On log-tape isomorphisms of complete sets

In this paper we study log n-tape computable reductions between sets and investigate conditions under which log n-tape reductions between sets can be extended to log n-tape computable isomorphisms of these sets. As an application of these results we obtain easy to check necessary and sufficient conditions that sets complete under log n-tape reductions in NL, CSL, P, NP, PTAPE, etc. are log n-tape isomorphic to the previously known complete sets in the respective classes. As a matter of fact, all the “known” complete sets for NL, CSL, P, NP, PTAPE, etc. are now easily seen to be, respectively, log n-tape isomorphic. These results strengthen and extend substantially the previously known results about polynomial time computable reductions and isomorphisms of NP and PTAPE complete sets. Furthermore, we show that any set complete in CSL, PTAPE, etc. must be dense and therefore, for example, cannot be over a single letter alphabet.