## Figures from this paper

## 108 Citations

Some recent developments on Shannon's General Purpose Analog Computer

- Mathematics, Computer ScienceMath. Log. Q.
- 2004

It is shown that if a new notion of computability is introduced 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.

Elementarily computable functions over the real numbers and R-sub-recursive functions

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2005

The Extended Analog Computer and Functions Computable in a Digital Sense

- Mathematics, Computer ScienceActa Cybern.
- 2010

It is shown that the EAC can generate any partial recursive function defined over N, and the classical halting problem for partial recursive functions is an equivalent of testing by EAC if sets are empty or not.

The General Purpose Analog Computer and Computable Analysis are Two Equivalent Paradigms of Analog Computation

- Computer Science, MathematicsTAMC
- 2006

This paper revisits one of the first models of analog computation, Shannon's General Purpose Analog Computer (GPAC), and shows that all real computable functions can be defined by such models.

An Analog Characterization of Elementarily Computable Functions over the Real Numbers

- Mathematics, Computer ScienceICALP
- 2004

It is proved that the Grzegorczyk Hierarchy functions correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema.

On the Functions Generated by the General Purpose Analog Computer

- Mathematics, Computer ScienceInf. Comput.
- 2017

Implicit complexity in recursive analysis

- Computer Science, Mathematics
- 2009

This paper provides a framework that allows to dive into complexity for functions over the reals and provides the first algebraic characterization of polynomial time computable functions overThe reals, inspired by Bellantoni and Cook's characterization.

Analog Computability with Differential Equations

- Mathematics, Computer Science
- 2017

This dissertation study of a pioneering model of analog computation called General Purpose Analog Computer, introduced by Shannon in 1941, presents a characterization which generalizes Shannon’s results and attempts to relate its model of computation to the notion of tracking computability as studied by Tucker and Zucker.

## References

SHOWING 1-10 OF 23 REFERENCES

The general purpose analog computer and recursive functionsover the reals

- Computer Science
- 2002

This work starts by analyzing the first known model of this kind, the General Purpose Analog Computer, and proposes an alternative approach that originates a more robust model than the GPAC, maintaining its principal properties such as the equivalence with differentially algebraic functions.

Recursion Theory on the Reals and Continuous-Time Computation

- Computer ScienceTheor. Comput. Sci.
- 1996

Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers)

- Computer Science, Mathematics
- 1974

This work presents a mathematical definition of an analog generable function of a real variable in terms of a simultaneous set of nonlinear differential equations possessing a "domain of generation," which includes functions generated by existing general-purpose analog computers.

Iteration, Inequalities, and Differentiability in Analog Computers

- MathematicsJ. Complex.
- 2000

It is shown that G, the set G of GPAC-computable functions is not closed under iteration, but a simple extension of it is, which includes all primitive recursive functions and has the additional closure property that if T(x) is in G+?k, then any function ofx computable by a Turing machine in T( x) time is also.

Upper and Lower Bounds on Continuous-Time Computation

- Computer Science, MathematicsUMC
- 2000

Shannon's General Purpose Analog Computer is considered, which is a model of computation by differential equations in continuous time, and it is shown that several classical computation classes have natural analog counterparts.

A concise introduction to the theory of integration

- Mathematics
- 1990

The choice of topics included in this book, as well as the presentation of those topics, has been guided by the author's experience in teaching this material to classes consisting of advanced…

Computability with Low-Dimensional Dynamical Systems

- Computer Science, MathematicsTheor. Comput. Sci.
- 1994