# The degree structure of 1-L reductions

## Abstract

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.

Relatedly, we prove that there exist two sets *A* and *B* that are ≤ _{1} ^{1−L} equivalent but not ≤ _{1honest} ^{1−L} equivalent. That is, there is a one-one 1-L degree that is no honest one-one 1-L degree.

## Keywords

Turing Machine Random Oracle Isomorphism Type Input Tape Logarithmic Space## Preview

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