## Upper and Lower Bounds on Continuous-Time Computation

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Citations: | 8 - 2 self |

### BibTeX

@MISC{Campagnolo_upperand,

author = {Manuel Lameiras Campagnolo and Cristopher Moore},

title = {Upper and Lower Bounds on Continuous-Time Computation},

year = {}

}

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### Abstract

We consider various extensions and modifications of Shannon's General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive functions, the elementary functions, the levels of the Grzegorczyk hierarchy, and the arithmetical and analytical hierarchies.

### Citations

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Citation Context ...th and 1st levels respectively, Σ0 0 and Σ0 1 .Sets in the jth level of this hierarchy can be defined with j alternating quantifiers over the set of integers, ∃ and ∀, applied to recursive predicates =-=[Odi89]-=-. We note that Bournez uses a similar recursion to show that systems with piecewise-constant derivatives can compute various levels of the hyperarithmetical hierarchy [Bou99,Bou99b]. If we quantify ov... |

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Citation Context ...nctions. In fact, it is known that flows in three dimensions, or iterated functions in two, can simulate arbitrary Turing machines. In two dimensions, these functions can be infinitely differentiable =-=[Moo90]-=-, piecewise-linear [Moo90,KCG94], or closed-form analytic and composed of a finite number of trigonometric terms [KM99]. (In [KM99] a simulation in one dimension is achieved at the cost of an exponent... |

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Citation Context ...the reals of En−1. While this produces a smaller set of functions on Ê, it produces extensions to Ê of the same set of functions on Æ as the class defined here [CMC00]. 6 Zero-finding on the reals In =-=[Moo96]-=- another definition of analog computation is proposed, the Ê-recursive functions. These are the functions that can be generated from the constants 0 and 1 from composition, integration of differential... |

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Citation Context ...dition y(x0) =y0. HereA and B are n × n and m × m matrices linear in 1 and the variables x1, ..., xm, y1, ..., yn, andy ′ is the n × m matrix of the derivatives of y with respect to x. Later, Pour-El =-=[PE74]-=- made this definition more precise by requiring the solution to be unique for all initial values belonging to a closed set with non-empty interior called the domain of generation of the initial condit... |

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Citation Context ... somewhat awkward definitions of E0 and E1 are the historical ones.) We will use the fact that for n ≥ 3, we can replace limited recursion in the definition of En with bounded sum and bounded product =-=[Ros84]-=-: Proposition 4 For n ≥ 3, E n is the smallest class containing zero, successor, the projections, cut-off subtraction, and En−1, which is closed under composition, bounded sum, and bounded product. On... |

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Citation Context ...g machines. In two dimensions, these functions can be infinitely differentiable [Moo90], piecewise-linear [Moo90,KCG94], or closed-form analytic and composed of a finite number of trigonometric terms =-=[KM99]-=-. (In [KM99] a simulation in one dimension is achieved at the cost of an exponential slowdown.) However, Proposition 12 is in some sense more elegant than these constructions, since it uses the operat... |

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Citation Context ...m of Babakhanian [Bab73]: Lemma 7 For n ≥ 0, exp [n] (x) satisfies no non-trivial algebraic differential equation of order less than n. Proposition 6, Lemma 7 and our previous remarks are combined in =-=[CMC99]-=- to prove that: Proposition 8 The class G is not closed under iteration. Specifically, there is no GPAC-computable function F (x, n) of two variables that matches the iterated exponential exp [n] (x) ... |

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Citation Context ...it that calculates sin t. Its initial conditions are sin(0) = 0 and cos(0) = 1. The output w of the integrator unit Ê obeys dw = udv where u and v are its upper and lower inputs respectively. Shannon =-=[Sha41]-=- showed that the class of functions generable in this abstract model is the set of solutions of systems of the following system of quasilinear differential equations, cos t − sin t Ê −1 sin t Ê A(x, y... |

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Citation Context ... sums, products, compositions and solutions of differential equations formed from these such as f ′ =sinf. Examples of functions which are not d.a. include Euler’s Γ function and Riemann’s ζ function =-=[Rub89b]-=- The General Purpose Analog Computer (GPAC) is a simple model of a computer evolving in continuous time. It was originally defined as a mathematical model of an analog device, the Differential Analyse... |

