On the number of P-isomorphism classes of NP-complete sets

All known NP-complete sets are P-isomorphic (i.e. there are polynomial time, one-to-one and onto, polynomial time invertible reductions between any two known NP-complete sets) [BH]. If all NP-complete sets are P-isomorphic, then. P ≠ NP. However it is not known if the existence of more than one P-isomorphism class of NP-complete sets has implications for the P = NP? problem. In the main result of this paper we prove: Theorem: If there is an NP-complete set that is not P-isomorphic to SAT, then there are infinitely many NP-complete sets that are mutually non-P-isomorphic. Thus, the number of P-isomorphism classes of NP-complete sets is either one or (countably) infinite. Two proof techniques are developed in this paper: we use delayed diagonalization [BCH, L] to construct new sets that are not P-isomorphic to existing sets; the diagonalization conditions are used to defeat P-isomorphisms. We also examine certain properties of 'generic' NP-complete sets and introduce techniques based on padding functions to assure that the sets constructed will be NP-complete. The results on P-isomorphisms and constructing non-P-isomorphic sets apply also to sets complete for PTAPE, EXPTIME, and EXPTAPE and other classes. We also examine the structure of NP-complete sets based on size increasing, invertible reductions, The degrees are P-isomorphism classes [BH]. We show that if there is more than one degree, then there is an ω chain of degrees with SAT representing a maximal element.