420 research outputs found
Asymptotic density and the coarse computability bound
For we say that a set is \emph{coarsely
computable at density} if there is a computable set such that has lower density at least . Let . We study the interactions of
these concepts with Turing reducibility. For example, we show that if there are sets such that
where is coarsely computable at density while is not coarsely
computable at density . We show that a real is equal to
for some c.e.\ set if and only if is left-. A
surprising result is that if is a -generic set, and with , then is coarsely computable at density
-Generic Computability, Turing Reducibility and Asymptotic Density
Generic computability has been studied in group theory and we now study it in
the context of classical computability theory. A set A of natural numbers is
generically computable if there is a partial computable function f whose domain
has density 1 and which agrees with the characteristic function of A on its
domain. A set A is coarsely computable if there is a computable set C such that
the symmetric difference of A and C has density 0. We prove that there is a
c.e. set which is generically computable but not coarsely computable and vice
versa. We show that every nonzero Turing degree contains a set which is not
coarsely computable. We prove that there is a c.e. set of density 1 which has
no computable subset of density 1. As a corollary, there is a generically
computable set A such that no generic algorithm for A has computable domain. We
define a general notion of generic reducibility in the spirt of Turing
reducibility and show that there is a natural order-preserving embedding of the
Turing degrees into the generic degrees which is not surjective
Generic algorithms for halting problem and optimal machines revisited
The halting problem is undecidable --- but can it be solved for "most"
inputs? This natural question was considered in a number of papers, in
different settings. We revisit their results and show that most of them can be
easily proven in a natural framework of optimal machines (considered in
algorithmic information theory) using the notion of Kolmogorov complexity. We
also consider some related questions about this framework and about asymptotic
properties of the halting problem. In particular, we show that the fraction of
terminating programs cannot have a limit, and all limit points are Martin-L\"of
random reals. We then consider mass problems of finding an approximate solution
of halting problem and probabilistic algorithms for them, proving both positive
and negative results. We consider the fraction of terminating programs that
require a long time for termination, and describe this fraction using the busy
beaver function. We also consider approximate versions of separation problems,
and revisit Schnorr's results about optimal numberings showing how they can be
generalized.Comment: a preliminary version was presented at the ICALP 2015 conferenc
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