Abstract
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−L1,qli -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−L1 equivalent but not ≤ 1−L1honest equivalent. That is, there is a one-one 1-L degree that is no honest one-one 1-L degree.
Research supported by a Deutsche Forschungsgesellschaft Postdoktorandenstipendium.
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References
E. Allender. Isomorphisms and 1-L reductions. Journal of Computer and System Sciences, 36(6):336–350, 1988.
J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. Springer Verlag, 1988.
L. Berman. Polynomial Reducibilities and Complete Sets. PhD thesis, Cornell University, Ithaca, USA, 1977.
L. Berman and J. Hartmanis. On isomorphisms and density of NP and other complete sets. SIAM Journal on Computing, 6(2):305–322, 1977.
H.-J. Burtschick and A. Hoene. The degree structure of 1-l reductions. Technical Report 1992/6, Technische Universität Berlin, February 1992.
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.
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.
L. Hemachandra and A. Hoene. Collapsing degrees via strong computation. Journal on Computer and System Sciences. To appear.
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.
J. Hartmanis and S. Mahaney. Languages simultaneously complete for one-way and two-way log-tape automata. SIAM Journal on Computing, 10(2):383–390, 1981.
J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, 1979.
D. Joseph and P. Young. Some remarks on witness functions for non-polynomial and non-complete sets in NP. Theoretical Computer Science, 39:225–237, 1985.
K. Ko, T. Long, and D. Du. On one-way functions and polynomial-time isomorphisms. Theoretical Computer Science, 47:263–276, 1986.
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.
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Burtschick, HJ., Hoene, A. (1992). The degree structure of 1-L reductions. In: Havel, I.M., Koubek, V. (eds) Mathematical Foundations of Computer Science 1992. MFCS 1992. Lecture Notes in Computer Science, vol 629. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-55808-X_13
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DOI: https://doi.org/10.1007/3-540-55808-X_13
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