Automorphisms in the PTIME-Turing degrees of recursive sets

Christine Ann Haught, Theodore A. Slaman
1997 Annals of Pure and Applied Logic  
We consider questions related to the rigidity of the structure W, the PTZME-Turing degrees of recursive sets of strings together with PTME-Turing reducibility, <pr , and related structures; do these structures have nontrivial automorphisms? We prove that there is a nontrivial automorphism of an ideal of 3. This can be rephrased in terms of partial relativizations. We consider the sets which are PTZME-Turing computable from a set A, and call this class PTIMEA. Our result can be stated as
more » ... There is an oracle, A, relative to which the PTIME-Turing degrees are not rigid (i.e. there is a nontrivial automorphism of the structure (PTZMEA, < "r ) ). Furthermore, the automorphism can be made to preserve the complexity classes DTZMEA(nk) (the collection of sets computable from A by a deterministic computation with time bound of order nk) for all k 2 1, or to move any DTZMEA(n") for n 3 2. From the existence of such an automorphism we conclude as a corollary that there is an oracle A relative to which the classes DTME(nk) are not definable over .R. We carry out the corresponding partial relativization for the many-one degrees to construct an oracle, A, relative to which the PTZME-many-one degrees relative to A have a nontrivial automorphism, and one relative to which the lattice of sets in PTMEA under inclusion have a nontrivial automorphism. The proof is phrased as a forcing argument; we construct the set A to meet a particular collection of dense sets in our notion of forcing. Roughly, the dense sets will guarantee that if A meets these sets and we split A into two pieces, A0 and A" in a simple way, and then switching the roles of A0 and A, in all computations from A will produce an automorphism of the ideal of PTIME-degrees below A. We force A0 and Al to have different PTIME-Turing degree; this will then make the automorphism nontrivial. An appropriately generic set A is constructed using a priority argument.
doi:10.1016/s0168-0072(95)00065-8 fatcat:63si4h43srcwrnqtp56dju5hum