Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Algorithmic information theory
Algorithmic information theory
A recursive introduction to the theory of computation
A recursive introduction to the theory of computation
An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
Recursively enumerable reals and Chaitin &OHgr; numbers
Theoretical Computer Science
An Introduction to the General Theory of Algorithms
An Introduction to the General Theory of Algorithms
The Non-Recursive Power of Erroneous Computation
Proceedings of the 19th Conference on Foundations of Software Technology and Theoretical Computer Science
Approximations to the halting problem
Journal of Computer and System Sciences
Filter-resistant code injection on ARM
Proceedings of the 16th ACM conference on Computer and communications security
Filter-resistant code injection on ARM
Journal in Computer Virology
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No algorithm can of course solve the Halting Problem, that is, decide within finite time always correctly whether a given program halts on a certain given input. It might however be able to give correct answers for ‘most' instances and thus solve it at least approximately. Whether and how well such approximations are feasible highly depends on the underlying encodings and in particular the Gödelization (programming system) which in practice usually arises from some programming language. We consider BrainF*ck (BF), a simple yet Turing-complete real-world programming language over an eight letter alphabet, and prove that the natural enumeration of its syntactically correct sources codes induces a both efficient and dense Gödelization in the sense of [Jakoby&Schindelhauer'99]. It follows that any algorithm M approximating the Halting Problem for BF errs on at least a constant fraction εM 0 of all instances of size n for infinitely many n. Next we improve this result by showing that, in every dense Gödelization, this constant lower bound ε to be independent of M; while, the other hand, the Halting Problem does admit approximation up to arbitrary fraction δ 0 by an appropriate algorithm Mδ handling instances of size n for infinitely many n. The last two results complement work by [Lynch'74].