Arithmetical complexity of some problems in computer science
We show that the set of all (indices of) Turing machines running in time n2 is a complete π 1 0 set and that the set of all (indices of Turing machines computing characteristic functions of) recursive sets A such that PA ≠ NPA is a complete π 2 0 set. As corollaries we obtain results saying that some assertions concerning running time of Turing machines and some instances of the relativized P = NP problem are independent of set theory (or of another theory containing arithmetic).
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