62 research outputs found
The probability distribution as a computational resource for randomness testing
When testing a set of data for randomness according to a probability
distribution that depends on a parameter, access to this parameter can be
considered as a computational resource. We call a randomness test Hippocratic
if it is not permitted to access this resource. In these terms, we show that
for Bernoulli measures , and the Martin-L\"of randomness
model, Hippocratic randomness of a set of data is the same as ordinary
randomness. The main idea of the proof is to first show that from
Hippocrates-random data one can Turing compute the parameter . However, we
show that there is no single Hippocratic randomness test such that passing the
test implies computing , and in particular there is no universal Hippocratic
randomness test
Kolmogorov structure functions for automatic complexity
For a finite word we define and study the Kolmogorov structure function
for nondeterministic automatic complexity. We prove upper bounds on
that appear to be quite sharp, based on numerical evidence.Comment: Preliminary version: "Kolmogorov structure functions for automatic
complexity in computational statistics", Lecture Notes in Comput. Sci., vol.
8881, Springer, Cham, 2014, 652--665, 8th International Conference on
Combinatorial Optimization and Applications (COCOA 2014
Models of the Chisholm set
We give a counter-example showing that Carmo and Jones' condition 5(e) may
conflict with other conditions on the models in their paper \emph{A new
approach to contrary-to-duty obligations}.Comment: Paper for Filosofi hovedfag spesialomr{\aa}de 1 exam, University of
Oslo, Fall 1996. First cited in Carmo and Jones, Deontic logic and
contrary-to-duties, Handbook of Philosophical Logic, 2002, footnote 2
On the complexity of automatic complexity
Generalizing the notion of automatic complexity of individual strings due to
Shallit and Wang, we define the automatic complexity of an equivalence
relation on a finite set of strings.
We prove that the problem of determining whether equals the number
of equivalence classes of is -complete. The problem of
determining whether for a fixed is complete for the
second level of the Boolean hierarchy for , i.e.,
-complete.
Let be the language consisting of all strings of maximal nondeterministic
automatic complexity. We characterize the complexity of infinite subsets of
by showing that they can be co-context-free but not context-free, i.e., is
-immune, but not -immune.
We show that for each , , where
is the set of all strings whose deterministic automatic complexity
satisfies
Automatic complexity of shift register sequences
Let be an -sequence, a maximal length sequence produced by a linear
feedback shift register. We show that has maximal subword complexity
function in the sense of Allouche and Shallit. We show that this implies that
the nondeterministic automatic complexity is close to maximal:
, where is the length of . In contrast, Hyde has
shown for all sequences of length .Comment: Preliminary version: "Shift registers fool finite automata", Lecture
Notes in Computer Science 10388 (2017), 170-181, Workshop on Logic, Language,
Information and Computation (WoLLIC) 201
Infinite subsets of random sets of integers
There is an infinite subset of a Martin-L\"of random set of integers that
does not compute any Martin-L\"of random set of integers. To prove this, we
show that each real of positive effective Hausdorff dimension computes an
infinite subset of a Martin-L\"of random set of integers, and apply a result of
Miller
Local initial segments of the Turing degrees
Recent results on initial segments of the Turing degrees are presented, and
some conjectures about initial segments that have implications for the
existence of non-trivial automorphisms of the Turing degrees are indicated
A strong law of computationally weak subsets
We show that in the setting of fair-coin measure on the power set of the
natural numbers, each sufficiently random set has an infinite subset that
computes no random set. That is, there is an almost sure event
such that if then has an infinite subset such that no
element of is Turing computable from
Effective Banach spaces
This thesis addresses Pour-El and Richards' fourth question from their book
"Computability in analysis and physics", concerning the relation between higher
order recursion theory and computability in analysis.
Among other things it is shown that there is a computability structure that
is uncountable. The example given is a structure on the Banach space of bounded
linear operators on the set of almost periodic functions.Comment: Master's thesis, University of Oslo, 1997. Adviser: Dag Normann.
Translated from Norwegian. Original title: "Effektive Banach-rom
Lattice initial segments of the Turing degrees
We characterize the isomorphism types of principal ideals of the Turing
degrees below 0' that are lattices as the lattices with a Sigma-0-3
presentation, by showing that each Sigma-0-3 presentable bounded upper
semilattice is isomorphic to such a principal ideal. We get a similar result
for the Turing degrees below any degree above 0".Comment: Doctoral dissertation, Logic and the Methodology of Science,
University of California, Berkeley, 200
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