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The complexity of analog computation
 in Math. and Computers in Simulation 28(1986
"... We ask if analog computers can solve NPcomplete 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 w ..."
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We ask if analog computers can solve NPcomplete 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 NPcomplete problem, 3SAT, 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 wellbehaved 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.
THE MATHEMATICS OF COMPUTING BETWEEN LOGIC AND PHYSICS
, 2010
"... Abstract. Do physical processes compute? And what is a computation? These questions have gained a revival of interest in recent years, due to new technologies in physics, new ideas in computer sciences (for example quantum computing, networks, nondeterministic algorithms) and new concepts in logic. ..."
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Abstract. Do physical processes compute? And what is a computation? These questions have gained a revival of interest in recent years, due to new technologies in physics, new ideas in computer sciences (for example quantum computing, networks, nondeterministic algorithms) and new concepts in logic. In this paper we examine a few directions, as well as the problems they bring to the surface. Contents