Polynomial Time Computation in the Context of Recursive Analysis

Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955] as an approach for investigating computation over the real numbers. It is based on enhancing the Turing machine model by introducing oracles that allow the machine to access finitary portions of the real infinite objects. Classes of computable real functions have been extensively studied as well as complexity-theoretic classes of functions restricted to compact domains. However, much less have been done regarding complexity of arbitrary real functions. In this article we give a definition of polynomial time computability of arbitrary real functions. Then we present two main applications based on that definition. The first one, which has already been published, concerns the relationships between polynomial time real computability and the corresponding notion over continuous rational functions. The second application, which is a new contribution to this article, concerns the construction of a function algebra that captures polynomial time real computability.

[1]  C. Michaux,et al.  A survey on real structural complexity theory , 1997 .

[2]  Manuel Lameiras Campagnolo,et al.  Characterizing Computable Analysis with Differential Equations , 2008, CCA.

[3]  Alexander Artikis,et al.  Specifying norm-governed computational societies , 2009, TOCL.

[4]  Walid Gomaa,et al.  A survey of recursive analysis and Moore’s notion of real computation , 2012, Natural Computing.

[5]  Ker-I Ko,et al.  Complexity Theory of Real Functions , 1991, Progress in Theoretical Computer Science.

[6]  Stephen A. Cook,et al.  A new recursion-theoretic characterization of the polytime functions , 1992, STOC '92.

[7]  A. Church.  Review: A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem , 1937 .

[8]  Olivier Bournez,et al.  Recursive Analysis Characterized as a Class of Real Recursive Functions , 2006, Fundam. Informaticae.

[9]  Cristopher Moore,et al.  Iteration, Inequalities, and Differentiability in Analog Computers , 2000, J. Complex..

[10]  Walid Gomaa.  Characterizing Polynomial Time Computability of Rational and Real Functions , 2009, DCM.

[11]  Andrzej Grzegorczyk.  On the definition of computable functionals , 1955 .

[12]  Cristopher Moore,et al.  An Analog Characterization of the Grzegorczyk Hierarchy , 2002, J. Complex..

[13]  John N. Tsitsiklis,et al.  A survey of computational complexity results in systems and control , 2000, Autom..

[14]  Stephen Smale,et al.  On a theory of computation over the real numbers; NP completeness, recursive functions and universal machines , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[15]  Klaus Weihrauch,et al.  A unified approach to constructive and recursive analysis , 1984 .

[16]  Jerzy Mycka,et al.  Computability on reals, infinite limits and differential equations , 2007, Appl. Math. Comput..

[17]  Manuel Lameiras Campagnolo,et al.  The Complexity of Real Recursive Functions , 2002, UMC.

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

[19]  Cristopher Moore,et al.  An Analog Characterization of the Subrecursive Functions , 2000 .

[20]  Akitoshi Kawamura,et al.  Differential recursion , 2007, TOCL.

[21]  Ugo Dal Lago.  The geometry of linear higher-order recursion , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

[22]  Olivier Bournez,et al.  Polynomial differential equations compute all real computable functions on computable compact intervals , 2007, J. Complex..

[23]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[24]  Yehoshua Bar-Hillel,et al.  The Intrinsic Computational Difficulty of Functions , 1969 .

[25]  Cristopher Moore,et al.  Recursion Theory on the Reals and Continuous-Time Computation , 1996, Theor. Comput. Sci..

[26]  Pekka Orponen,et al.  A Survey of Continous-Time Computation Theory , 1997, Advances in Algorithms, Languages, and Complexity.

[27]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[28]  Manuel L. Campagnolo,et al.  Computational complexity of real valued recursive functions and analog circuits , 2001 .

[29]  Christoph Kreitz,et al.  Theory of Representations , 1985, Theor. Comput. Sci..

[30]  Manuel Lameiras Campagnolo,et al.  The elementary computable functions over the real numbers: applying two new techniques , 2008, Arch. Math. Log..

[31]  Extended Grzegorczyk hierarchy in the BSS model of computability , 1997 .

[32]  Benedikt Löwe,et al.  New Computational Paradigms , 2005 .

[33]  Michael M. Richter,et al.  Computation and Proof Theory , 1984 .

[34]  Olivier Bournez,et al.  A Survey on Continuous Time Computations , 2009, ArXiv.