Polynomial Terse Sets

Abstract Let A be a set and k ∈ N be such that we wish to know the answers to x1 ∈ A?, x2 ∈ A?, …, xk ∈ A? for various k-tuples 〈x1, x2, …, xk〉. If this problem requires k queries to A in order to be solved in polynomial time then A is called polynomial terse or pterse. We show the existence of both arbitrarily complex pterse and non-pterse sets; and that P ≠ NP iff every NP-complete set is pterse. We also show connections with p-immunity, p-selective, p-generic sets, and the boolean hierarchy. In our framework unique satisfiability (and a variation of it called kSAT is, in some sense, “close” to satisfiability.

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