Some recent developments on Shannon's General Purpose Analog Computer

This paper revisits one of the rst 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 re ned. With this we prove the following: (i) the previous model can be simpli ed; (ii) it admits extensions having close connec- tions 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

Computability and dynamical systems: a perspective

In this paper we look at dynamical systems from a computability perspective. We survey some topics and themes of research for dynamical systems and then see how they can be fitted in a computational framework. We will recall some selected results, and enounce problems that might lay possible routes for further research

Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length

We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous (time and space) elegant and simple characterization of $\operatorname{PTIME}$. This is the first time such classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of computable analysis. This extends to deterministic complexity classes above polynomial time. This may provide a new perspective on classical complexity, by giving a way to define complexity classes, like $\operatorname{PTIME}$, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations, i.e.~by using the framework of analysis

Non-computability, unpredictability, and financial markets

One of the most significant achievements from theoretical computer science was to show that there are non-computable problems, which can- not be solved through algorithms. Although the formulation of such prob- lems is mathematical, they often can be interpreted as problems derived from other elds, like physics or computer science. However no non- computable problem with economical or financial inspiration has been presented before.1 Here we study the problem of valuation: given some adequate data, find the value of an asset. Valuation is modeled mathemat- ically by the discounted cash ow operator. We show, using surprisingly simple arguments, that this operator is not computable. Since, theoreti- cally, financial markets should trade assets based on their fair value, our result suggests that unpredictability of such markets may partially stem from inherent non-computable behavior. A discussion of this result is also included

Computability with polynomial differential equations

Tese dout., Matemática, Inst. Superior Técnico, Univ. Técnica de Lisboa, 2007Nesta dissertação iremos analisar um modelo de computação analógica, baseado em equações diferenciais polinomiais. Começa-se por estudar algumas propriedades das equações diferenciais polinomiais, em particular a sua equivalência a outro modelo baseado em circuitos analógicos (GPAC), introduzido por C. Shannon em 1941, e que é uma idealização de um dispositivo físico, o Analisador Diferencial. Seguidamente, estuda-se o poder computacional do modelo. Mais concretamente, mostra-se que ele pode simular máquinas de Turing, de uma forma robusta a erros, pelo que este modelo é capaz de efectuar computações de Tipo-1. Esta simulação é feita em tempo contínuo. Mais, mostramos que utilizando um enquadramento apropriado, o modelo é equivalente à Análise Computável, isto é, à computação de Tipo-2. Finalmente, estudam-se algumas limitações computacionais referentes aos problemas de valor inicial (PVIs) definidos por equações diferenciais ordinárias. Em particular: (i) mostra-se que mesmo que o PVI seja definido por uma função analítica e que a mesma, assim como as condições iniciais, sejam computáveis, o respectivo intervalo maximal de existência da solução não é necessariamente computável; (ii) estabelecem-se limites para o grau de não-computabilidade, mostrando-se que o intervalo maximal é, em condições muito gerais, recursivamente enumerável; (iii) mostra-se que o problema de decidir se o intervalo maximal é ou não limitado é indecídivel, mesmo que se considerem apenas PVIs polinomiais

Computability via analog circuits

In this paper we are interested in a particular model of analog computation, the General Purpose Analog Computer (GPAC). In particular, we provide more solid foundations for this model and we show that it can be used to introduce a notion of computability for smooth continuous dynamical systems over Rn. We also show that hierarchies over these dynamical systems can be established, thereby defining a notion of relative computability

On the functions generated by the general purpose analog computer

PreprintWe consider the General Purpose Analog Computer (GPAC), introduced by Claude Shannon in 1941 as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later one electronic) machines of that time. The GPAC generates as output univariate functions (i.e. functions f:R→R). In this paper we extend this model by: (i) allowing multivariate functions (i.e. functions f:Rn→Rm); (ii) introducing a notion of amount of resources (space) needed to generate a function, which allows the stratification of GPAC generable functions into proper subclasses. We also prove that a wide class of (continuous and discontinuous) functions can be uniformly approximated over their full domain. We prove a few stability properties of this model, mostly stability by arithmetic operations, composition and ODE solving, taking into account the amount of resources needed to perform each operation. We establish that generable functions are always analytic but that they can nonetheless (uniformly) approximate a wide range of nonanalytic functions. Our model and results extend some of the results from [19] to the multidimensional case, allow one to define classes of functions generated by GPACs which take into account bounded resources, and also strengthen the approximation result from [19] over a compact domain to a uniform approximation result over unbounded domains.info:eu-repo/semantics/acceptedVersio