A Note on Parallel Queries and the Symmetric-Difference Hierarchy

For classes K1,K2,…,Km of languages and m,r1,r2,…,rm ⩾ 1 let the class P∥K1[r1],K2[r2],…,Km[rm] be defined as the class of languages which can be accepted by a deterministic polynomial time machine which makes (simultaneously for j = 1, 2,…,m) rj parallel queries to an oracle Bj ϵ Kj. For classes K1,K2,…, Km ϵ {bE1p, βE2p, βE3p,… of the polynomial time hierarchy we prove P∥K1[r1],K2[r2],…,Km[rm] = P β + K1 β + … β + K1 β + K2 β + … β + Km β + … β + Km, where K β + M = def {A Δ B: A ϵ K and B ϵ M} (Kj appears rj times). Hence the classes of the form P∥K1[r1],K2[r2],…,Km[rm] are exactly the classes of Selivanov's plus hierarchy when the term Pβ + is added. This result is valid also for a large variety of other families of classes. Finally, applications to query order classes from Hemaspaandra et al. (1997) and Beigel and Chang (1997) are given.

[1]  Klaus W. Wagner,et al.  The Difference and Truth-Table Hierarchies for NP , 1987, RAIRO Theor. Informatics Appl..

[2]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

[3]  Mark W. Krentel.  The complexity of optimization problems , 1986, STOC '86.

[4]  R. Beigel,et al.  Bounded Queries to SAT and the Boolean Hierarchy , 1991, Theor. Comput. Sci..

[5]  Klaus W. Wagner,et al.  Bounded Query Classes , 1990, SIAM J. Comput..

[6]  Richard Chang,et al.  Commutative queries , 1997, Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems.

[7]  Juris Hartmanis,et al.  The Boolean Hierarchy I: Structural Properties , 1988, SIAM J. Comput..

[8]  Edith Hemaspaandra,et al.  A Downward Translation in the Polynomial Hierarchy , 1997, STACS.

[9]  Edith Hemaspaandra,et al.  Query Order in the Polynomial Hierarchy , 1997, FCT.

[10]  Samuel R. Buss,et al.  On truth-table reducibility to SAT and the difference hierarchy over NP , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[11]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[12]  C. Papadimitriou,et al.  Two remarks on the power of counting , 1983 .

[13]  Jin-Yi Cai,et al.  The Boolean Hierarchy: Hardware over NP , 1986, SCT.

[14]  Klaus W. Wagner.  More Complicated Questions About Maxima and Minima, and Some Closures of NP , 1987, Theor. Comput. Sci..

[15]  Victor L. Selivanov.  Two Refinements of the Polynomial Hierarcht , 1994, STACS.

[16]  Lane A. Hemaspaandra,et al.  Query Order , 1998, SIAM J. Comput..