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Some Thoughts on Hypercomputation
Hypercomputation is a relatively new branch of computer science that emerged
from the idea that the Church--Turing Thesis, which is supposed to describe
what is computable and what is noncomputable, cannot possible be true. Because
of its apparent validity, the Church--Turing Thesis has been used to
investigate the possible limits of intelligence of any imaginable life form,
and, consequently, the limits of information processing, since living beings
are, among others, information processors. However, in the light of
hypercomputation, which seems to be feasibly in our universe, one cannot impose
arbitrary limits to what intelligence can achieve unless there are specific
physical laws that prohibit the realization of something. In addition,
hypercomputation allows us to ponder about aspects of communication between
intelligent beings that have not been considered befor
Most Programs Stop Quickly or Never Halt
Since many real-world problems arising in the fields of compiler
optimisation, automated software engineering, formal proof systems, and so
forth are equivalent to the Halting Problem--the most notorious undecidable
problem--there is a growing interest, not only academically, in understanding
the problem better and in providing alternative solutions. Halting computations
can be recognised by simply running them; the main difficulty is to detect
non-halting programs. Our approach is to have the probability space extend over
both space and time and to consider the probability that a random -bit
program has halted by a random time. We postulate an a priori computable
probability distribution on all possible runtimes and we prove that given an
integer k>0, we can effectively compute a time bound T such that the
probability that an N-bit program will eventually halt given that it has not
halted by T is smaller than 2^{-k}. We also show that the set of halting
programs (which is computably enumerable, but not computable) can be written as
a disjoint union of a computable set and a set of effectively vanishing
probability. Finally, we show that ``long'' runtimes are effectively rare. More
formally, the set of times at which an N-bit program can stop after the time
2^{N+constant} has effectively zero density.Comment: Shortened abstract and changed format of references to match Adv.
Appl. Math guideline
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