2023/07/21 by Manon Blanc, Blanc, Manon, Olivier Bournez +1
Computer Science · #Cellular Automata and Applications #Computability, Logic, AI Algorithms #Computational Complexity (cs.CC) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Logic (math.LO) #Logic in Computer Science (cs.LO) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2307.11747
openalex publication_date 2023/07/21 · openalex created_date 2023/07/25 · openalex updated_date 2026/07/28
We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals. In particular, this covers space complexity, while existing characterisations were only able to cover time complexity, and were restricted to functions over the integers. We prove furthermore that no artificial sign or test function is needed even for time complexity. At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity.