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Consider an inductive definition of a version of Ackermann$$s function. This function was named after Wilhelm Ackermann, a German mathematician who was a student of the great mathematician David Hilbert. Ackermann$$s function plays an important role in the theory of recursive functions and in the study of the complexity of certain algorithms involving set unions. $($There are several different variants of this function. All are called Ackermann$$s function and have similar properties even though their values do not always agree.$)$