230 research outputs found
Quantum Kolmogorov Complexity
In this paper we give a definition for quantum Kolmogorov complexity. In the
classical setting, the Kolmogorov complexity of a string is the length of the
shortest program that can produce this string as its output. It is a measure of
the amount of innate randomness (or information) contained in the string.
We define the quantum Kolmogorov complexity of a qubit string as the length
of the shortest quantum input to a universal quantum Turing machine that
produces the initial qubit string with high fidelity. The definition of Vitanyi
(Proceedings of the 15th IEEE Annual Conference on Computational Complexity,
2000) measures the amount of classical information, whereas we consider the
amount of quantum information in a qubit string. We argue that our definition
is natural and is an accurate representation of the amount of quantum
information contained in a quantum state.Comment: 14 pages, LaTeX2e, no figures, \usepackage{amssymb,a4wide}. To appear
in the Proceedings of the 15th IEEE Annual Conference on Computational
Complexit
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
Universal Prediction
In this thesis I investigate the theoretical possibility of a universal method of prediction. A prediction method is universal if it is always able to learn from data: if it is always able to extrapolate given data about past observations to maximally successful predictions about future observations. The context of this investigation is the broader philosophical question into the possibility of a formal specification of inductive or scientific reasoning, a question that also relates to modern-day speculation about a fully automatized data-driven science.
I investigate, in particular, a proposed definition of a universal prediction method that goes back to Solomonoff (1964) and Levin (1970). This definition marks the birth of the theory of Kolmogorov complexity, and has a direct line to the information-theoretic approach in modern machine learning. Solomonoff's work was inspired by Carnap's program of inductive logic, and the more precise definition due to Levin can be seen as an explicit attempt to escape the diagonal argument that Putnam (1963) famously launched against the feasibility of Carnap's program.
The Solomonoff-Levin definition essentially aims at a mixture of all possible prediction algorithms. An alternative interpretation is that the definition formalizes the idea that learning from data is equivalent to compressing data. In this guise, the definition is often presented as an implementation and even as a justification of Occam's razor, the principle that we should look for simple explanations.
The conclusions of my investigation are negative. I show that the Solomonoff-Levin definition fails to unite two necessary conditions to count as a universal prediction method, as turns out be entailed by Putnam's original argument after all; and I argue that this indeed shows that no definition can. Moreover, I show that the suggested justification of Occam's razor does not work, and I argue that the relevant notion of simplicity as compressibility is already problematic itself
The Machine as Data: A Computational View of Emergence and Definability
Turing’s (Proceedings of the London Mathematical Society 42:230–265, 1936) paper on computable numbers has played its role in underpinning different perspectives on the world of information. On the one hand, it encourages a digital ontology, with a perceived flatness of computational structure comprehensively hosting causality at the physical level and beyond. On the other (the main point of Turing’s paper), it can give an insight into the way in which higher order information arises and leads to loss of computational control—while demonstrating how the control can be re-established, in special circumstances, via suitable type reductions. We examine the classical computational framework more closely than is usual, drawing out lessons for the wider application of information–theoretical approaches to characterizing the real world. The problem which arises across a range of contexts is the characterizing of the balance of power between the complexity of informational structure (with emergence, chaos, randomness and ‘big data’ prominently on the scene) and the means available (simulation, codes, statistical sampling, human intuition, semantic constructs) to bring this information back into the computational fold. We proceed via appropriate mathematical modelling to a more coherent view of the computational structure of information, relevant to a wide spectrum of areas of investigation
Universal Prediction
In this dissertation I investigate the theoretical possibility of a universal method of prediction. A prediction method is universal if it is always able to learn what there is to learn from data: if it is always able to extrapolate given data about past observations to maximally successful predictions about future observations. The context of this investigation is the broader philosophical question into the possibility of a formal specification of inductive or scientific reasoning, a question that also touches on modern-day speculation about a fully automatized data-driven science.
I investigate, in particular, a specific mathematical definition of a universal prediction method, that goes back to the early days of artificial intelligence and that has a direct line to modern developments in machine learning. This definition essentially aims to combine all possible prediction algorithms. An alternative interpretation is that this definition formalizes the idea that learning from data is equivalent to compressing data. In this guise, the definition is often presented as an implementation and even as a justification of Occam's razor, the principle that we should look for simple explanations.
The conclusions of my investigation are negative. I show that the proposed definition cannot be interpreted as a universal prediction method, as turns out to be exposed by a mathematical argument that it was actually intended to overcome. Moreover, I show that the suggested justification of Occam's razor does not work, and I argue that the relevant notion of simplicity as compressibility is problematic itself
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