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:
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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.
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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.
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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.
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Please see the technical report [9] for proofs omitted due to space restrictions.
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Glaßer, C., Nguyen, D.T., Reitwießner, C., Selman, A.L., Witek, M. (2013). Autoreducibility of Complete Sets for Log-Space and Polynomial-Time Reductions. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_40
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DOI: https://doi.org/10.1007/978-3-642-39206-1_40
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