DSPACE(n)=?NSPACE(n): a degree theoretic characterization

It is shown that the following are equivalent: 1. DSPACE(n)=NSPACE(n). 2. There is a /spl les//sub m//sup 1-NL/-degree that coincides with a /spl les//sub m//sup 1-L/ degree. 3. For every class C closed under logspace reductions the /spl les//sub m//sup 1-NL/-complete degree of C coincides with the /spl les//sub m//sup 1-L/-complete degree of C.