Three Syntactic Theories for Combinatory Graph Reduction

We present a purely syntactic theory of graph reduction for the canonical combinators S, K, and I, where graph vertices are represented with evaluation contexts and let expressions. We express this syntactic theory as a reduction semantics. We then factor out the introduction of let expressions to denote as many graph vertices as possible upfront instead of on demand, resulting in a second syntactic theory, this one of term graphs in the sense of Barendregt et al. We then interpret let expressions as operations over a global store (thus shifting, in Strachey's words, from denotable entities to storable entities), resulting in a third syntactic theory, which we express as a reduction semantics. This store-based reduction semantics corresponds to a store-based abstract machine whose architecture coincides with that of Turner's original reduction machine. The three syntactic theories presented here therefore properly account for combinatory graph reduction As We Know It.