A Lightweight Double-negation Translation

Deciding whether a classical theorem can be proved constructively is a well-known undecidable problem. As a consequence, any computable double-negation translation inserts some unnecessary double negations. This paper shows that most of these unnecessary insertions can be avoided without any use of constructive proof search techniques. For this purpose, we restrict the analysis to syntax-directed double-negation translations, which translate a proposition through a single traversal – and include most of the usual translations such as Kolmogorov's, Godel-Gentzen's, and Kuroda's. A partial order among translations are presented to select translations avoiding as many double negations as possible. This order admits a unique minimal syntax-directed translation with noticeable properties.