Abstract
It is well known that not all algorithms are feasible; whether an algorithm is feasible or not depends on how many computational steps this algorithm requires. The problem with the existing definitions of feasibility is that they are rather ad hoc. Our goal is to use the maximum entropy (MaxEnt) approach and get more motivated definitions.
If an algorithm is feasible, then, intuitively, we would expect the following to be true: If we have a flow of problems with finite average length \( \bar{l} \), then we expect the average time \( \bar{t} \) to be finite as well.
Thus, we can say that an algorithm is necessarily feasible if \( \bar{t} \) is finite for every probability distribution for which \( \bar{l} \) is finite, and possibly feasible if \( \bar{t} \) is finite for some probability distribution for which \( \bar{l} \) is finite.
If we consider all possible probability distributions, then these definitions trivialize: every algorithm is possibly feasible, and only linear-time algorithms are necessarily feasible.
To make the definitions less trivial, we will use the main idea of MaxEnt and consider only distributions for which the entropy is the largest possible. Since we are interested in the distributions for which the average length is finite, it is reasonable to define MaxEnt distributions as follows: we fix a number l 0 and consider distributions for which the entropy is the largest among all distributions with the average length \( \bar{l} = {{l}_{0}} \).
If, in the above definitions, we only allow such “MaxEnt” distributions, then the above feasibility notions become non-trivial: an algorithm is possibly feasible if it takes exponential time (to be more precise, if and only if its average running time \( \bar{t} \)(n) over all inputs of length n grows slower than some exponential function C n), and necessarily feasible if it is sub-exponential (i.e., if \( \bar{t} \)(n) grows slower than any exponential function).
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References
D. E. Cooke, Scientific Research: From the Particular to the General, El Paso Energy Award for Research Excellence Presentation, University of Texas at El Paso, April 9, 1997.
J. H. Davenport and J. Heintz, “Real quantifier elimination is doubly exponential”, Journal of Symbolic Computations, 1988, Vol. 5, No. 1/2, pp. 29–35.
M. R. Garey and D. S. Johnson, Computers and intractability: a guide to the theory of NP-completeness, W. F. Freeman, San Francisco, 1979.
M. Gell-Mann, The Quark and the Jaguar: Adventures in the Simple and the Complex, Freeman, N.Y., 1994.
K. M. Hanson and R. N. Silver (Eds.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, Dordrecht, 1996.
E. T. Jaynes, “Information theory and statistical mechanics”, Phys. Rev., 1957, Vol. 108, pp. 171–190.
E. T. Jaynes, “Where do we stand on maximum entropy?”, In: R. D. Levine and M. Tribus (Eds.) The maximum entropy formalism, MIT Press, Cambridge, MA, 1979.
V. Kreinovich, “Maximum entropy and interval computations”, Reliable Computing, 1996, Vol. 2, No. 1, pp. 63–79.
H. R. Lewis and C. H. Papadimitriou, Elements of the Theory of Computation, Prentice-Hall, Inc., New Jersey, 1981.
J. C. Martin, Introduction to languages and the theory of computation, McGraw-Hill, N.Y., 1991.
G. E. Moore, “Cramming more components onto integrated circuits”, Electronics Magazine, 1965, Vol. 38, No. 8, pp. 114–117.
G. E. Moore, “Lithography and the Future of Moore’s Law”, In Proceedings of the SPIE Conference on Optical/Laser Microlithography, February 1995, SPIE Publ., Vol. 2440,1995, pp. 2–17.
D. Morgenstein and V. Kreinovich, “Which algorithms are feasible and which are not depends on the geometry of space-time”, Geombinatorics, 1995, Vol. 4, No. 3, pp. 80–97.
H. T. Nguyen and V. Kreinovich, “When is an algorithm feasible? Soft computing approach”, Proceedings of the Joint 4th IEEE Conference on Fuzzy Systems and 2nd IFES, Yokohama, Japan, March 20–24, 1995, Vol. IV, pp. 2109–2112.
H. T. Nguyen and V. Kreinovich, “Towards theoretical foundations of soft computing applications”, International Journal on Uncertainty, Fuzziness, and Knowledge-Based Systems, 1995, Vol. 3, No. 3, pp. 341–373.
R. R. Schaller, “Moore’s law: past, present, and future”, IEEE Spectrum, June 1997, pp. 53–59; see also discussion on p. 8 of the August 1997 issue of IEEE Spectrum.
D. Schirmer and V. Kreinovich, “Towards a More Realistic Definition of Feasibility”, Bulletin of the European Association for Theoretical Computer Science (EATCS), 1996, Vol. 90, pp. 151–153.
A. Tarski, A decision method for elementary algebra and geometry, 2nd ed., Berkeley and Los Angeles, 1951.
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Cooke, D.E., Kreinovich, V., Longpré, L. (1998). Which Algorithms are Feasible? Maxent Approach. In: Erickson, G.J., Rychert, J.T., Smith, C.R. (eds) Maximum Entropy and Bayesian Methods. Fundamental Theories of Physics, vol 98. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5028-6_3
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