Combinator graph reduction

In computer science, graph reduction implements an efficient version of non-strict evaluation, an evaluation strategy where the arguments to a function are not immediately evaluated. This form of non-strict evaluation is also known as lazy evaluation and used in functional programming languages. The technique was first developed by Chris Wadsworth in 1971.

A simple example of evaluating an arithmetic expression follows:

The above reduction sequence employs a strategy known as outermost tree reduction. The same expression can be evaluated using innermost tree reduction, yielding the reduction sequence:

Notice that the reduction order is made explicit by the addition of parentheses. This expression could also have been simply evaluated right to left, because addition is an associative operation.

Represented as a tree, the expression above looks like this:

This is where the term tree reduction comes from. When represented as a tree, we can think of innermost reduction as working from the bottom up, while outermost works from the top down.

The expression can also be represented as a directed acyclic graph, allowing sub-expressions to be shared:

As for trees, outermost and innermost reduction also applies to graphs. Hence we have graph reduction.

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