Undecidable Extensions of Büchi Arithmetic and Cobham-Semënov Theorem

International audienceno abstrac

Undecidable Extensions of Büchi Arithmetic and Cobham-Semënov Theorem

Let k and l be two multiplicatively independent integers, and let L IN be a l recognizable set which is not de nable in hIN; +i. We prove that the elementary theory of hIN; +; Vk ; Li, where Vk (x) denotes the greatest power of k dividing x, is undecidable. This result leads to a new proof of the Cobham-Sem\u7fenov theorem. 1