618 research outputs found
Asymptotic Proportion of Hard Instances of the Halting Problem
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: does a given program halt on the empty
input, does a given program halt when given itself as its input, or does a
given program halt 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.Comment: 18 pages. The differences between this version and arXiv:1307.7066v1
are significant. They have been listed in the last paragraph of Section 1.
Excluding layout, this arXiv version is essentially identical to the Acta
Cybernetica versio
Equivalence-Checking on Infinite-State Systems: Techniques and Results
The paper presents a selection of recently developed and/or used techniques
for equivalence-checking on infinite-state systems, and an up-to-date overview
of existing results (as of September 2004)
Computational universes
Suspicions that the world might be some sort of a machine or algorithm
existing ``in the mind'' of some symbolic number cruncher have lingered from
antiquity. Although popular at times, the most radical forms of this idea never
reached mainstream. Modern developments in physics and computer science have
lent support to the thesis, but empirical evidence is needed before it can
begin to replace our contemporary world view.Comment: Several corrections of typos and smaller revisions, final versio
A reformulation of Hilbert's tenth problem through Quantum Mechanics
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the
domain of integer arithmetics into either a problem involving a set of
infinitely coupled differential equations or a problem involving a Shr\"odinger
propagator with some appropriate kernel. Either way, Mathematics and Physics
could be combined for Hilbert's tenth problem and for the notion of effective
computability
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