be mapped to their more precise equivalent operations.…”

“…), and ψ is the size for array variables (including the number of dimensions and the extent along each dimension). Various other pieces of work have addressed the issue of inferring types in MATLAB, and equivalently Octave, and that will not be discussed further in this paper [7,2,11]. The work described here assumes the availability of the variable types.…”

“…Other recent attempts at compiling MATLAB have largely focused on type inference and specialization, rather than efficient generation of calls to the underlying libraries [7,11,3]. However, it is worth comparing our approach with the techniques that were explored in an extension of the FALCON system at the University of Illinois [7,12].…”