Completeness for nondeterministic complexity classes

We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomial-time bounded (even logarithmic-space bounded) reducibilities turn out to be different for any class containingNE. For space classes the completeness notions under logspace reducibilities can be separated for any class properly containingLOGSPACE. Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes.The work of S. Homer was supported in part by National Science Foundation Grants MIP-8608137 and CCR-8814339 and a Fulbright-Hays Research Fellowship. Some of this research was done while he was a Guest Professor at the Mathematics Institute of Heidelberg University.