Eeciency of -calculi with Explicit Substitutions

We introduce a criterion of eeciency to simulate-reduction in calculi of explicit substitutions and we apply it to several calculi: , * , , s, t and u. The latter is presented here for the rst time and may be considered as an eecient variant of s. The results of this paper imply that calculi a la s are usually more eecient at simulating-reduction than calculi in the-style. In fact, we prove that t is more eecient than and that u is more eecient than , * and s. We also give counterexamples to show that all other comparisons are impossible.