Recursion Theoretic Characterizations of Complexity Classes of Counting Functions

Abstract There has been a great effort in giving machine-independent, algebraic characterizations of complexity classes, especially of functions. Astonishingly, no satisfactory characterization of the prominent class #P is known up to now. Here, we characterize #P as the closure of a set of simple arithmetical functions under summation and weak product. Based on that result, the hierarchy of counting functions, which is the closure of #P under substitution, is characterized, remarkably without using the operator of substitution, since we can show that in the context of this hierarchy the operation of modified subtraction is as powerful as substitution. This leads us to a number of consequences concerning closure of #P under certain arithmetical operations. Analogous results are achieved for the class Gap-P which is the closure of #P under subtraction.

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

[2]  Patrick Suppes,et al.  Logic, Methodology and Philosophy of Science , 1963 .

[3]  Peter Clote,et al.  Bounded arithmetic and computational complexity , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[4]  Bernard R. Hodgson,et al.  A Normal Form for Arithmetical Representation of NP-Sets , 1983, J. Comput. Syst. Sci..

[5]  P. Clote.  Sequential, machine-independent characterizations of the parallel complexity classes AlogTIME, AC k , NC k and NC , 1990 .

[6]  Neil Immerman,et al.  Expressibility and Parallel Complexity , 1989, SIAM J. Comput..

[7]  Stuart A. Kurtz,et al.  Gap-Definable Counting Classes , 1994, J. Comput. Syst. Sci..

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

[9]  Leslie G. Valiant,et al.  Relative Complexity of Checking and Evaluating , 1976, Inf. Process. Lett..

[10]  John Gill,et al.  Computational Complexity of Probabilistic Turing Machines , 1977, SIAM J. Comput..

[11]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[12]  Bernard R. Hodgson,et al.  An Arithmetical Characterization of NP , 1982, Theor. Comput. Sci..

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

[14]  Martin D. Davis,et al.  Arithmetical problems and recursively enumerable predicates , 1953, Journal of Symbolic Logic.

[15]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[16]  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.

[17]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[18]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[19]  Michael Sipser,et al.  The history and status of the P versus NP question , 1992, STOC '92.

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

[21]  Lane A. Hemachandra,et al.  A complexity theory for feasible closure properties , 1993 .

[22]  K. V. Subrahmanyam,et al.  Descriptive Complexity of #P Functions , 1995, J. Comput. Syst. Sci..

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

[24]  ’ Ronald Fagin,et al.  - Order Spectra and Polynomial-Time Recognizable Sets , 1974 .

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