We show that for most complexity classes of interest, all sets complete under first-order projections are isomorphic under first-order isomorphisms. That is, a very restricted version of the Berman-Hartmanis Conjecture holds.

/pdf/a-first-order-isomorphism-theorem-4ofhil0ski.pdf

A First-Order Isomorphism Theorem

An equivalence class with respect to a given type of reduction is called a degree. It is natural to expect that using more flexible reduction types will increase the size of a degree. When using more flexible reduction types fails to increase the size of a degree, this is described as a collapsing degree; for example, it is known that all many-one-complete sets for exponential time are indeed equivalent with respect to many-one length-increasing reductions.

/pdf/collapsing-degrees-via-strong-computation-1i95mypp8t.pdf

Collapsing degrees via strong computation

Let C be any complexity class closed under log-lin reductions. It is shown that all complete sets for C under 1-L reductions are polynomial time isomorphic to one other. It is indicated how to generalize the result to reductions computed by finite-crossing machines. >

Polynomial isomorphism of 1-L-complete sets

A 1-L function is one that is computable by a logspace Turing machine that moves its input head only in one direction. We show that there exist 1-L complete sets for PSPACE that are not 1-L isomorphic. In other words, the 1-L complete degree for PSPACE does not collapse. This contrasts a result of Allender who showed that all 1-L complete sets for PSPACE are polynomial-time isomorphic. Since all 1-L complete sets for PSPACE are equivalent under 1-L reductions that are one-one and quadratically length-increasing this also provides an example of a ≤ 1,qli 1−L -degree that does not collapse to a single 1-L isomorphism type.

The Degree Structure of 1-L Reductions

An equivalence class with respect to a given type of reduction is called a degree. It is natural to expect that using more flexible reduction types will increase the size of a degree. When using more flexible reduction types fails to increase the size of a degree, this is described as a collapsing degree; for example, it is known that M1 many-one-complete sets for exponential time are indeed equivalent with respect to many-one length-increasing reductions. Over the years, powerful techniques have been developed to collapse nondeterministic degrees at and above nondeterministic linear space, and Immerman and Szelepcs~nyi have provided techniques that collapse even sublinear nondeterministic space classes. However, it has remained an open problem whether any collapses could be proven for sublinear nondeterministic space degrees. This paper provides the first such collapses. For nondeterministic space classes C above NL, we show that all <~L-complete sets for C collapse to a single _<~l "L degree (i.e., all _<~L-complete sets for C are <~fi-equivalent), and that all <~NL-complete sets for C are NL-isomorphic (and thus P-isomorphic). Our techniques also improve previous results for PSPACE.

Albrecht Hoeno

Papers

Collapsing degrees via strong computation