## Analog computers and recursive functions over the reals (2003)

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Venue: | Journal of Complexity |

Citations: | 40 - 19 self |

### BibTeX

@ARTICLE{Graça03analogcomputers,

author = {Daniel Silva Graça and José Félix Costa},

title = {Analog computers and recursive functions over the reals},

journal = {Journal of Complexity},

year = {2003},

volume = {19},

pages = {644--664}

}

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### Abstract

In this paper we show that Shannon’s General Purpose Analog Computer (GPAC) is equivalent to a particular class of recursive functions over the reals with the flavour of Kleene’s classical recursive function theory. We first consider the GPAC and several of its extensions to show that all these models have drawbacks and we introduce an alternative continuous-time model of computation that solve these problems. We also show that this new model preserve all the significant relations involving the previous models (namely, the equivalence with the differentially algebraic functions). We then continue with the topic of recursive functions over the reals, and we show full connections between functions generated by the model introduced so far and a particular class of recursive functions over the reals. 1

### Citations

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155 |
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(Show Context)
Citation Context ... between recursive functions over the naturals and recursive functions over the reals are established. This work does not rely in the coding of Turing machines into some real numbers as in [8], [10], =-=[15]-=-, and hence presents an alternative path to the traditional work in the field. Acknowledgments. We would like to thank Cris Moore and Manuel Campagnolo for several useful discussions related to this w... |

101 |
Unpredictability and undecidability in dynamical systems
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(Show Context)
Citation Context ...ctions between recursive functions over the naturals and recursive functions over the reals are established. This work does not rely in the coding of Turing machines into some real numbers as in [8], =-=[10]-=-, [15], and hence presents an alternative path to the traditional work in the field. Acknowledgments. We would like to thank Cris Moore and Manuel Campagnolo for several useful discussions related to ... |

81 | Recursion Theory on the Reals and Continuous-Time Computation, in "Theor
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(Show Context)
Citation Context ... than the GPAC, but that still preserves the fundamental properties (equivalence with differentially algebraic functions). Our objective is to develop some of the ideas presented in the seminal paper =-=[Moo96]-=-. In this paper, Moore introduced a recursion theory over the reals and proved some results establishing links with classical recursion theory and also with the GPAC. Unfortunately, these results pres... |

71 | Special functions of applied mathematics - Carlson - 1977 |

70 |
Abstract Computability and Its Relation to the General Purpose Analog Computer (Some Connections Between Logic, Differential Equations and Analog Computers
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(Show Context)
Citation Context ...nct input for a GPAC, then they all depend on a parameter t that we will call the time (this is not very clear in existing literature. Shannon [Sha41] assumes independent variables as inputs. Pour-El =-=[Pou74]-=- talks on inputs depending on the time, but then assumes independent variables). For the one variable case, these approaches are equivalent: if x is the input of a GPAC, simply take x as the independe... |

60 |
Mathematical Theory of the Differential Analyzer
- Shannon
- 1941
(Show Context)
Citation Context ...sts on a general theory of analog computation, and this still seems far away. In this paper we will go to the roots of analog computation by recalling Shannon’s General Purpose Analog Computer (GPAC) =-=[Sha41]-=-. We will point out some problems that appear in the scope of this model and also in its subsequent versions. We will then present an alternative model (the feedforward GPAC: FF-GPAC) that solves the ... |

54 |
Computability with lowdimensional dynamical systems
- Koiran, Cosnard, et al.
- 1994
(Show Context)
Citation Context ...connections between recursive functions over the naturals and recursive functions over the reals are established. This work does not rely in the coding of Turing machines into some real numbers as in =-=[8]-=-, [10], [15], and hence presents an alternative path to the traditional work in the field. Acknowledgments. We would like to thank Cris Moore and Manuel Campagnolo for several useful discussions relat... |

