We ask if analog computers can solve NP-complete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP we can draw conclusions about the operation of physical devices used for computation. An NP-complete problem, 3-SAT, is reduced to the problem of checking whether a feasible point is a local optimum of an optimization problem. A mechanical device is proposed for the solution of this problem. It encodes variables as shaft angles and uses gears and smooth cams. If we grant Strong Church’s Thesis, that P ≠ NP, and a certain ‘‘Downhill Principle’ ’ governing the physical behavior of the machine, we conclude that it cannot operate successfully while using only polynomial resources. We next prove Strong Church’s Thesis for a class of analog computers described by well-behaved ordinary differential equations, which we can take as representing part of classical mechanics. We conclude with a comment on the recently discovered connection between spin glasses and combinatorial optimization. 1.

Abstract — A class of neural networks that solve linear programming problems is analyzed. The neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact (nondifferentiable) penalty functions. It is proved that for a given linear programming problem and sufficiently large penalty parameters, any trajectory of the neural network converges in finite time to its solution set. For the analysis, Lyapunov-type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets. The results are illustrated via numerical simulation examples. Index Terms—Invariant sets, linear programming, neural networks, nondifferentiable optimization, penalty functions, sliding modes. I.

Sliding modes are used to analyze a class of dynamical systems that solve convex programming problems. The analysis is carried out using concepts from the theory of differential equations with discontinuous right-hand sides and Lyapunov stability theory. It is shown that the equilibrium points of the system coincide with the minimizers of the convex programming problem, and that irrespective of the initial state of the system the state trajectory converges to the solution set of the problem. The dynamic behavior of the systems is illustrated by two numerical examples.

This thesis develops an engineering practice and design methodology to enable us to use CMOS analog VLSI chips to perform more accurate and precise computation. These techniques form the basis of an approach that permits us to build computer graphics and neural network applications using analog VLSI. The nature of the design methodology focuses on defining goals for circuit behavior to be met as part of the design process. To increase the accuracy of analog computation, we develop techniques for creating compensated circuit building blocks, where compensation implies the cancellation of device variations, offsets, and nonlinearities. These compensated building blocks can be used as components in larger and more complex circuits, which can then also be compensated. To this end, we develop techniques for automatically determining appropriate parameters for circuits, using constrained optimization. We also fabricate circuits that implement multi-dimensional gradient estimation for a grad...