The Chain Method to Separate Counting Classes

We introduce a new method to separate counting classes of a special type by oracles. Among the classes, for which this method is applicable, are NP, coNP, US (also called 1-NP), ⊕ P, all other MOD-classes, PP, and C= P, classes of Boolean Hierarchies over the named classes, classes of finite acceptance type, and many more. As an important special case, we completely characterize all relativizable inclusions between classes NP(k) from the Boolean Hierarchy over NP and other classes defined by what we call bounded counting.

[1]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[2]  Ulrich Hertrampf.  Classes of Bounded Counting Type and their Inclusion Relations , 1995, STACS.

[3]  Ulrich Hertrampf.  Complexity Classes with Finite Acceptance Types , 1993, STACS.

[4]  Stephen A. Fenner,et al.  Gap-deenable Counting Classes , 1991 .

[5]  J. Tor,et al.  Complexity classesde ned by counting quanti ers * , 1991 .

[6]  Gerd Wechsung,et al.  On the Boolean closure of NP , 1985, FCT.

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

[8]  Gerd Wechsung,et al.  A survey on counting classes , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[9]  Klaus W. Wagner,et al.  Some Observations on the Connection Between Counting an Recursion , 1986, Theor. Comput. Sci..

[10]  Thomas Schwentick,et al.  On the power of polynomial time bit-reductions , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[11]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[12]  Gerd Wechsung,et al.  Counting classes with finite acceptance types , 1987 .

[13]  N. Vereshchagin.  RELATIVIZABLE AND NONRELATIVIZABLE THEOREMS IN THE POLYNOMIAL THEORY OF ALGORITHMS , 1994 .

[14]  Heribert Vollmer,et al.  On the power of number-theoretic operations with respect to counting , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[15]  Janos Simon.  On some central problems in computational complexity , 1975 .

[16]  Denis Thérien,et al.  Logspace and logtime leaf languages , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[17]  A. Stephen,et al.  Gap-De nable Counting Classes , 1991 .

[18]  Jacobo Torr.  Complexity Classes Deened by Counting Quantiiers* , 2022 .

[19]  Andreas Blass,et al.  On the Unique Satisfiability Problem , 1982, Inf. Control..

[20]  Christos H. Papadimitriou,et al.  Computational complexity , 1993 .

[21]  Stuart A. Kurtz,et al.  Gap-definable counting classes , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[22]  Eric Allender,et al.  Oracles versus Proof Techniques that Do Not Relativize , 1990, SIGAL International Symposium on Algorithms.

[23]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[24]  Eric Allender,et al.  Counting Hierarchies: Polynomial Time and Constant , 1990, Bull. EATCS.

[25]  Jacobo Torán,et al.  Complexity classes defined by counting quantifiers , 1991, JACM.

[26]  Eric Allender,et al.  Counting Hierarchies: Polynomial Time and Constant Depth Circuits , 1993, Current Trends in Theoretical Computer Science.

[27]  Pierluigi Crescenzi,et al.  A Uniform Approach to Define Complexity Classes , 1992, Theor. Comput. Sci..