Uniformly Defining Complexity Classes of Functions

We introduce a general framework for the definition of function classes. Our model, which is based on polynomial time nondeterministic Turing transducers, allows uniform characterizations of FP, FPNP, counting classes (#·P, #·NP, #·coNP, GapP, GapPNP), optimization classes (max·P, min·P, max·NP, min·NP), promise classes (NPSV, #few·P, c#·P), multivalued classes (FewFP, NPMV) and many more. Each such class is defined in our model by a certain family of functions. We study a reducibility notion between such families, which leads to a necessary and sufficient criterion for relativizable inclusion between function classes. As it turns out, this criterion is easily applicable and we get as a consequence e.g. that there are oracles A, B, such that min.PA \(\nsubseteq\) #·NPA, and max.NPB \(\nsubseteq\) c#·coNPB (note that no structural consequences are known to follow from the corresponding positive inclusions).

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

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

[3]  Heribert Vollmer,et al.  Complexity Classes of Optimization Functions , 1995, Inf. Comput..

[4]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[5]  José L. Balcázar,et al.  Structural Complexity I , 1995, Texts in Theoretical Computer Science An EATCS Series.

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

[7]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[8]  Heribert Vollmer,et al.  The Complexity of Finding Middle Elements , 1993, Int. J. Found. Comput. Sci..

[9]  Bernd Borchert,et al.  Predicate classes and promise classes , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[10]  Johannes Köbler.  Strukturelle Komplexität von Anzahlproblemen , 1989 .

[11]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[12]  Harald Hempel,et al.  The Operators min and max on the Polynomial Hierarchy , 1997, STACS.

[13]  Heribert Vollmer,et al.  The satanic notations , 1995, SIGACT News.

[14]  Mark W. Krentel.  The Complexity of Optimization Problems , 1988, J. Comput. Syst. Sci..

[15]  Heribert Vollmer.  On Different Reducibility Notions for Function Classes , 1994, STACS.

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

[17]  Jacobo Torán,et al.  Computing Functions with Parallel Queries to NP , 1995, Theor. Comput. Sci..

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

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

[20]  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.

[21]  Mark W. Krentel.  Generalizations of Opt P to the Polynomial Hierarchy , 1992, Theor. Comput. Sci..

[22]  Bernd Borchert.  Predicate classes, promise classe, and the acceptance power of regular languages , 1994 .

[23]  Alan L. Selman,et al.  A Taxonomy of Complexity Classes of Functions , 1994, J. Comput. Syst. Sci..

[24]  Timothy J. Long,et al.  Quantitative Relativizations of Complexity Classes , 1984, SIAM J. Comput..

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

[26]  Pierluigi Crescenzi,et al.  Introduction to the theory of complexity , 1994, Prentice Hall international series in computer science.