## The Complexity of Real Recursive Functions (2002)

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Venue: | Unconventional Models of Computation (UMC'02), LNCS 2509 |

Citations: | 10 - 5 self |

### BibTeX

@INPROCEEDINGS{Campagnolo02thecomplexity,

author = {Manuel Lameiras Campagnolo},

title = {The Complexity of Real Recursive Functions},

booktitle = {Unconventional Models of Computation (UMC'02), LNCS 2509},

year = {2002},

pages = {1--14},

publisher = {Springer-Verlag}

}

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### Abstract

We explore recursion theory on the reals, the analog counterpart of recursive function theory. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions, such as composition and various forms of differential equations. We define classes of real recursive functions, in a manner similar to the classical approach in recursion theory, and we study their complexity. In particular, we prove both upper and lower bounds for several classes of real recursive functions, which lie inside the primitive recursive functions and, therefore, can be characterized in terms of standard computational complexity.

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Citation Context ..., we can characterize them in terms of standard space or time complexity, and consider the Turing machine as the underlying computational model. This approach differs from others, namely BSS-machines =-=[BSS89]-=- or informationbased complexity [TW98], since it focus on effective computability and complexity. There are two main reasons to this. First, the Turing machine model allows us to represent the concept... |

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Citation Context ...on. In fact, the standard techniques for numerical integration (Euler’s method) require a number of steps which is exponential in the bounds on the derivatives of the functions we want to approximat=-=e [Hen62]-=-. Since the bounds for functions in S0 are polynomial, the required number of steps N in the numerical integration is exponential. Thus all functions in S0 can be approximated in exponential space. Fi... |

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Citation Context ... function theory provides the standard notion of computable function [Cut80,Odi89]. Moreover, many time and space complexity classes have recursive characterizations [Clo99]. As far as we know, Moore =-=[Moo96]-=- was the first to extend recursion theory to real valued functions. We will explore this and show that all main concepts in recursion theory like basic functions, operators, function algebras, or func... |

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Citation Context ...nder proper integration, and it contains the non-analytic function θk or θ∞. We call this property closure under iteration in a weak sense. For instance, it can be shown, using a technique similar=-= to [Bra95], tha-=-t Proposition 8. [0, 1, −1, θ∞, U; COMP, I] is closed under iteration in a weak sense. 4 Computational complexity In this section we explore connections among real recursive classes and standard ... |

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Citation Context ...consider C ′ as a lower bound for C. On the other hand, we consider the computational complexity of real functions. We use the notion of [Ko91], which is equivalent to the one proposed by Grzegorczy=-=k [Grz55]-=-, and whose underlying computational model is the functionoracle Turing machine. Intuitively, the time (resp. space) complexity of f is the number of moves (resp. the amount of tape) required by a fun... |

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Citation Context ...mbers. Then, in the presence of some appropriate non-analytic function like θk, proper integration and even linear integration are sufficient to simulate bounded products. In particular, we proved in=-= [CMC] the fo-=-llowing: 1 The constant ck is a rational or a rational multiplied by π. 7sProposition 7. For all k ∈ N, [0, 1, −1, π, θk, U; COMP, LI] is closed under bounded products in a weak sense. We can a... |

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(Show Context)
Citation Context ... between real recursive functions and dynamical systems. It is known that the unary functions in [0, 1, −1, U; COMP, I] are precisely the solutions of equations y ′ = p(y, x), where p is a polynom=-=ial [Gra02]. We-=- conjecture that [0, 1, −1, U; COMP, LI] corresponds to the family of dynamical systems y ′ = f(y, x), where each fi is linear and depends at most on x, y1, . . . , yi. Given such canonical repres... |

2 |
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Citation Context ... differentially algebraic [Moo96], that is, they satisfy a polynomial differential equation of finite order. So, for any fixed t, F is differentially algebraic in x. But, from a result of Babakhanian =-=[Bab73]-=-, we know that exp [t] satisfies no non-trivial polynomial differential equation of order less than t. This means that the number of integrations that are necessary to define exp [t] has to grow with ... |

1 |
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(Show Context)
Citation Context ...a weak sense. But since all functions in B0 have polynomials bounds, then this can be done with techniques similar to [Bra95] using bounded integration instead of integration. Details can be found in =-=[Cam01]. ��-=-�⊔ The Ritchie hierarchy [Rit63] is one of the first attempts to classify recursive functions in terms of computational complexity. The Ritchie classes, which range from FLINSPACE to the elementary ... |