The Extended Analog Computer and Turing machine

In this paper we compare computational power of two models of analog and classicalcomputers. As a model of analog computer we use the model proposed by Rubel in 1993 called theExtended Analog Computer (EAC) while as a model of classical computer, the Turing machines.Showing that the Extended Analog Computer can robustly generate result of any Turing machinewe use the method of simulation proposed by D.S. Graça, M.L. Campagnolo and J. Buescu [1] in2005

Computability with polynomial differential equations

In this paper, we show that there are Initial Value Problems de ned with polynomial ordinary di erential equations that can simulate univer- sal Turing machines in the presence of bounded noise. The polynomial ODE de ning the IVP is explicitly obtained and the simulation is per- formed in real time

A foundation for real recursive function theory

The class of recursive functions over the reals, denoted by REC(R), was introduced by Cristopher Moore in his seminal paper written in 1995. Since then many subsequent investigations brought new results: the class REC(R) was put in relation with the class of functions generated by the General Purpose Analogue Computer of Claude Shannon; classical digital computation was embedded in several ways into the new model of computation; restrictions of REC(R) were proved to represent different classes of recursive functions, e.g., recursive, primitive recursive and elementary functions, and structures such as the Ritchie and the Grzergorczyk hierarchies. The class of real recursive functions was then stratified in a natural way, and REC(R) and the analytic hierarchy were recently recognised as two faces of the same mathematical concept. In this new article, we bring a strong foundational support to the Real Recursive Function Theory, rooted in Mathematical Analysis, in a way that the reader can easily recognise both its intrinsic mathematical beauty and its extreme simplicity. The new paradigm is now robust and smooth enough to be taught. To achieve such a result some concepts had to change and some new results were added

Applications of real recursive infinite limits

Doutor in Informatics, speciality of Theory of ComputationUsando a teoria das funções reais recursivas, que deriva da proposta original em [Moo96], mostramos como cada função periódica definida por partes, que admite um desenvolvimento em série de Fourier, pode ser definida como uma destas funções reais recursivas. Demonstramos, também, que o poder computacional de um certo tipo de autómatos finitos em tempo contínuo está limitado à computação de sinais que são descritos por funções lineares parcialmente periódicas definidas por partes, as quais constituem um subconjunto muito restrito de sinais que podem ser gerados por funções reais recursivas. Uma função real recursiva com limites infinitos é apresentada para simular máquinas de Turing em tempo infinito, restrito a w2, bem como o seu poder computacional, nomeadamente para decidir as respectivas aproximações w2 aos problemas da paragem e, ainda, a hierarquia da aritmética recorrendo a um número finito de limites. Para isso, é introduzido um novo esquema de iteração nos ordinais até w2, que simula as máquinas de Turing em tempo infinito com a codificação para inputs binários finitos, introduzida por Christopher Moore, e o sistema de equações diferenciais da simulação da máquina de Turing, introduzido, recentemente, por Jerzy Mycka e José Félix Costa

A nonlinear dynamic system for spread spectrum code acquisition

Thesis (S.M.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 1999.Includes bibliographical references (leaves 88-89).Nonlinear differential equations and iterated maps can perform any computation. Sometimes, the most difficult part of performing a useful computation, however, is writing the program. Furthermore, in practice, we often need to build special purpose computing hardware suited to run a particular program. Nonlinear dynamics provides a novel and useful language for constructing "algorithms" and "computer architectures." We apply the language of nonlinear dynamics to solve a fast coding problem which has previously been implemented by a Digital Signal Processor chip in digital wireless receivers. We eventually hope to produce a novel physical system which exhibits the nonlinear dynamics we require, thereby creating one of the first nonlinear dynamic systems engineered to perform a practical computation. This system, called an Analog Feedback Shift Register (AFSR), should be a faster, more reliable, less expensive, and lower power Spread Spectrum (SS) code acquisition system for wireless receivers. A prohibitive factor in creating ubiquitous short range, digital radio transceivers is the difficulty and expense of creating a mechanism for locking onto the incoming Spread Spectrum code sequence. AFSR is also potentially useful in other applications where low cost, low power channel sharing or addressing is required, for example in wireless auto-identification tags.by Benjamin William Vigoda.S.M

Iteration, Inequalities, and Differentiability in Analog Computers

Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G such that F (x; t) = f t (x) for non-negative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x k (x) that sense inequalities in a differentiable way, the resulting class, which we call G + k , is closed under iteration. Furthermore, G + k includes all primitive recursive functions, and has the additional closure property that if T (x) is in G+ k , then any function of x computable by a Turing machine in T (x) time is also