A Note on P-Selective Sets and Closeness

Abstract We investigate the class of sets that form sparse symmetric differences with P-selective sets. Intuitively, this class (denoted by PSEL-close) comprises of sets that can, in a certain sense, be approximated by P-selective sets. A primary motivation behind the introduction of this new class is to unify the separate approaches that have been undertaken for sparse and P-selective sets. In order to establish PSEL-close as a distinct class, we first prove a theorem separating it from both the encompassing class of P/poly and the subclasses of P-selective and sparse sets. We then prove that no ⩽mp-hard set for E can be in PSEL-close. The proof of this theorem relies on techniques from the work of Berman and Hartmanis (1977) and Schoning (1986), and generalizes their results in a straightforward manner.

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