Baire measurable functions and automorphisms of the truth-table degrees

Abstract

We show that if an automorphism $\pi$ of the truth-table degrees is induced by a Baire measurable function $F:2^{\omega }\to 2^{\omega }$, or merely induced on some comeager set by a continuous function, then $\pi$ is the identity. The same results hold for the weak truth-table degrees. The argument is an adaptation to truth-table reducibility of OpenAI's recent proof that the Turing degrees are rigid.