K. De Leeuw, E. F. Moore, C. E. Shannon, N. Shapiro. Computability by probabilistic machines, in C. E. Shannon, J. McCarthy (eds.). Automata Studies, Princeton University Press, Princeton, N.J., 1956, 183-212.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... all mathematical proofs or computer programs are ultimately probabilistic, see Davis [9] De Millo, Lipton, Perlis [11] but goes beyond the capability of any classical computation: 4 even the best probabilistic algorithms are not able to achieve this computational power (by a classical result [10], probabilistic algorithms are equivalent to Turing machines) Let us finally notice that by virtue of the same information theoretic argument, the possibility of time travel (see, for example, Nahin [17] would not solve the halting problem, unless one could travel back and forth in time at a ....

K. De Leeuw, E. F. Moore, C. E. Shannon, N. Shapiro. Computability by probabilistic machines, in C. E. Shannon, J. McCarthy (eds.). Automata Studies, Princeton University Press, Princeton, N.J., 1956, 183-212.

....are B(1) 1, B(2) 4, B(3) 6, B(4) 13, and B(5) # 1915. 6 probabilistic, see Davis [7] De Millo, Lipton, Perlis [8] but go beyond the capability of any classical computation: even the best probabilistic algorithms are not able to achieve this computational power (by a classical result [11], probabilistic algorithms are equivalent to Turing machines) ....

K. De Leeuw, E. F. Moore, C. E. Shannon, N. Shapiro. Computability by probabilistic machines, in C. E. Shannon, J. McCarthy (eds.). Automata Studies, Princeton University Press, Princeton, N.J., 1956, 183-212.

....states the machine uses the output of a random experiment to decide among the p possible next states. So, a probabilistic Turing machine can make mistakes; the output is not truly correct , but correct within a probability . Classical results due to De Leuuw, Moore, Shannon, and Shapiro [17] and Gill [29] show that the class of functions computed by probabilistic algorithms coincides with the class of recursive functions. The difference is only in complexity: if we do not insist on a guarantee, then sometimes it is possible to compute faster. All results pertaining incompleteness, ....

K. De Leuuw, E. F. Moore, C. E. Shannon, N. Shapiro. Computability by probabilistic machines, in C. E. Shannon, J. McCarthy (eds.). Automata Studies, Princeton University Press, Princeton, 1956, 183-212.

....theory is the probability that a computing machine enumerates a given set when its program is manufactured by coin flipping. The entropy of a set is defined to be Gamma log 2 of this probability. 2 G. J. Chaitin 1. Introduction In a classical paper on computability by probabilistic machines [1], de Leeuw et al. showed that if a machine with a random element can enumerate a specific set of natural numbers with positive probability, then there is a deterministic machine that also enumerates this set. We propose to throw further light on this matter by bringing into play the concepts of ....

....unending computations. The computer is used to enumerate a set of objects instead of a single one. An important difference between this paper and [3] is that here it is possible for the machine to read the entire program tape, so that in a sense infinite programs are permitted. However, following [1] it is better to think of these as cases in which a nondeterministic machine uses coin flipping infinitely often. Here, as in [3] we pick a universal computer that makes the probability of obtaining any given machine output as high as possible. We are thus led to define three concepts: P (A) the ....

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K. de Leeuw, E. F. Moore, C. E. Shannon and N. Shapiro, Computability by probabilistic machines, in Automata Studies, C. E. Shannon and J. McCarthy (Eds.), pp. 183--212. Princeton University Press, N.J. (1956).

....states the machine uses the output of a random experiment to decide among the p possible next states. So, a probabilistic Turing machine can make mistakes; the output is not truly correct , but correct within a probability . Classical results due to De Leuuw, Moore, Shannon, and Shapiro [17] and Gill [29] show that the class of functions computed by probabilistic algorithms coincides with the class of recursive functions. The di#erence is only in complexity: if we do not insist on a guarantee, then sometimes it is possible to compute faster. All results pertaining incompleteness, ....

K. De Leuuw, E. F. Moore, C. E. Shannon, N. Shapiro. Computability by probabilistic machines, in C. E. Shannon, J. McCarthy (eds.). Automata Studies, Princeton University Press, Princeton, 1956, 183-212.

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K. de Leeuw, E. F. Moore, C. E. Shannon and N. Shapiro, "Computability by probabilistic machines", in Automata Studies (C. E. Shannon and J. McCarthy, eds.), Princeton Univ. Press, Princeton, NJ, 1955, 183--212.

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