41 research outputs found
Short lists for shortest descriptions in short time
Is it possible to find a shortest description for a binary string? The
well-known answer is "no, Kolmogorov complexity is not computable." Faced with
this barrier, one might instead seek a short list of candidates which includes
a laconic description. Remarkably such approximations exist. This paper
presents an efficient algorithm which generates a polynomial-size list
containing an optimal description for a given input string. Along the way, we
employ expander graphs and randomness dispersers to obtain an Explicit Online
Matching Theorem for bipartite graphs and a refinement of Muchnik's Conditional
Complexity Theorem. Our main result extends recent work by Bauwens, Mahklin,
Vereschchagin, and Zimand
On approximate decidability of minimal programs
An index in a numbering of partial-recursive functions is called minimal
if every lesser index computes a different function from . Since the 1960's
it has been known that, in any reasonable programming language, no effective
procedure determines whether or not a given index is minimal. We investigate
whether the task of determining minimal indices can be solved in an approximate
sense. Our first question, regarding the set of minimal indices, is whether
there exists an algorithm which can correctly label 1 out of indices as
either minimal or non-minimal. Our second question, regarding the function
which computes minimal indices, is whether one can compute a short list of
candidate indices which includes a minimal index for a given program. We give
some negative results and leave the possibility of positive results as open
questions
How powerful are integer-valued martingales?
In the theory of algorithmic randomness, one of the central notions is that
of computable randomness. An infinite binary sequence X is computably random if
no recursive martingale (strategy) can win an infinite amount of money by
betting on the values of the bits of X. In the classical model, the martingales
considered are real-valued, that is, the bets made by the martingale can be
arbitrary real numbers. In this paper, we investigate a more restricted model,
where only integer-valued martingales are considered, and we study the class of
random sequences induced by this model.Comment: Long version of the CiE 2010 paper
Things that can be made into themselves
One says that a property of sets of natural numbers can be made into
itself iff there is a numbering of all left-r.e.
sets such that the index set satisfies has the property
as well. For example, the property of being Martin-L\"of random can be made
into itself. Herein we characterize those singleton properties which can be
made into themselves. A second direction of the present work is the
investigation of the structure of left-r.e. sets under inclusion modulo a
finite set. In contrast to the corresponding structure for r.e. sets, which has
only maximal but no minimal members, both minimal and maximal left-r.e. sets
exist. Moreover, our construction of minimal and maximal left-r.e. sets greatly
differs from Friedberg's classical construction of maximal r.e. sets. Finally,
we investigate whether the properties of minimal and maximal left-r.e. sets can
be made into themselves
Translating the Cantor set by a random
We determine the constructive dimension of points in random translates of the
Cantor set. The Cantor set "cancels randomness" in the sense that some of its
members, when added to Martin-Lof random reals, identify a point with lower
constructive dimension than the random itself. In particular, we find the
Hausdorff dimension of the set of points in a Cantor set translate with a given
constructive dimension
Directed Multicut with linearly ordered terminals
Motivated by an application in network security, we investigate the following
"linear" case of Directed Mutlicut. Let be a directed graph which includes
some distinguished vertices . What is the size of the
smallest edge cut which eliminates all paths from to for all ? We show that this problem is fixed-parameter tractable when parametrized in
the cutset size via an algorithm running in time.Comment: 12 pages, 1 figur