2024/05/29 by Gozzi Riccardo, Gozzi, Riccardo, Olivier Bournez +1 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Fuzzy Systems and Optimization #Logic in Computer Science (cs.LO)
paper · pdf · doi:10.48550/arxiv.2405.19304
openalex publication_date 2024/05/29 · openalex created_date 2024/05/31 · openalex updated_date 2026/07/28
In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique evolution. They correspond to a class of systems for which a transfinite method exist to compute the solution. 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 solvable functions 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.