We investigate the autoreducibility and mitoticity of complete sets for several classes with respect to different polynomial-time and logarithmic-space reducibility notions.

Previous work in this area focused on polynomial-time reducibility notions. Here we obtain new mitoticity and autoreducibility results for the classes EXP and NEXP with respect to some restricted truth-table reductions (e.g., polynomial-time 2-tt, ctt, dtt reducibility).

Moreover, we start a systematic study of logarithmic-space autoreducibility and mitoticity, which enables us to also consider P and smaller classes. Among others, we obtain the following results:- Regarding log-space many-one, 2-tt, dtt and ctt reducibility, complete sets for PSPACE and EXP are mitotic, and complete sets for NEXP are autoreducible.- All log-space 1-tt-complete sets for NL and P are log-space 2-tt-autoreducible, and all log-space btt-complete sets for NL, P and the delta-levels of the PH are log-space Turing-autoreducible.- There is a log-space 3-tt-complete set for PSPACE that is not even log-space btt-autoreducible.Using the last result, we conclude that some of our results are hard or even impossible to improve