Abstract
We investigate the complexity of obtaining sparse descriptions for sets in various reduction classes to sparse sets. Let A be a set in a certain reduction class R r (SPARSE). Then we are interested in finding upper bounds for the complexity (relative to A) of sparse sets S such that A ∃ R r (S). By establishing such upper bounds we are able to derive the lowness of A. In particular, we show that if a set A is in the class R p hd (R p c (SPARSE)) then A is in R c p (R p hd (S)) for a sparse set S ∃ NP(A). As a consequence we can locate R p hd (R p c (SPARSE)) in the EL Θ3 level of the extended low hierarchy. Since R p hd (R p c (SPARSE)) \(\supseteq\) R p b (R p c (SPARSE)) this solves the open problem of locating the closure of sparse sets under bounded truth-table reductions optimally in the extended low hierarchy. Furthermore, we show that for every A ∃ R p d (SPARSE) there exists a sparse set S ∃ NP(A ⊕ SAT)/Fθ p2 (A) such that A ∃ R p d (S). Based on this we show that R p1−tt (R p d (SPARSE)) is in EL Θ3 .
Finally, we construct for every set A ∃ R p c (TALLY)∩R p d (TALLY) (or equivalently, A ∃ IC[log, poly], as shown in [AHH+92]) a tally set T ∃ P(A ⊕ SAT) such that A ∃ R p c (T) ∩ R p d (T). This implies that the class IC[log, poly] of sets with low instance complexity is contained in EL Σ1 .
Work done while visiting UniversitÄt Ulm. Supported in part by an Alexander von Humboldt research fellowship.
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References
E. Allender and L. Hemachandra. Lower bounds for the low hierarchy. Journal of the ACM, 39(1):234–250, 1992.
V. Arvind, Y. Han, L. Hemachandra, J. Köbler, A. Lozano, M. Mundhenk, M. Ogiwara, U. Schöning, R. Silvestri, and T. Thierauf. Reductions to sets of low information content. Proceedings of the 19th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science, #623:162–173, Springer Verlag, 1992.
E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, to appear.
V. Arvind, J. Köbler, and M. Mundhenk. On bounded truth-table, conjunctive, and randomized reductions to sparse sets. To appear in Proceedings 12th Conference on the Foundations of Software Technology & Theoretical Computer Science, 1992.
J. Balcázar and R. Book. Sets with small generalized Kolmogorov complexity. Acta Informatica, 23(6):679–688, 1986.
J.L. Balcázar, R. Book, and U. Schöning. Sparse sets, lowness and highness. SIAM Journal on Computing, 23:679–688, 1986.
J.L. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1988.
H. Buhrman, L. Longpré, and E. Spaan. Sparse reduces conjunctively to tally. Technical Report NU-CCS-92-8, Northeastern University, Boston, 1992.
R. Gavaldà and O. Watanabe. On the computational complexity of small descriptions. Proceedings of the 6th Structure in Complexity Theory Conference, 89–101, IEEE Computer Society Press, 1991.
F. Hausdorff. Grundzüge der Mengenlehre. Leipzig, 1914.
J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities. Theoretical Computer Science, 34:17–32, 1984.
L. Hemachandra. The strong exponential hierarchy collapses. Proceedings of the 19th ACM Symposium on Theory of Computing, 110–122, 1987.
J. Kadin. PNP[log n] and sparse Turing-complete sets for NP. Journal of Computer and System Sciences, 39(3):282–298, 1989.
R. Karp and R. Lipton. Some connections between nonuniform and uniform complexity classes. Proceedings of the 12th ACM Symposium on Theory of Computing, 302–309, April 1980.
K. Ko and U. Schöning. On circuit-size complexity and the low hierarchy in NP. SIAM Journal on Computing, 14:41–51, 1985.
J. Köbler. Locating P/poly optimally in the low hierarchy. Ulmer Informatik-Bericht 92-05, UniversitÄt Ulm, August 1992.
J. Köbler, U. Schöning, and K.W. Wagner. The difference and truth-table hierarchies of NP. Theoretical Informatics and Applications, 21 (4):419–435, 1987.
J. Köbler and T. Thierauf. Complexity classes with advice. Proceedings 5th Structure in Complexity Theory Conference, 305–315, IEEE Computer Society, 1990.
R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time reducibilities. Theoretical Computer Science, 1(2):103–124, 1975.
T.J. Long and M.-J. Sheu. A refinement of the low and high hierarchies. Technical Report OSU-CISRC-2/91-TR6, The Ohio State University, 1991.
A. Lozano and J. Torán. Self-reducible sets of small density. Mathematical Systems Theory, 24:83–100, 1991.
S. Mahaney. Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25(2):130–143, 1982.
A. Meyer, M. Paterson. With what frequency are apparently intractable problems difficult? Tech. Report MIT/LCS/TM-126, Lab. for Computer Science, MIT, Cambridge, 1979.
P. Orponen, K. Ko, U. Schöning, and O. Watanabe. Instance complexity. Journal of the ACM, to appear.
M. Ogiwara and O. Watanabe. On polynomial-time bounded truth-table reducibility of NP sets to sparse sets. SIAM Journal on Computing, 20(3):471–483, 1991.
U. Schöning. A low hierarchy within NP. Journal of Computer and System Sciences, 27:14–28, 1983.
U. Schöning. Complexity and Structure, Lecture Notes in Computer Science, #211, Springer-Verlag, 1985
M.-J. Sheu and T.J. Long. The extended low hierarchy is an infinite hierarchy. Proceedings of 9th Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, #577:187–189, Springer-Verlag 1992.
S. Tang and R. Book. Reducibilities on tally and sparse sets. Theoretical Informatics and Applications, 25:293–302, 1991.
K.W. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51:53–80, 1987.
K.W. Wagner. Bounded query classes. SIAM Journal on Computing, 19(5):833–846, 1990.
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Arvind, V., Köbler, J., Mundhenk, M. (1992). Lowness and the complexity of sparse and tally descriptions. In: Ibaraki, T., Inagaki, Y., Iwama, K., Nishizeki, T., Yamashita, M. (eds) Algorithms and Computation. ISAAC 1992. Lecture Notes in Computer Science, vol 650. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56279-6_78
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DOI: https://doi.org/10.1007/3-540-56279-6_78
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