Inverting the Turing jump in complexity theory

Abstract

The paper investigates the invertibility of certain analogs of the Turing jump operator in the polynomial-time Turing degrees. If /spl Cscr/ is some complexity class, the /spl Cscr/-jump of a set A is the canonical /spl Cscr/-complete set relative to A. It is shown that the PSPACE-jump and EXP-jump operators are not invertible, i.e., there is a PSPACE-hard (resp. EXP-hard) set that is not p-time Turing equivalent to the PSPACE-jump (resp. EXP-jump) of any set. It is also shown that if PH collapses to /spl Sigma//sub k//sup p/, then the /spl Sigma//sub k//sup p/-jump is not invertible. In particular, if NP=co-NP, then the NP-jump is not invertible, witnessed, in particular, by G/spl oplus/SAT, where G is any 1-generic set. These results run contrary to the Friedberg (1957) Completeness Criterion in recursion theory, which says that every (recursive) Turing degree above 0' is the Turing jump of another degree. The sets used in the paper to witness /spl Cscr/-jump noninvertibility are all of the form G/spl oplus/C, cohere G is 1-generic and C is some /spl Cscr/-complete set. Other facts regarding 1-generics G and G/spl oplus/SAT are also explored. In particular, G always lies in NP/sup A/-P/sup A/ for some A/spl les/G/sub tt//sup P/, but if A/spl les//sub /spl oplus/P-tt//sup p/ G/sub /spl oplus//SAT/spl les//sub T//sup P/SAT/sup A/ for some A, then G/spl les//sub T//sup P/A/spl oplus/SAT, which in turn implies either G/spl les//sub T//sup P/ A or P/spl ne/NP.