24,487 research outputs found
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
Finding subsets of positive measure
An important theorem of geometric measure theory (first proved by Besicovitch
and Davies for Euclidean space) says that every analytic set of non-zero
-dimensional Hausdorff measure contains a closed subset of
non-zero (and indeed finite) -measure. We investigate the
question how hard it is to find such a set, in terms of the index set
complexity, and in terms of the complexity of the parameter needed to define
such a closed set. Among other results, we show that given a (lightface)
set of reals in Cantor space, there is always a
subset on non-zero -measure definable from
Kleene's . On the other hand, there are sets of reals
where no hyperarithmetic real can define a closed subset of non-zero measure.Comment: This is an extended journal version of the conference paper "The
Strength of the Besicovitch--Davies Theorem". The final publication of that
paper is available at Springer via
http://dx.doi.org/10.1007/978-3-642-13962-8_2
Pure patterns of order 2
We provide mutual elementary recursive order isomorphisms between classical
ordinal notations, based on Skolem hulling, and notations from pure elementary
patterns of resemblance of order , showing that the latter characterize the
proof-theoretic ordinal of the fragment - of second
order number theory, or equivalently the set theory . As a
corollary, we prove that Carlson's result on the well-quasi orderedness of
respecting forests of order implies transfinite induction up to the ordinal
of . We expect that our approach will facilitate analysis of
more powerful systems of patterns.Comment: corrected Theorem 4.2 with according changes in section 3 (mainly
Definition 3.3), results unchanged. The manuscript was edited, aligned with
reference [14] (moving former Lemma 3.5 there), and argumentation was
revised, with minor corrections in (the proof of) Theorem 4.2; results
unchanged. Updated revised preprint; to appear in the APAL (2017
Graph-Based Shape Analysis Beyond Context-Freeness
We develop a shape analysis for reasoning about relational properties of data
structures. Both the concrete and the abstract domain are represented by
hypergraphs. The analysis is parameterized by user-supplied indexed graph
grammars to guide concretization and abstraction. This novel extension of
context-free graph grammars is powerful enough to model complex data structures
such as balanced binary trees with parent pointers, while preserving most
desirable properties of context-free graph grammars. One strength of our
analysis is that no artifacts apart from grammars are required from the user;
it thus offers a high degree of automation. We implemented our analysis and
successfully applied it to various programs manipulating AVL trees,
(doubly-linked) lists, and combinations of both
Tracking chains revisited
The structure , introduced and first
analyzed in Carlson and Wilken 2012 (APAL), is shown to be elementary
recursive. Here, denotes the proof-theoretic ordinal of the fragment
- of second order number theory, or equivalently the
set theory , which axiomatizes limits of models of
Kripke-Platek set theory with infinity. The partial orderings and
denote the relations of - and -elementary
substructure, respectively. In a subsequent article we will show that the
structure comprises the core of the structure of pure
elementary patterns of resemblance of order . In Carlson and Wilken 2012
(APAL) the stage has been set by showing that the least ordinal containing a
cover of each pure pattern of order is . However, it is not
obvious from Carlson and Wilken 2012 (APAL) that is an elementary
recursive structure. This is shown here through a considerable disentanglement
in the description of connectivity components of and . The key
to and starting point of our analysis is the apparatus of ordinal arithmetic
developed in Wilken 2007 (APAL) and in Section 5 of Carlson and Wilken 2012
(JSL), which was enhanced in Carlson and Wilken 2012 (APAL) specifically for
the analysis of .Comment: The text was edited and aligned with reference [10], Lemma 5.11 was
included (moved from [10]), results unchanged. Corrected Def. 5.2 and Section
5.3 on greatest immediate -successors. Updated publication
information. arXiv admin note: text overlap with arXiv:1608.0842
Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma
Dickson's Lemma is a simple yet powerful tool widely used in termination
proofs, especially when dealing with counters or related data structures.
However, most computer scientists do not know how to derive complexity upper
bounds from such termination proofs, and the existing literature is not very
helpful in these matters.
We propose a new analysis of the length of bad sequences over (N^k,\leq) and
explain how one may derive complexity upper bounds from termination proofs. Our
upper bounds improve earlier results and are essentially tight
A fast semi-direct least squares algorithm for hierarchically block separable matrices
We present a fast algorithm for linear least squares problems governed by
hierarchically block separable (HBS) matrices. Such matrices are generally
dense but data-sparse and can describe many important operators including those
derived from asymptotically smooth radial kernels that are not too oscillatory.
The algorithm is based on a recursive skeletonization procedure that exposes
this sparsity and solves the dense least squares problem as a larger,
equality-constrained, sparse one. It relies on a sparse QR factorization
coupled with iterative weighted least squares methods. In essence, our scheme
consists of a direct component, comprised of matrix compression and
factorization, followed by an iterative component to enforce certain equality
constraints. At most two iterations are typically required for problems that
are not too ill-conditioned. For an HBS matrix with
having bounded off-diagonal block rank, the algorithm has optimal complexity. If the rank increases with the spatial dimension as is
common for operators that are singular at the origin, then this becomes
in 1D, in 2D, and
in 3D. We illustrate the performance of the method on
both over- and underdetermined systems in a variety of settings, with an
emphasis on radial basis function approximation and efficient updating and
downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App
On the information carried by programs about the objects they compute
In computability theory and computable analysis, finite programs can compute
infinite objects. Presenting a computable object via any program for it,
provides at least as much information as presenting the object itself, written
on an infinite tape. What additional information do programs provide? We
characterize this additional information to be any upper bound on the
Kolmogorov complexity of the object. Hence we identify the exact relationship
between Markov-computability and Type-2-computability. We then use this
relationship to obtain several results characterizing the computational and
topological structure of Markov-semidecidable sets
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