Abstract
We investigate properties of the relativized NC and AC hierarchies in their DLOGTIME-, respectively, ALOGTIME-uniform setting and show that these hierarchies can be characterized in terms of adaptive reducibility in deterministic (poly)logarithmic time, i.e. in time O(log n)i for i≥0. Using this characterization, we substantially generalize various previous results concerning the structure of the NC and AC hierarchies.
Partially supported by DAAD and Spanish Government (Acción Integrada 131-B, 313-AI-e-es/zk).
Research supported by the ESPRIT Basic Research Actions Program of the EC under contract No. 7141 (project ALCOM II).
On leave until March 1995, visiting Dept. L.S.I. of Univ. Politècnica de Catalunya supported by a Habilitationsstipendium of Deutsche Forschungsgemeinschaft (DFG-Je 154 2/1).
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References
M. Ajtai, σ 11 formulae on finite structures, Ann. Pure Appl. Logic, 24 (1983), 1–48.
C. àlvarez, Polylogtime and Logspace Adaptive Reductions, Tech. Report LSI-93-42-R, Dept. LSI, Universitat Politècnica de Catalunya, 1993.
C. àlvarez, J.L. Balcázar and B. Jenner, Adaptive logspace reducibility and parallel time, to appear in Math. Systems Theory. (preliminary version in Proc. 8th STACS, LNCS 480 (Springer, Berlin, 1991), 422–433.)
C. àlvarez and B. Jenner, A very hard log-space counting class, Theoret. Computer Science 107 (1993), 3–30.
C. àlvarez and B. Jenner, On adaptive dlogtime and polylogtime Reductions, Tech. Report LSI-93-41-R, Dept. LSI, Universitat Politècnica de Catalunya, 1993.
J.L. Balcázar, Adaptive logspace and depth-bounded reducibilities, Proc. 6th Structure in Complexity Theory Conference (1991), 240–254.
J.L. Balcázar, A. Lozano, J. Torán, The Complexity of Algorithmic Problems on Succinct Instances, R. Baeza-Yates, U. Manber (eds.), Computer Science, Plenum Press, New York, (1992), 351–377.
D.A. Mix Barrington, N. Immerman, H. Straubing, On uniformity within NC1, J. of Comp. and System Sci. 41, 3 (1990), 274–306.
A. Borodin, S.A. Cook, P.W. Dymond, W.L. Ruzzo, M. Tompa, Two applications of inductive counting for complementation problems, SIAM J. of Comput. 18, 3 (1989), 559–578.
S.R. Buss, The formula value problem is in ALOGTIME, Proc. 19th ACM STOC Symp., 1987, 123–131.
S. Buss, S. Cook, A. Gupta, V Ramachandran, An optimal parallel algorithm for formula evaluation, typescript, Univ. of Toronto, 1989.
J. Castro, C. Seara, Characterizations of some complexity classes between θ p2 and δ p2 , Proc. 9th STACS, LNCS 577 (Springer, Berlin, 1992), 305–319.
A.K. Chandra, D. Kozen, L.J. Stockmeyer, Alternation, J. of the ACM 28 (1981), 114–133.
A.K. Chandra, L.J. Stockmeyer, U. Vishkin, Constant depth reducibility, SIAM J. of Comput. 13 (1984), 423–439.
S.A. Cook, A taxonomy of problems, with fast parallel algorithms, Information and Control 64 (1985), 2–22.
M. Furst, J.B. Saxe, M. Sipser, Parity, circuits, and the polynomial-time hierarchy, Math. Systems Theory 17 (1984), 13–27.
B. Jenner, J. Torán, Parallel queries to NP, Proc. 8th Structure in Complexity Theory Conference (1993), 280–291. (to appear in Theoret. Computer Science)
R. Ladner, The circuit value problem is log space complete for P, SIGACT News 7 (1975), 18–20.
W.L. Ruzzo, On uniform circuit complexity, J. of Comput. and System Sci. 22 (1981), 365–383.
M. Sipser, Borel sets and circuit complexity, Proc. 15th ACM STOC Symp. (1983), 61–69.
L. Stockmeyer, The polynomial-time hierarchy, Theoret. Computer Science 3 (1977), pp 1–22.
I.H. Sudborough, On the tape complexity of deterministic context-free languages, J. of the ACM 25 (1978), 405–414.
K.W. Wagner, Bounded Query Classes, SIAM J. of Comput. 19, 5 (1990), 833–846.
C.B. Wilson, Relativized NC, Math. Systems Theory 20 (1987), 13–29.
C.B. Wilson, Decomposing NC and AC, SIAM J. of Comput. 19, 2 (1990), pp. 384–396.
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àlvarez, C., Jenner, B. (1994). On adaptive dlogtime and polylogtime reductions. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57785-8_150
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DOI: https://doi.org/10.1007/3-540-57785-8_150
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