Autoreducibility of Complete Sets for Log-Space and Polynomial-Time Reductions

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., ≤p2-tt,≤ p ctt,≤ p dtt). 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 ≤ m , ≤ log 2-tt, ≤ log dtt and ≤ log ctt , complete sets for PSPACE and EXP are mitotic, and complete sets for NEXP are autoreducible. • All ≤ 1-tt-complete sets for NL and P are ≤ log 2-tt-autoreducible, and all ≤ log btt-complete sets for NL, P and ∆pk are ≤ log log-T-autoreducible. • There is a ≤ 3-tt-complete set for PSPACE that is not even ≤ log btt-autoreducible. Using the last result, we conclude that some of our results are hard or even impossible to improve.