Glaßer et al. (SIAMJCOMP 2008 and TCS 2009 (The two papers have slightly different sets of authors)) proved existence of two sparse sets A and B in EXP, where A is 3-tt (truth-table) polynomial-time autoreducible but not weakly polynomial-time Turing mitotic and B is polynomial-time 2-tt autoreducible but not weakly polynomial-time 2-tt mitotic. We unify and strengthen both of those results by showing that there is a sparse set in EXP that is polynomial-time 2-tt autoreducible but not even weakly polynomial-time Turing mitotic. All these results indicate that polynomial-time autoreducibilities in general do not imply polynomial-time mitoticity at all with the only exceptions of the many-one and 1-tt reductions. On the other hand, however, we proved that every autoreducible set for the polynomial-time bounded disjunctive or conjunctive tt reductions is weakly mitotic for the polynomial-time tt reduction that makes logarithmically many queries only. This shows that autoreducible sets for reductions making more than one query could still be mitotic in some way if they possess certain special properties.
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