Reductions among polynomial isomorphism types

Abstract

A set A is polynomial many-one reducible to a set B ( A is Karp-reducible to B) if there is a polynomially computable function f such that, for all x, x ϵ A iff f( x) ϵ B. Arbitrary sets A and B are of the same polynomial many-one degree if each is polynomial many-one reducible to the other. A and B are (polynomially) isomorphic if the function f can be taken one-to-one, onto, and polynomially invertible. In classical recursive function theory, all many-one complete sets are recursively isomorphic. Berman and Hartmanis have observed that all known NP-complete sets are polynomially isomorphic, Berman and Hartmanis have observed that all known NP-complete sets are polynomially isomorphic, and have conjectured that all NP-complete sets (complete under Karp-reducibility) are isomorphic. In this paper we show that not just the complete degree, but every polynomial many-one degree consists either of a single isomorphism type or else contains infinitely many isomorphism types densely ordered under one-one, size-increasing, polynomially invertible reductions and also contains infinitely many isomorphisms types which are incomparable under one-one invertible reductions. In fact, we show that every countable partial ordering can be embedded in any such many-one degree. We also exhibit polynomial degrees which have infinitely many isomorphism types. No examples are known of degrees consisting of a single isomorphism type.