Asymptotic Proportion of Hard Instances of the Halting Problem
نویسنده
چکیده
Although the halting problem is undecidable, imperfect testers that fail on some instances are possible. Such instances are called hard for the tester. One variant of imperfect testers replies “I don’t know” on hard instances, another variant fails to halt, and yet another replies incorrectly “yes” or “no”. Also the halting problem has three variants. A tester may test halting on the empty input or when the tested program is given to itself as its input. It may also test whether a given program halts on a given input. The failure rate of a tester for some size is the proportion of hard instances among all instances of that size. This publication investigates the behaviour of the failure rate as the size grows without limit. Earlier results are surveyed and new results are proven. Some of them use C++ on Linux as the computational model. It turns out that the behaviour is sensitive to the details of the programming language or computational model, but in many cases it is possible to prove that the proportion of hard instances does not vanish. ACM Computing Classification System 1998: F.1.1 Models of Computation–Computability theory Mathematics Subject Classification 2010: 68Q17 Computational difficulty of problems
منابع مشابه
The Asymptotic Behaviour of the Proportion of Hard Instances of the Halting Problem (Extended Version)
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عنوان ژورنال:
- Acta Cybern.
دوره 21 شماره
صفحات -
تاریخ انتشار 2014