On log-Time Alternating Turing Machines of Alternation Depth k (Extended Abstract)

Several input read-modes for alternating Turing machines have been proposed in the literature. For each input read-mode and for each fixed integer k ≥ 1, a precise circuit characterization is established for log-time alternating Turing machines of k alternations, which is a nontrivial refinement of Ruzzo's circuit characterization of alternating Turing machines. Complete languages in strong sense for each level of the log-time hierarchy are presented, refining a result by Buss. The class GC(s(n), Π k B ) is investigated, which is the class of languages accepted by log-time alternating Turing machines of k alternations enhanced by an extra ability of guessing a string of length s(n). A systematic technique is developed to show that for many functions s(n) and for every integer k>1, the class GC(s(n), Π k B ) has natural complete languages. Connections of these results to computational optimization problems are exhibited.

[1]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[2]  Jianer Chen.  Characterizing Parallel Hierarchies by Reducibilities , 1991, Inf. Process. Lett..

[3]  Marty J. Wolf.  Nondeterministic Circuits, Space Complexity and Quasigroups , 1994, Theor. Comput. Sci..

[4]  Liming Cai,et al.  On Input Read-Modes of Alternating Turing Machines , 1995, Theor. Comput. Sci..

[5]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[6]  Michael R. Fellows,et al.  Fixed-parameter intractability , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[7]  Liming Cai,et al.  On the Amount of Nondeterminism and the Power of Verifying (Extended Abstract) , 1993, MFCS.

[8]  Michael Sipser,et al.  Borel sets and circuit complexity , 1983, STOC.

[9]  Samuel R. Buss,et al.  The Boolean formula value problem is in ALOGTIME , 1987, STOC.

[10]  Uzi Vishkin,et al.  On Finding a Minimum Dominating Set in a Tournament , 1988, Theor. Comput. Sci..

[11]  Liming Cai,et al.  On fixed-parameter tractability and approximability of NP-hard optimization problems , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

[12]  Mihalis Yannakakis,et al.  On limited nondeterminism and the complexity of the V-C dimension , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[13]  Ravi B. Boppana,et al.  The Complexity of Finite Functions , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[14]  Alok Aggarwal,et al.  Hierarchical memory with block transfer , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[15]  Samuel R. Buss,et al.  An Optimal Parallel Algorithm for Formula Evaluation , 1992, SIAM J. Comput..

[16]  Liming Cai,et al.  On the Structure of Parameterized Problems in NP , 1995, Inf. Comput..

[17]  Liming Cai,et al.  On the Amount of Nondeterminism and the Power of Verifying , 1997, SIAM J. Comput..

[18]  Judy Goldsmith,et al.  Nondeterminism Within P , 1993, SIAM J. Comput..

[19]  Walter L. Ruzzo.  On Uniform Circuit Complexity , 1981, J. Comput. Syst. Sci..