The complexity of analog computation

Anastasios Vergis, Kenneth Steiglitz, Bradley Dickinson
1986 Mathematics and Computers in Simulation  
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
more » ... 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.
doi:10.1016/0378-4754(86)90105-9 fatcat:ugbfkmn455hixab3cvphmhhf3u