Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions

In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition x + y x+y , multiplication x ⋅ y x \cdot y , and two-place exponentiation x y x^y . For a restricted class of such functions, we show that every true identity follows from the natural set of eleven axioms. The rates of growth of such functions, in the case of a single independent variable x x , as x → ∞ x \to \infty , are also studied, and we give an algorithm for the Hardy relation of eventual domination, again for a restricted class of functions. Value distribution of analytic functions of one and of several complex variables, especially the Nevanlinna characteristic, plays a major role in our proofs.