Classes of equational programs that compile into efficient machine code

Abstract

Huet and Lévy [HL79] showed that, if an equational program E is strongly sequential, there exists an automaton that, given a term in the language L(E), finds a redex in that term.

The most serious problem with their approach becomes evident when one tries to use their result in a programming system. Once a redex has been found, it must be replaced by a term built from the structure of the right-hand side corresponding to the redex, and from parts of the old term. Then, the reduction process must be restarted so that other redexes can be found. With their approach, a large part of the term tree may have to be rescanned.

Hoffmann and O'Donnell [HO82a] improved the situation by defining the class of strongly left-sequential programs. For this class, a particularly simple reduction algorithm exists. A stack is used to hold information about the state of the reduction process. When a redex has been found and replaced by the corresponding right-hand side, the stack holds all the relevant information needed to restart the reduction process in a well defined state such that no unnecessary rescanning of the term is done.

However, it turns out that the approach of Hoffmann and O'Donnell is unnecessarily restrictive. In this paper, we define a new class of Equational Programs, called the forward branching programs. This class is much larger than the class of strongly left-sequential programs. Together with a new reduction algorithm, briefly discussed in this paper, our approach allows us to use the hardware stack to hold reduction information in a way similar to the way a block structured programming language uses the stack to hold local variables. In effect, our approach allows us to use innermost stabilization, while preserving the overall outermost reduction strategy.