35 |
The differential analyzer. A new machine for solving differential equations
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- 1931
(Show Context)
Citation Context ...f the GPAC was needed for our purposes. 2 The GPAC model In 1941, Shannon introduced the General Purpose Analog Computer [Sha41] as a mathematical model of an analog device, the Differential Analyser =-=[Bus31]-=-. This model basically consists of circuits composed of (finite number of) “black boxes” as indicated in fig. 2.1 (the so-called analog units). It is required that two inputs and two outputs can never... |

35 | An analog characterization of the Grzegorczyk hierarchy
- Campagnolo, Moore, et al.
(Show Context)
Citation Context ...e was able to show that the most usual functions used in analysis are R-recursive. For the sake of completeness, we also note that Campagnolo and Costa added several contributions to this theory [3], =-=[4]-=-, [5]. It was also indicated in [11] that a subclass of R-recursive functions corresponded exactly to functions generated by a T-GPAC [11, proposition 9]. Unfortunately, we believe that this result pr... |

35 |
A dierentially algebraic replacement theorem, and analog computation
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- 1987
(Show Context)
Citation Context ...in order to distinguish this model from Shannon’s GPAC). The original paper [Pou74] can be referred for more details on the T-GPAC. Pour-El proved (with some corrections made by Lipshitz and Rubel in =-=[LR87]-=-) the following results: Theorem 3 (Pour-El) Let y be a differentially algebraic function on I. Then there exists a closed subinterval I ′ ⊆ I with non-empty interior such that, on I ′ , y can be gene... |

34 | Iteration, inequalities, and differentiability in analog computers - Campagnolo, Moore, et al. |

10 |
A Concise Introduction to the Theory of Integration
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(Show Context)
Citation Context ...integrator units, one problem still arises: if I = [a, b] is a closed interval, the RiemannStieljes integral � ϕ(t)dψ(t) is not defined for every pair of functions ϕ, ψ, even I if they are continuous =-=[Str99]-=-. However, it is possible to show that if ϕ, ψ are continuously differentiable on I, then � I ϕ(x)dψ(x) is defined. So, from now on, we will always assume that the inputs are continuously differentiab... |

8 | Upper and lower bounds on continuous-time computation
- Campagnolo, Moore
- 2001
(Show Context)
Citation Context ... Moore was able to show that the most usual functions used in analysis are R-recursive. For the sake of completeness, we also note that Campagnolo and Costa added several contributions to this theory =-=[3]-=-, [4], [5]. It was also indicated in [11] that a subclass of R-recursive functions corresponded exactly to functions generated by a T-GPAC [11, proposition 9]. Unfortunately, we believe that this resu... |

7 |
The General Purpose Analog Computer and recursive functions over the reals
- Graça
(Show Context)
Citation Context ...lowing results and may be seen as physical constraints to which all units are subjected. 4 Properties of the FF-GPAC We now state the following theorems. Their proofs can be found in the dissertation =-=[Gra02]-=-. Theorem 9 Suppose that the input functions of a FF-GPAC are of class C r on some interval I, for some r ≥ 1, possibly ∞. Then the outputs are also of class C r on I. Theorem 10 Consider a FF-GPAC wi... |

5 | Mathematical theory of the di erential analyzer - Shannon - 1941 |

4 |
Iteration, inequalities, and di#erentiability in analog computers
- Campagnolo, Moore, et al.
- 2000
(Show Context)
Citation Context ... able to show that the most usual functions used in analysis are R-recursive. For the sake of completeness, we also note that Campagnolo and Costa added several contributions to this theory [3], [4], =-=[5]-=-. It was also indicated in [11] that a subclass of R-recursive functions corresponded exactly to functions generated by a T-GPAC [11, proposition 9]. Unfortunately, we believe that this result present... |

3 |
The di#erential analyzer. A new machine for solving di#erential equations
- Bush
- 1931
(Show Context)
Citation Context ...on of the GPAC was needed for our purposes. 2 The GPAC model In 1941, Shannon introduced the General Purpose Analog Computer [14] as a mathematical model of an analog device, the Di#erential Analyzer =-=[1]. This mod-=-el basically consists of circuits composed of (a finite number of) "black boxes" as indicated in fig. 2.1 (the so-called analog units). 1 It is required that two inputs and two outputs can n... |