15,307 research outputs found
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
The cohesive principle and the Bolzano-Weierstra{\ss} principle
The aim of this paper is to determine the logical and computational strength
of instances of the Bolzano-Weierstra{\ss} principle (BW) and a weak variant of
it.
We show that BW is instance-wise equivalent to the weak K\"onig's lemma for
-trees (-WKL). This means that from every bounded
sequence of reals one can compute an infinite -0/1-tree, such that
each infinite branch of it yields an accumulation point and vice versa.
Especially, this shows that the degrees d >> 0' are exactly those containing an
accumulation point for all bounded computable sequences.
Let BW_weak be the principle stating that every bounded sequence of real
numbers contains a Cauchy subsequence (a sequence converging but not
necessarily fast). We show that BW_weak is instance-wise equivalent to the
(strong) cohesive principle (StCOH) and - using this - obtain a classification
of the computational and logical strength of BW_weak. Especially we show that
BW_weak does not solve the halting problem and does not lead to more than
primitive recursive growth. Therefore it is strictly weaker than BW. We also
discuss possible uses of BW_weak.Comment: corrected typos, slightly improved presentatio
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