A Weak Version of the Blum, Shub, and Smale Model

We propose a weak version of the Blum?Shub?Smale model of computation over the real numbers. In this weak model only a “moderate” usage of multiplications and divisions is allowed. The class of boolean languages recognizable in polynomial time is shown to be the complexity class P/poly. The main tool is a result on the existence of small rational points in semi-algebraic sets which is of independent interest. As an application, we generalize recent results of Siegelmann and Sontag on recurrent neural networks, and of Maass on feedforward nets. A preliminary version of this paper was presented at the1993 IEEE Symposium on Foundations of Computer Science. Additional results include: an efficient simulation of order-free real Turing machines by probabilistic Turing machines in the full Blum?Shub?Smale model; the strict inclusion of the real polynomial hierarchy in weak exponential time.

[1]  R. Loos.  Computing in Algebraic Extensions , 1983 .

[2]  Arnold Schönhage,et al.  On the Power of Random Access Machines , 1979, ICALP.

[3]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part III: Quantifier Elimination , 1992, J. Symb. Comput..

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

[5]  Stephen Smale,et al.  Some Remarks on the Foundations of Numerical Analysis , 1990, SIAM Rev..

[6]  Michel Coste,et al.  Thom's Lemma, the Coding of Real Algebraic Numbers and the Computation of the Topology of Semi-Algebraic Sets , 1988, J. Symb. Comput..

[7]  J. Milnor.  On the Betti numbers of real varieties , 1964 .

[8]  J. Risler,et al.  Real algebraic and semi-algebraic sets , 1990 .

[9]  J. Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I , 1989 .

[10]  Wolfgang Maass,et al.  Bounds for the computational power and learning complexity of analog neural nets , 1993, SIAM J. Comput..

[11]  Felipe Cucker,et al.  Pr != Ncr , 1992, Journal of Complexity.

[12]  H. P. Williams.  THEORY OF LINEAR AND INTEGER PROGRAMMING (Wiley-Interscience Series in Discrete Mathematics and Optimization) , 1989 .

[13]  R. Tennant.  Algebra , 1941, Nature.

[14]  Felipe Cucker,et al.  Separation of Complexity Classes in Koiran's Weak Model , 1994, Theor. Comput. Sci..

[15]  Felipe Cucker,et al.  On algorithms for real algebraic plane curves , 1991 .

[16]  José L. Balcázar,et al.  Structural Complexity II , 2012, EATCS.

[17]  Jehoshua Bruck,et al.  On the Power of Threshold Circuits with Small Weights , 1991, SIAM J. Discret. Math..

[18]  Bruno Buchberger,et al.  Computer algebra symbolic and algebraic computation , 1982, SIGS.

[19]  James Renegar,et al.  On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I: Introduction. Preliminaries. The Geometry of Semi-Algebraic Sets. The Decision Problem for the Existential Theory of the Reals , 1992, J. Symb. Comput..

[20]  M-F Roy,et al.  Géométrie algébrique réelle , 1987 .

[21]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[22]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[23]  John B. Goode.  Accessible Telephone Directories , 1994, J. Symb. Log..

[24]  Hava T. Siegelmann,et al.  Analog computation via neural networks , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

[25]  Pascal Koiran.  A weak version of the Blum, Shub and Smale model , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.