On Inverting Onto Functions
نویسندگان
چکیده
We study the complexity of inverting many-one, honest, polynomial-time computable onto functions. Asserting that every polynomial-time computable, honest, onto function is invertible is equivalent to the following proposition that we call Q: For all NP machines M that accept , there exists a polynomial-time computable function gM such that for all x, gM(x) outputs an accepting computation of M on x. We show that Q is equivalent to several well-studied propositions in complexity theory. For example, we show that Q is equivalent to the proposition that, for all NP machines M that accept SAT , there exists a polynomial-time algorithm gM that transforms any accepting computation of M on input x into a satisfying assignment of x. We compare Q with its following weaker version that we call Q0: for all NP machines accepting there is a polynomial-time computable function gM that computes the rst bit of an accepting computation of M . As a rst step in comparing Q and Q0, we show that if every 0-1-valued total NPMV function has poly-time computable re nements, then for all k 0, every k-valued total NPMV function has re nements in PF. We relate both Q and Q0 to the question of whether the class NPMVt has re nements in TFNP, a class of functions studied by Beame et al. Finally, we study the relationship of Q and Q0 with other complexity hypothesis. We show that Q0 implies that AM \ coAM BPP, and NP \ coAM RP. Also, Q0 and NP = UP implies that the polynomial hierarchy collapses to ZPPNP, and Q implies that every one-one paddable degree collapses to a one-one length-increasing degree.
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