Collapsing degrees via strong computation

Abstract

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. Over the years, powerful techniques have been developed to collapse non-deterministic degrees at and above nondeterministic linear space, and Immerman and Szelepcsényi 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 (formula presented)-complete sets for C collapse to a single (formula presented) degree (i.e., all (formula presented)-complete sets for C are (formula presented)-equivalent), and that all (formula presented) -complete sets for C are NL-isomorphic (and thus P-isomorphic). Our techniques also improve previous results for PSPACE.