Approximations to the Halting Problem

@article{Lynch1974ApproximationsTT,
  title={Approximations to the Halting Problem},
  author={Nancy A. Lynch},
  journal={J. Comput. Syst. Sci.},
  year={1974},
  volume={9},
  pages={143-150}
}
  • N. Lynch
  • Published 1 October 1974
  • Mathematics, Computer Science
  • J. Comput. Syst. Sci.
A probabilistic anytime algorithm for the halting problem
TLDR
This paper works with running times to define a class of computable probability distributions on the set of halting programs in order to construct an anytime algorithm for the Halting problem with a probabilistic evaluation of the error of the decision.
What Percentage of Programs Halt?
TLDR
It is shown that the ratio of the number of halting programs of length at most n by the total number of such programs does not have a limit value, and the reals which can be the limsup of such a sequence are further characterised.
Generic algorithms for halting problem and optimal machines revisited
TLDR
It is shown that the fraction of terminating programs cannot have a limit, and all limit points are Martin-L\"of random reals", which means that the halting problem cannot be solved for "most" inputs.
On the Complexity of Random Strings (Extended Abstract)
  • M. Kummer
  • Mathematics, Computer Science
    STACS
  • 1996
TLDR
It is shown that the set R of Kolmogorov random strings is truth-table complete and how the halting problem can be encoded into the distribution of random strings rather than using the time complexity of non-random strings.
On Approximating Real-World Halting Problems
TLDR
BrainF*ck (BF), a simple yet Turing-complete real-world programming language over an eight letter alphabet, is considered, and it is proved that the natural enumeration of its syntactically correct sources codes induces a both efficient and dense Godelization.
M ay 2 01 6 Optimal asymptotic bounds on the oracle use in computationsfr m Chaitin ’ s Omega ∗
TLDR
The asymptotic upper bounds on the use of Cha itin’s Ω in oracle computations of halting probabilities are characterised and show that the following two conditions are equivalent for any computable functionh such thath(n)−n is non-decreasing.
A statistical anytime algorithm for the Halting Problem
TLDR
An efficient statistical anytime algorithm for the Halting Problem which can be implemented without any prior information about the running times on the specific model of computation and the cut-off temporal bound is reasonably small.
Dense computability, upper cones, and minimal pairs
TLDR
It is shown that nontrivial upper cones in the generic, dense, and effective dense degrees are of measure $0$ and use this fact to show that there are minimal pairs in the dense degrees.
The Asymptotic Behaviour of the Proportion of Hard Instances of the Halting Problem (Extended Version)
TLDR
It turns out that the behaviour of the failure rate as the size grows without limit is sensitive to the details of the programming language or computational model, but in some cases it is possible to prove that the proportion of hard instances does not vanish.
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If searching for the ebook by Hartley Rogers Theory of Recursive Functions and Effective Computability in pdf format, then you've come to the faithful site. We presented the complete version of this
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Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular and generalizations of recursion theory.
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