Set descriptive complexity of solvable functions

Abstract

In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems that present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms. These systems are interesting because, when they exhibit a unique evolution, a transfinite method always exists to define such evolution as a limit of a sequence of continuous functions. We also presented several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems might describe ordinal Turing computations. In the current article, we study in more depth solvable systems, using tools from descriptive set theory. By establishing a correspondence with the class of well-founded trees, we construct a coanalytic ranking over the set of real functions with bounded, solvable derivatives and discuss its relation with other existing rankings for differentiable functions, in particular with the Kechris–Woodin, Denjoy, and Zalcwasser ranking. We prove that our ranking is unbounded below the first uncountable ordinal.