We are very proud and honoured to dedicate this volume to Corrado Böhm, who has been for us a teacher, mentor, colleague, and friend. But most of all, he was a brilliant role-model to follow.

The concept of syntactical duality is central in logic. In particular, the duality defined by classical negation, or more syntactically by left and right in sequents, has been widely used to relate logic and computations. We study the proof/test duality proposed by Girard in his 1999 paper on the meaning of logical rules. In detail, starting from the notion of “test” proposed by Girard, we develop a notion of test for intuitionistic logic and we give a complete deductive system whose computational interpretation is the target language of the call-by-value and call-by-name continuation passing style translations.

We reformulate Danos contractibility criterion in terms of a sort of unification. As for term unification, a direct implementation of the unification criterion leads to a quasi-linear algorithm. Linearity is obtained after observing that the disjoint-set union-find at the core of the unification criterion is a special case of union-find with a real linear time solution.