On the computational complexity of algebraic numbers : the Hartmanis-Stearns problem revisited

Boris Adamczewski (1, 2) , Julien Cassaigne (3) , Marion Le Gonidec (4)

1 ICJ - Institut Camille Jordan
2 CTN - Combinatoire, théorie des nombres
3 I2M - Institut de Mathématiques de Marseille
4 LIM - Laboratoire d'Informatique et de Mathématiques

— We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis–Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-b expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one.

Mots clés: Hartmanis-Stearns problem; pushdown automata; transcendence; stack machines; integer base expansions; tag machines

Domaines: Théorie des nombres [math.NT]; Combinatoire [math.CO]; Théorie et langage formel [cs.FL]

Boris Adamczewski, Julien Cassaigne, Marion Le Gonidec. On the computational complexity of algebraic numbers : the Hartmanis-Stearns problem revisited. Transactions of the American Mathematical Society, 2020, 373, pp.3085-3115. ⟨10.1090/tran/7915⟩. ⟨hal-01254293⟩