An Overview of -calculus Sharing Graph Reduction

Reduction Stefano Guerrini July 20, 1999 In the -calculus, the -reduction ( x:MN) ! M[N=x] replaces each occurrence of the bound variable x in M by a copy of the argument N. This operation could obviously duplicate work, since any reduction required to simplify the argument will be repeated on each copy. At rst glance, it could seem that by choosing an innermost reduction strategy (i.e., reducing arguments rst) we would easily solve this problem, simply because we would avoid copying redexes. Unfortunately, this is not true. First of all, we should also avoid reducing useless redexes (i.e., by the standardization theorem, redexes not reduced in the leftmost-outermost reduction), but they cannot be found e ectively in the calculus. Moreover, and this is the crucial point, even a reduction strategy that always reduces needed internal redexes is not guaranteed to reach the normal form (if it exists) in a minimal number of steps. As a matter of fact, in the case of the I-calculus, where all redexes are needed for computing the normal form, innermost reductions cannot be optimal.