In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time.

“Hypercomputation ” is often defined as transcending Turing computation in the sense of computing a larger class of functions than can Turing machines. While this possibility is important and interesting, this paper argues that there are many other important senses in which we may “transcend Turing computation. ” Turing computation, like all models, exists in a frame of relevance, which underlies the assumptions on which it rests and the questions that it is suited to answer. Although appropriate in many circumstances, there are other important applications of the idea of computation for which this model is not relevant. Therefore we should supplement it with new models based on different assumptions and suited to answering different questions. In alternative frames of relevance, including natural computation and nanocomputation, the central issues include real-time response, continuity, indeterminacy, and parallelism. Once we understand computation in a broader sense, we can see new possibilities for using physical processes to achieve computational goals, which will increase in importance as we approach the limits of electronic binary logic. Key words: hypercomputation, Church-Turing thesis, natural computation, theory of computation, model of computation, Turing computation,

Up to Turing power, all computations are describable by suitable programs, which correspond to the prescription by finite means of some rational parameters of the system or some computable reals. ¿From Turing power up we have computations that are not describable by finite means: computation without a program. When we observe natural phenomena and endow them with computational significance, it is not the algorithm we are observing but the process. Some objects near us may be performing hypercomputation: we observe them, but we will never be able to simulate their behaviour on a computer. What is then the profit of such a theory of computation to Science? The theory of analog computation, where the internal states of a computer are continuous rather than discrete, has enjoyed a recent resurgence of interest. This stems partly from a wider program of exploring alternative approaches to computation, such as neural and quantum computation; partly as an abstraction

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

eigenproblem ∗

Lee A. Rubel defined the extended analog computer to remove the limitations of Shannon’s general purpose analog computer. Partial differential equation solvers were a “quintessential ” part of Rubel’s theoretical machine. These components have been implemented with “empty space ” (VLSI circuits without transistors), as well as conductive plastic. For the past decade research at Indiana University has explored the design and applications of extended analog computers. The machines have become increasingly sophisticated and flexible. The “empty ” computational area solves partial differential equations. The rest of the space includes fuzzy logic elements, configuration memory and input/output channels. This paper describes the theoretical definition, architecture and implementation of these unconventional computers. Two applications are described in detail. Rubel’s model can be viewed as an abstract specification for a distributed supercomputer. We close with a description of this inexpensive 64-node supercomputer that is based on our current single processor, and which has been placed into the public domain. The next step is to implement the improved architecture in VLSI, and seek computation speeds approaching trillions of partial differential equations per second.

We argue that post-Moore’s Law computing technology will require the exploitation of new physical processes for computational purposes, which will be facilitated by new models of computation. After a brief discussion of computation in the broad sense, we present a model of generalized computation, and a corresponding machine model, which can be applied to massively-parallel nanocomputation in bulk materials. The machine is able to implement quite general transformations on a broad class of topological spaces by means of Hilbert-space representations. Neural morphogenesis provides a model for the physical structure of the machine and means by which it may be configured, a process that involves the definition of signal pathways between two-dimensional data areas and the setting of interconnection strengths within them. This approach also provides a very flexible means of reconfiguring of the internal structure of the machine.

My goal in this report is to recontextualize the concept of computation. I review the historical roots of Church-Turing computation to show that the theory exists in a frame of relevance, which underlies the assumptions on which it rests and the questions it is suited to answer. Although this frame of relevance is appropriate in many circumstances, there are many important applications of the idea of computation for which it is not relevant. These include natural computation (computation occurring in or inspired by nature), nanocomputation (computation based on nanoscale objects and processes), and computation based on quantum theory. As a consequence we need, not so much to abandon the Church-Turing model of computation, as to supplement it with new models based on different assumptions and suited to answering different questions. Therefore I will discuss alternative frames of relevance more suited to the interrelated application areas of natural computation, emergent computation, and nanocomputation. Central issues include continuity, indeterminacy, and parallelism. Finally, I will argue that once we understand computation in a broader sense than the Church-Turing model, we begin to see new possibilities for using natural processes to achieve our computational goals. These possibilities will increase in importance as we approach the limits of electronic binary logic as a basis for computation. They will also help us to understand computational processes in nature. * This report is based on an invited presentation at the workshop “Natural Processes & Models of

Up to Turing power, all computations are describable by suitable programs, which correspond to the prescription by finite means of some rational parameters of the system or some computable reals. ¿From Turing power up we have computations that are not describable by finite means: computation without a program. When we observe natural phenomena and endow them with computational significance, it is not the algorithm we are observing but the process. Some objects near us may be performing hypercomputation: we observe them, but we will never be able to simulate their behaviour on a computer. What is then the profit of such a theory of computation to Science?

In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations. We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines. We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56. Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.