The degree structure of 1-L reductions
A 1-L function is one that is computable by a logspace Turing machine that moves its input head only in one direction. We show that there exist 1-L complete sets for PSPACE that are not 1-L isomorphic. In other words, the 1-L complete degree for PSPACE does not collapse. This contrasts a result of Allender who showed that all 1-L complete sets for PSPACE are polynomial-time isomorphic. Since all 1-L complete sets for PSPACE are equivalent under 1-L reductions that are one-one and quadratically length-increasing this also provides an example of a ≤ 1,qli 1−L -degree that does not collapse to a single 1-L isomorphism type.
Relatedly, we prove that there exist two sets A and B that are ≤ 1 1−L equivalent but not ≤ 1honest 1−L equivalent. That is, there is a one-one 1-L degree that is no honest one-one 1-L degree.
KeywordsTuring Machine Random Oracle Isomorphism Type Input Tape Logarithmic Space
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- [BDG88]J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. Springer Verlag, 1988.Google Scholar
- [Ber77]L. Berman. Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University, Ithaca, USA, 1977.Google Scholar
- [BH92]H.-J. Burtschick and A. Hoene. The degree structure of 1-l reductions. Technical Report 1992/6, Technische Universität Berlin, February 1992.Google Scholar
- [FKR89]S. Fenner, S. Kurtz, and J. Royer. Every polynomial 1-degree collapses iff P=PSPACE. In Proceedings 30th IEEE Symposium on Foundations of Computer Science, pages 624–629. IEEE Computer Society Press, 1989.Google Scholar
- [GH89]K. Ganesan and S. Homer. Complete problems and strong polynomial reducibilities. In STACS 1989: 6th Annual Symposium on Theoretical Aspects of Computer Science, pages 240–250. Springer-Verlag Lecture Notes in Computer Science #349, 1989.Google Scholar
- [HH]L. Hemachandra and A. Hoene. Collapsing degrees via strong computation. Journal on Computer and System Sciences. To appear.Google Scholar
- [HIM78]J. Hartmanis, N. Immerman, and S. Mahaney. One-way log-tape reductions. In Proceedings 19th IEEE Symposium on Foundations of Computer Science, pages 65–71. IEEE Computer Society Press, 1978.Google Scholar
- [HU79]J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 1979.Google Scholar
- [KMR89]S. Kurtz, S. Mahaney, and J. Royer. The isomorphism conjecture fails relative to a random oracle. In Proceedings of the 21st ACM Symposium on Theory of Computing, pages 157–166. ACM Press, 1989.Google Scholar