Computational Complexity for Continuous Time Dynamics

Dissipative flows model a large variety of physical systems. In this Letter the evolution of such systems is interpreted as a process of computation; the attractor of the dynamics represents the output. A framework for an algorithmic analysis of dissipative flows is presented, enabling the comparison of the performance of discrete and continuous time analog computation models. A simple algorithm for finding the maximum of n numbers is analyzed, and shown to be highly efficient. The notion of tractable (polynomial) computation in the Turing model is conjectured to correspond to computation with tractable (analytically solvable) dynamical systems having polynomial complexity. The computation of a digital computer, and its mathematical abstraction, the Turing machine is described by a map on a discrete configuration space. In recent years scientists have developed new approaches to computation, some of them based on continuous time analog systems. The most promising are neuromorphic systems [1], models of human memory [2], and experimentally realizable quantum computers [3]. Although continuous time systems are widespread in experimental realizations, no theory exists for their algorithmic analysis. The standard theory of computation and computational complexity [4] deals with computation in discrete time and in a discrete configuration space, and is inadequate for the description of such systems. This Letter describes an attempt to fill this gap. Our model of a computer is based on dissipa