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Citation Context ...imate GPAC-computable functions with recursive ones. However, strictly speaking this approximation only works when a bound on the derivatives is known a priori [VSD86] or on arbitrarily small domains =-=[Rub89]-=-. If this conjecture is false, then Proposition 13 shows that G + θk contains a wide variety of non-primitive recursive functions. We close this section by noting that since all functions in G + θk ar... |

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Citation Context ...ally defined as a mathematical model of an analog device, the Differential Analyser, thesfundamental principles of which were described first by Lord Kelvin in 1876 [Kel76] and later by Vannevar Bush =-=[Bow96]-=-. The outputs are generated from the inputs by means of a dependence defined by a finite directed graph (not necessarily acyclic) where each node is either an adder, a unit that outputs the sum of its... |

12 |
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Citation Context ...g in continuous time. It was originally defined as a mathematical model of an analog device, the Differential Analyser, thesfundamental principles of which were described first by Lord Kelvin in 1876 =-=[Kel76]-=- and later by Vannevar Bush [Bow96]. The outputs are generated from the inputs by means of a dependence defined by a finite directed graph (not necessarily acyclic) where each node is either an adder,... |

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Citation Context ...ion and minimization. 4. The recursive functions are the partial recursive functions that are total. A number of our results regard the class E of elementary functions, which was introduced by Kálmar =-=[Kál43]-=-. For example, multiplication and exponentiation over Æ are both in E, since they can be written as bounded sums and products respectively: xy = � z<y x and xy = � z<y x. Since E is closed under compo... |

6 | F.: An Analog Characterization of the Subrecursive Functions
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(Show Context)
Citation Context ...unctions. First, note that by Proposition 16 all functions in L + θk are total. In addition, their growth is bounded by a finitely iterated exponential, exp [m] for some m. The following is proved in =-=[CMC00]-=-, using the fact that if f and g are bounded by a finite tower of exponentials then their composition and linear integration h = f + � gh dy as well: Proposition 18 Let h be a function in L + θk of ar... |

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3 |
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Citation Context ...unction which grows more quickly than any primitive recursive function, this proposition shows that G + θk contains many other non-primitive recursive functions as well. It is believed, but not known =-=[Hay96]-=-, that all differentially algebraic functions on the complex plane are bounded by some elementary function, i.e. exp [n] (x) forsomen, whenever they are defined for all x > 0. For real solutions of d.... |

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Equations Différentielles Ordinaires. EditionsMir,5ème edition
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Citation Context ...= f(y) whenf is Lipshitz. For linear differential equations, we can strengthen this to global existence whenever A(x) is continuous, and establish a bound on y that depends on �A(x)�: Proposition 16 (=-=[Arn96]-=-) If A(x) is defined and continuous on an interval I =[a, b] where a ≤ 0 ≤ b, then the solution of a homogeneous linear differential equation with initial condition y(0) = y0 is defined and unique on ... |

2 |
Exponentials in differentially algebraic extension fields
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(Show Context)
Citation Context ...ifferential algebra on the iterated exponential function exp [n] (x) defined by exp [0] (x) =x and exp [n] (x) =e exp[n−1] (x) . The following lemma follows from a more general theorem of Babakhanian =-=[Bab73]-=-: Lemma 7 For n ≥ 0, exp [n] (x) satisfies no non-trivial algebraic differential equation of order less than n. Proposition 6, Lemma 7 and our previous remarks are combined in [CMC99] to prove that: P... |

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1 |
The complexity of analog computation. Mathematics and computers in simulation
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Citation Context ...by using numerical integration to approximate GPAC-computable functions with recursive ones. However, strictly speaking this approximation only works when a bound on the derivatives is known a priori =-=[VSD86]-=- or on arbitrarily small domains [Rub89]. If this conjecture is false, then Proposition 13 shows that G + θk contains a wide variety of non-primitive recursive functions. We close this section by noti... |

1 | Sur la croissance des fonctions de par les equations dierentielles - Vijayaraghavan - 1932 |