Reducing Tarski to Unique Tarski (in the Black-box Model)

We study the problem of finding a Tarski fixed point over the k -dimensional grid [ n ] k . We give a black-box reduction from the Tarski problem to the same problem with an additional promise that the input function has a unique fixed point. It implies that the Tarski problem and the unique Tarski problem have exactly the same query complexity. Our reduction is based on a novel notion of partial-information functions which we use to fool algorithms for the unique Tarski problem as if they were working on a monotone function with a unique fixed point