The extended analog computer and functions computable in a digital sense

In this paper we compare the computational power of the Extended Analog Computer (EAC) with partial recursive functions. We first give a survey of some part of computational theory in discrete and in real space. In the last section we show that the EAC can generate any partial recursive function defined over N. Moreover we conclude that the classical halting problem for partial recursive functions is an equivalent of testing by EAC if sets are empty or not

Approximability in the GPAC

Most of the physical processes arising in nature are modeled by either ordinary or partial differential equations. From the point of view of analog computability, the existence of an effective way to obtain solutions of these systems is essential. A pioneering model of analog computation is the General Purpose Analog Computer (GPAC), introduced by Shannon as a model of the Differential Analyzer and improved by Pour-El, Lipshitz and Rubel, Costa and Gra\c{c}a and others. Its power is known to be characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations. We address one of the limitations of this model, concerning the notion of approximability, a desirable property in computation over continuous spaces that is however absent in the GPAC. In particular, the Shannon GPAC cannot be used to generate non-differentially algebraic functions which can be approximately computed in other models of computation. We extend the class of data types using networks with channels which carry information on a general complete metric space $X$; for example $X=C(R,R)$, the class of continuous functions of one real (spatial) variable. We consider the original modules in Shannon's construction (constants, adders, multipliers, integrators) and we add \emph{(continuous or discrete) limit} modules which have one input and one output. We then define an L-GPAC to be a network built with $X$-stream channels and the above-mentioned modules. This leads us to a framework in which the specifications of such analog systems are given by fixed points of certain operators on continuous data streams. We study these analog systems and their associated operators, and show how some classically non-generable functions, such as the gamma function and the zeta function, can be captured with the L-GPAC

Mathematik ohne Ziffern - Analoge Rechengeräte

Der Verfasser ist bestrebt, in dieser Arbeit zur Geschichte der Rechentechnik – nach ‚Rechnen mit Maschinen‘ (Vieweg, 1968) und der dreibändigen, von Dr. Friedrich Genser überarbeiteten und erweiterten Neufassung ‚Vom Zahnrad zum Chip‘ (Superbrain-Verlag, 2004) – nun auch die einstigen analogen mathematischen Instrumente bis zu den Analogrechnern / Integrieranlagen zu beschreiben

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