Relating graph and term rewriting via Böhm models

Abstract

Dealing properly with sharing is important for expressing some of the common compiler optimizations, such as common subexpressions elimination, lifting of free expressions and removal of invariants from a loop, as source-to-source transformations. Graph rewriting is a suitable vehicle to accommodate these concerns. In [4] we have presented a term model for graph rewriting systems (GRSs) without interfering rules, and shown the partial correctness of the aforementioned optimizations. In this paper we define a different model for GRSs, which allows us to prove total correctness of those optimizations. Differently from [4] we will discard sharing from our observations and introduce more restrictions on the rules. We will introduce the notion of Böhm tree for GRSs, and show that in a system without interfering and non-left linear rules (orthogonal GRSs), Böhm tree equivalence defines a congruence. Total correctness then follows in a straightforward way from showing that if a program M contains less sharing than a program N, then both M and N have the same Böhm tree.

We will also show that orthogonal GRSs are a correct implementation of orthogonal TRSs. The basic idea of the proof is to show that the behavior of a graph can be deduced from its finite approximations, that is, graph rewriting is a continuous operation. Our approach differs from that of other researchers [6, 9], which is based on infinite rewriting.