This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes P d , Co-RP d , NP d

We propose certain non-Turing models of computation, but our intent is not to advocate models that surpass the power of Turing Machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of application and are ill-suited to use outside that domain. Hence we review the presuppositions and context of the TM model and show that it is unsuited to natural computation (computation occurring in or inspired by nature). Therefore we must consider an expanded definition of computation that includes alternative (especially analog) models as well as the TM. Finally we present an alternative model, of continuous computation, more suited to natural computation. We conclude with remarks on the expressivity of formal mathematics. Key words: analog computation, analog computer, biocomputation, computability, computation on reals, continuous computation, formal system, hypercomputation,

. We present an algorithm to determine the essential singular components of an algebraic differential equation. Geometrically, this corresponds to determining the singular solution that have enveloping properties. The algorithm is practical and efficient because it is factorization free, unlike the previous such algorithm. 1. Introduction We present an algorithm to determine the set of essential singular solutions of a differential equation. Essential singular solutions can be informally described as follows: the general solution of a differential equation is usually described as a solution depending on a number of arbitrary constants equal to the order of the differential equation. Then essential singular solutions are those that cannot be obtained by substituting numerical values to the arbitrary constants in the general solution. Adherence, defined in (Ritt 1950, VI.2), is the correct concept: singular solutions that are not essential are adherent to the general solution or to one...

this paper we apply model-fitting to

Proposed running head: Attractors: topology and dynamics Many a dissipative evolution equation possesses a global attractor A with finite Hausdorff dimension d. In this paper it is shown there is an embedding X of A into IR N, with N = [2d + 2], such that X is the global attractor of some finite-dimensional system on IR N with trivial dynamics on X. This allows the construction of a discrete dynamical system on IR N which reproduces the dynamics of the time T map on A, and has an attractor within an arbitrarily small neighbourhood of X. If the Hausdorff dimension is replaced by the fractal dimension, a similar construction can be shown to hold good even if one restricts to orthogonal projections rather than arbitrary embeddings.

We analyse the complexity of a simple algorithm for computing asymptotic solutions of an algebraic differential equation. This analysis is based on a computation of the number of possible asymptotic monomials of a certain order, and on the study of the growth of this number as the order of the equation grows. Comportements asymptotiques et 'equations diff'erentielles alg'ebriques R'esum'e Nous analysons la complexit'e d'un algorithme simple de calcul des solutions asymptotiques d"equations diff'erentielles alg'ebriques. L'analyse est bas'ee sur la d'etermination du nombre de monomes asymptotiques d'ordre donn'e, et sur l"etude de la croissance de ce nombre lorsque l'ordre de l"equation croit. To appear in the Journal of Symbolic Computation. Asymptotic Forms and Algebraic Differential Equations JOHN SHACKELL (jrs@ukc.ac.uk) University of Kent at Canterbury, Canterbury, Kent CT2 7NF, England BRUNO SALVY (Bruno.Salvy@inria.fr) INRIA Rocquencourt, 78153 Le Chesnay Cedex, France We ...

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

algorithms

Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.