Results 1  10
of
23
Analog computers and recursive functions over the reals
 Journal of Complexity
, 2003
"... 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 mode ..."
Abstract

Cited by 34 (19 self)
 Add to MetaCart
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 continuoustime 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
A Survey of ContinuousTime Computation Theory
 Advances in Algorithms, Languages, and Complexity
, 1997
"... Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists o ..."
Abstract

Cited by 29 (6 self)
 Add to MetaCart
Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists on the general theory of continuoustime models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. 1 Introduction After a long period of oblivion, interest in analog computation is again on the rise. The immediate cause for this new wave of activity is surely the success of the neural networks "revolution", which has provided hardware designers with several new numerically based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. [30, 45]). However, the more fundamental...
Polynomial differential equations compute all real computable functions on computable compact intervals
, 2007
"... ..."
Some recent developments on Shannon’s general purpose analog computer
 Mathematical Logic Quarterly
"... This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be ..."
Abstract

Cited by 18 (7 self)
 Add to MetaCart
This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be simplified; (ii) it admits extensions having close connections with the class of smooth continuous time dynamical systems. As a consequence, we conclude that some of these extensions achieve Turing universality. Finally, it is shown that if we introduce a new notion of computability for the GPAC, based on ideas from computable analysis, then one can compute transcendentally transcendental functions such as the Gamma function or Riemann’s Zeta function. 1
The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation
 Theory and Applications of Models of Computation, Third International Conference, TAMC 2006
, 2006
"... Abstract. In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Abstract. In this paper we revisit one of the first models of analog computation, Shannon’s General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, we compute exactly all real computable functions (in the sense of computable analysis). Moreover, since GPACs are equivalent to systems of polynomial differential equations then we show that all real computable functions can be defined by such models. 1
A network model of analogue computation over metric algebras
 Torenvliet (Eds.), Computability in Europe, 2005, Springer Lecture Notes in Computer Science
, 2005
"... Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the network with system parameters from A is modelled by a function Φ: C[T,A] p ×A r →C[T,A] q (p, q> 0,r ≥ 0), where C[T,A] is the set of all continuous streams equipped with the compactopen topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T,A] using a contraction principle. We analyse a case study involving a mechanical system. Finally, we introduce a custommade concrete computation theory over C[T,A] and show that if the modules are concretely computable then so is the function Φ. 1
Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains
"... Abstract. In this paper we consider the computational complexity of solving initialvalue problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of R n and C n, under the Computable Analysis setting. We show that the solution can be computed in polynomial time o ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. In this paper we consider the computational complexity of solving initialvalue problems defined with analytic ordinary differential equations (ODEs) over unbounded domains of R n and C n, under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of definition, provided it satisfies a very generous bound on its growth, and that the function admits an analytic extension to the complex plane. 1
The Differential Analyzer of Vannevar Bush is a machine for solving
, 2007
"... differential equations with reasonable boundary conditions. The original differential analyzer at MIT was capable of solving ”ordinary differential equations of any order up to the sixth, and with any amount of complexity within reason. ” (Bush, 451) An example of an equation that can be solved usin ..."
Abstract
 Add to MetaCart
differential equations with reasonable boundary conditions. The original differential analyzer at MIT was capable of solving ”ordinary differential equations of any order up to the sixth, and with any amount of complexity within reason. ” (Bush, 451) An example of an equation that can be solved using the differential analyzer follows (a machine configuration for the equation will be presented in section 2): d2x dx + k + g = 0 (1) dt2 dt An analog machine like the differential analyzer is uniquely suited for the solution of such equations, since the solutions involve the integration of continuous functions. In fact, the machine is actually configured as a mechanical representation of the equation to be modelled. The machine consists of a number of bus shafts, which can provide the input or accept the output of a number of functional units. Functional units are attached to the bus shafts by the use of spiral gear boxes, which allows the machine to be specialized for a wide variety of equations. The machine is ”programmed ” by applying appropriate interconnections of bus shafts, functional units, and input and output tables. 2 Example Machine Configuration Recall equation one (1) from above; rearranging the equation yields: dx = − k dt dx + g dt (2) dt The schematic in FIG. 1 follows more naturally from this arrangement. Note for the input to that the output of the integrator labelled II provides dx dt integrator II: � k dx dt + g �. The output also drives the input to integrator I, which has as its output the function x. Here the constant g is provided using an input table, so that it can be easily changed, while the constant multiplicative factor k has been introduced using a spur gear box. The output table has been set to record the dependent variable x and its derivative as functions of the independent variable t. 1 Figure 1. It was customary in schematics such as these, which represent the basic conceptual layout of the machine, to omit indications of sign and relative scales. In fact, a lefthand spiral gear box should connect the output of II with the input of I, to accurately reflect the equation. 3