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

Abstract

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., $\leq^{p}_{2-tt},\leq^{p}_{ctt},\leq^{p}_{dtt}$).

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 $\leq^{log}_{m}, \leq^{log}_{2-tt}, \leq^{log}_{dtt}$ and $\leq^{log}_{ctt}$, complete sets for PSPACE and EXP are mitotic, and complete sets for NEXP are autoreducible.

• All $\leq^{log}_{1-tt}$-complete sets for NL and P are $\leq^{log}_{2-tt}$-autoreducible, and all $\leq^{log}_{btt}$-complete sets for NL, P and $\Delta^{p}_{k}$ are $\leq^{log}_{log-T}$-autoreducible.

• There is a $\leq^{log}_{3-tt}$-complete set for PSPACE that is not even $\leq^{log}_{btt}$-autoreducible.

Using the last result, we conclude that some of our results are hard or even impossible to improve.

Please see the technical report [9] for proofs omitted due to space restrictions.