Optimal code generation for expression trees: an application BURS theory

  • Authors:

  • E. Pelegrí-Llopart;S. L. Graham

  • Affiliations:

  • Computer Science Division, EECS Department, University of California, Berkeley;Computer Science Division, EECS Department, University of California, Berkeley

  • Venue:
  • POPL '88 Proceedings of the 15th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
  • Year:
  • 1988

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Abstract

A Rewrite System is a collection of rewrite rules of the form &agr; &bgr; where &agr; and &bgr; are tree patterns. A rewrite system can be extended by associating a cost with each rewrite rule, and by defining the cost of a rewrite sequence as the sum of the costs of all the rewrite rules in the sequence. The REACHABILITY problem for a rewrite system R is, given an input tree T and a fixed goal tree G, to determine if there exists a rewrite sequence in R, rewriting T into G and, if so, to obtain one such sequence. The C-REACHABILITY problem is similar except that the obtained sequence must have minimal cost among all those sequences writing T into G.This paper introduces a class of rewrite systems called Bottom-Up Rewrite Systems (BURS), and a table-driven algorithm to solve REACHABILITY for member of the class. This algorithm is then modified to solve C-REACHABILITY and specialized for a subclass of BURS so that all cost manipulation is encoded into the tables and is not performed explicitly at solving time. The subclass extends the simple machine grammars [AGH84], rewrite systems used to describe target machine architectures for code generation, by allowing additional types of rewrite rules such as commutativity transformations.A table-driven code generator based on solving C-REACHABILITY has been implemented and tested with several machine descriptions. The code generator solves C-REACHABILITY faster than a comparable solver based on Graham-Glanville techniques [AGH84] (a non-optimal technique), yet requires only slightly larger tables. The code generator runs much faster than recent proposals to solve C-REACHABILITY that use pattern matching and deal with costs explicitly at solving time [AGT86, HeD87, WeW86]. The BURS theory generalizes and unifies the bottom-up approaches of Henry/Damron [HeD87] and Weisgerber/Wilhelm [WeW86].