# Lowness and the complexity of sparse and tally descriptions

## 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* _{ hd } ^{ p } (*R* _{ c } ^{ p } (SPARSE)) then *A* is in *R* _{ p } ^{ c } (*R* _{ hd } ^{ p } *(S))* for a sparse set *S* ∃ NP(*A*). As a consequence we can locate *R* _{ hd } ^{ p } (*R* _{ c } ^{ p } (SPARSE)) in the *EL* _{3} ^{Θ} level of the extended low hierarchy. Since *R* _{ hd } ^{ p } (*R* _{ c } ^{ p } (SPARSE)) \(\supseteq\)*R* _{ b } ^{ p } (*R* _{ c } ^{ p } (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* _{ d } ^{ p } (SPARSE) there exists a sparse set *S* ∃ NP(*A* ⊕ SAT)/Fθ _{2} ^{ p } (*A*) such that *A* ∃ *R* _{ d } ^{ p } *(S)*. Based on this we show that *R* _{1−tt} ^{ p } (*R* _{ d } ^{ p } (SPARSE)) is in *EL* _{3} ^{Θ} .

Finally, we construct for every set *A* ∃ *R* _{ c } ^{ p } (TALLY)∩*R* _{ d } ^{ p } (TALLY) (or equivalently, *A* ∃ IC[log, poly], as shown in [AHH^{+}92]) a tally set *T* ∃ P(*A* ⊕ SAT) such that *A* ∃ *R* _{ c } ^{ p } (*T*) ∩ *R* _{ d } ^{ p } (*T*). This implies that the class IC[log, poly] of sets with low instance complexity is contained in *EL* _{1} ^{Σ} .

## Keywords

Polynomial Time Simplicity Result Reduction Class Polynomial Time Hierarchy SlAM Journal## Preview

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## References

- [AH92]E. Allender and L. Hemachandra. Lower bounds for the low hierarchy.
*Journal of the ACM*, 39(1):234–250, 1992.CrossRefGoogle Scholar - [AHH+92]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.Google Scholar - [AHOW]E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets.
*SIAM Journal on Computing*, to appear.Google Scholar - [AKM92]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.Google Scholar - [BBS6]J. Balcázar and R. Book. Sets with small generalized Kolmogorov complexity.
*Acta Informatica*, 23(6):679–688, 1986.CrossRefGoogle Scholar - [BBS86]J.L. Balcázar, R. Book, and U. Schöning. Sparse sets, lowness and highness.
*SIAM Journal on Computing*, 23:679–688, 1986.Google Scholar - [BDG]J.L. Balcázar, J. Díaz, and J. Gabarró.
*Structural Complexity I.*EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1988.Google Scholar - [BLS92]H. Buhrman, L. Longpré, and E. Spaan. Sparse reduces conjunctively to tally.
*Technical Report NU-CCS-92-8*, Northeastern University, Boston, 1992.Google Scholar - [GW91]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.Google Scholar - [Hau14]F. Hausdorff.
*Grundzüge der Mengenlehre*. Leipzig, 1914.Google Scholar - [HY84]J. Hartmanis and Y. Yesha. Computation times of NP sets of different densities.
*Theoretical Computer Science*, 34:17–32, 1984.CrossRefGoogle Scholar - [Hem87]L. Hemachandra. The strong exponential hierarchy collapses.
*Proceedings of the 19th ACM Symposium on Theory of Computing*, 110–122, 1987.Google Scholar - [Kad87]J. Kadin. P
^{NP[log n]}and sparse Turing-complete sets for NP.*Journal of Computer and System Sciences*, 39(3):282–298, 1989.Google Scholar - [KL80]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.Google Scholar - [KoSc85]K. Ko and U. Schöning. On circuit-size complexity and the low hierarchy in NP.
*SIAM Journal on Computing*, 14:41–51, 1985.Google Scholar - [Köb92]J. Köbler. Locating P/poly optimally in the low hierarchy. Ulmer Informatik-Bericht 92-05, UniversitÄt Ulm, August 1992.Google Scholar
- [KSW87]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.Google Scholar - [KT90]J. Köbler and T. Thierauf. Complexity classes with advice.
*Proceedings 5th Structure in Complexity Theory Conference*, 305–315, IEEE Computer Society, 1990.Google Scholar - [LLS75]R. Ladner, N. Lynch, and A. Selman. A comparison of polynomial time reducibilities.
*Theoretical Computer Science*, 1(2):103–124, 1975.CrossRefGoogle Scholar - [LS91]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.Google Scholar
- [LT91]A. Lozano and J. Torán. Self-reducible sets of small density.
*Mathematical Systems Theory*, 24:83–100, 1991.Google Scholar - [Mah82]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.CrossRefGoogle Scholar - [MP79]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.Google Scholar
- [OKSW]P. Orponen, K. Ko, U. Schöning, and O. Watanabe. Instance complexity.
*Journal of the ACM*, to appear.Google Scholar - [OW91]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.CrossRefGoogle Scholar - [Sch83]U. Schöning. A low hierarchy within NP.
*Journal of Computer and System Sciences*, 27:14–28, 1983.CrossRefGoogle Scholar - [Sch86]U. Schöning.
*Complexity and Structure*, Lecture Notes in Computer Science, #211, Springer-Verlag, 1985Google Scholar - [SL92]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.Google Scholar - [TB91]S. Tang and R. Book. Reducibilities on tally and sparse sets.
*Theoretical Informatics and Applications*, 25:293–302, 1991.Google Scholar - [Wag87]K.W. Wagner. More complicated questions about maxima and minima, and some closures of NP.
*Theoretical Computer Science*, 51:53–80, 1987.CrossRefGoogle Scholar - [Wag90]K.W. Wagner. Bounded query classes.
*SIAM Journal on Computing*, 19(5):833–846, 1990.CrossRefGoogle Scholar