Collapsing Degrees via Strong Computation

Though Berman and others have provided powerful techniques to collapse nondeterministic degrees at and above nondeterministic linear space, and Immerman and Szelepcsenyi have provided techniques that collapse even sublinear nondeterministic space classes, 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 S:';L-complete sets for C collapse to a single s[;f degree (Le., all S:,;L-complete sets for Care S[i.L-equivalent), and that all S:';NL_ complete sets for Care NL-isomorphic (and thus P-isomorphic). Our techniques sharply improve previous results for PSPACE.