Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computer-generated images of such sets. The algorithms used to draw these pictures vary; the most naive work by iterating the center of a pixel to determine if it lies in the Julia set. Milnor's distance-estimator algorithm [Mil] uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution. Moreover, there are examples, even in the family of quadratic polynomials, when no satisfactory pictures of the Julia set exist. In this paper we study computability properties of Julia sets of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show:

/pdf/non-computable-julia-sets-4qj0cgr70h.pdf

Non-computable Julia sets

In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally restricted feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.

pdf/lipschitz-continuous-ordinary-differential-equations-are-1xzzkjs23p.pdf

Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete

In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally weak feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.

The computational complexity of finding a shortest path in a two-dimensional domain is studied in the Turing machine-based computational model and in the discrete complexity theory. This problem is studied with respect to two formulations of polynomial-time computable two-dimensional domains: (A) domains with polynomialtime computable boundaries, and (B) polynomial-time recognizable domains with polynomial-time computable distance functions. It is proved that the shortest path problem has the polynomial-space upper bound for domains of both type (A) and type (B); and it has a polynomial-space lower bound for the domains of type (B), and has a #P lower bound for the domains of type (A). (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the complexity of finding paths in a two-dimensional domain I: Shortest paths

On the Complexity of Finding Paths in a Two-Dimensional Domain II

Publisher Summary The chapter discusses the analysis of Polynomial-Time Computability. Polynomial-time complexity theory identifies the intuitive notion of feasible computability with the formal notion of polynomial-time computability, and allows studying the problems that are feasible and those that are infeasible. This theory closely ties the tools in abstract mathematics with practical computational problems. It has made strong impact on almost every area of discrete computation, including graph theory, combinatorial optimization, computational geometry, computational number theory, and modern cryptography. This chapter presents a survey on one of the polynomial-time complexity theories of numerical computation based on the model of recursive analysis. The advantage of using this model of computation is that it is compatible with the model of discrete complexity theory, and so the complexity of numerical problems may be studied and compared with discrete problems. A hierarchical classification of the computational complexity of numerical problems is presented in terms of the relations among discrete complexity classes. The chapter discusses different approaches to the polynomial-time complexity theory of real functions, including the works of Blum et al.

Chapter 19 Polynomial-time computability in analysis

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Chapter 19 Polynomial-time computability in analysis