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The complexity of analog computation. (English) Zbl 0594.68040
Summary: 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 efficienty (in polynomial time) by a digital computer. From this assumption and the assumption that $$P\neq NP$$ 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 $$P\neq NP$$, 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.

##### MSC:
 68Q25 Analysis of algorithms and problem complexity 68N99 Theory of software 68U20 Simulation (MSC2010) 68Q05 Models of computation (Turing machines, etc.) (MSC2010)
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##### References:
 [1] Barahona, F., On the computational complexity of Ising spin Glass models, J. phys., A 15, 3241-3253, (1982) [2] Bennet, C.H., The thermodynamics of computation—a review, Internat. J. theoret. phys., 21, 905-940, (1982) [3] C.H. Bennett, On the logical ‘depth’ of sequences and their reducibilities to random sequences, Inform. and Control, to appear. [4] Bush, V., The differential analyzer, J. franklin inst., 212, 447-488, (1931) · Zbl 0003.06504 [5] Chua, L.O.; Lin, G-N., Nonlinear programming without computation, IEEE trans. circuits and systems, 31, 182-188, (1984) [6] Church, A., An unsolvable problem of elementary number theory, Amer. J. math., 58, 345-363, (1936), (Reprinted in [7].) · JFM 62.0046.01 [7] Davis, M., The undecidable, (1965), Raven Press Hewlett, NY [8] Feynman, R.P., Simulating physics with computers, Internat. J. theoret. phys., 21, 467-488, (1982) [9] Garey, M.R.; Johnson, D.S., Computers and intractability: A guide to the theory of NP-completeness, (1979), Freeman San Francisco, CA · Zbl 0411.68039 [10] Geman, S.; Geman, D., Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE trans. pattern analysis machine intell., 6, 721-741, (1984) · Zbl 0573.62030 [11] Henrici, P., Discrete variable methods in ordinary differential equations, (1962), Wiley New York · Zbl 0112.34901 [12] Hopfield, J.J.; Tank, D.W., ‘neural’ computation of decisions in optimization problems, Biol. cybernet., 52, 1-12, (1985) · Zbl 0572.68041 [13] J.J. Hopfield and D.W. Tank, Collective computation with continuous variables, in: Disordered Systems and Biological Organization, to appear. · Zbl 1356.92005 [14] Jackson, A., Analog computation, (1960), McGraw-Hill New York · Zbl 0098.10406 [15] Johnson, D.S., The NP-completeness column: an ongoing guide, J. algorithms, 4, 87-100, (1983) · Zbl 0509.68034 [16] Karplus, W.; Soroka, W., Analog methods, (1959), McGraw Hill New York [17] Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P., Optimization by simulated annealing, Science, 220, 671-680, (1983) · Zbl 1225.90162 [18] Miehle, W., Link-length minimization in networks, Oper res., 6, 232-243, (1958) · Zbl 1414.90068 [19] Papadimitriou, C.H.; Steiglitz, K., Combinatorial optimization: algorithms and complexity, (1982), Prentice-Hall Englewood Cliffs, NJ · Zbl 0503.90060 [20] Phelan, R.M., Fundamentals of mechanical design, (), 433-434 [21] Plaisted, D., Some polynomial and integer divisibility problems are NP-hard, Proc. 17th ann, symp. on foundations of computer science, 264-267, (1976) [22] Pour-El, M.B., Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations, and analog computers), Trans. amer. math. soc., 199, 1-28, (1974) · Zbl 0296.02022 [23] Pour-El, M.B.; Richards, I., The wave equation with computable initial data such that its unique solution is not computable, Adv. in math., 39, 215-239, (1981) · Zbl 0465.35054 [24] Pour-El, M.B.; Richards, I., Noncomputability in models of physical phenomena, Internat. J. theoret. phys., 21, 553-555, (1982) · Zbl 0493.35057 [25] Pyne, I.B., Linear programming on an electronic analogue computer, Trans. AIEE, part 1, 75, 139-143, (1956) [26] Shannon, C.E., Mathematical theory of the differential analyzer, J. math. phys., 20, 337-354, (1941) · Zbl 0061.29410 [27] Tank, D.W.; Hopfield, J.J., Simple ‘neural’ optimization networks: an A/D converter, signal decision circuit and a linear programming circuit, (1985), preprint · Zbl 0572.68041 [28] Turing, A.M., On computable numbers, with an application to the entscheidungsproblem, Proc. London math. soc., series 2, Proc. London math. soc., series 2, 43, 544-546, (1937), (Reprinted in [7].) · Zbl 0018.19304
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