Computational complexity theory and the philosophy of mathematics

Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the P≠NP problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof

A Unifying review of linear gaussian models

Factor analysis, principal component analysis, mixtures of gaussian clusters, vector quantization, Kalman filter models, and hidden Markov models can all be unified as variations of unsupervised learning under a single basic generative model. This is achieved by collecting together disparate observations and derivations made by many previous authors and introducing a new way of linking discrete and continuous state models using a simple nonlinearity. Through the use of other nonlinearities, we show how independent component analysis is also a variation of the same basic generative model.We show that factor analysis and mixtures of gaussians can be implemented in autoencoder neural networks and learned using squared error plus the same regularization term. We introduce a new model for static data, known as sensible principal component analysis, as well as a novel concept of spatially adaptive observation noise. We also review some of the literature involving global and local mixtures of the basic models and provide pseudocode for inference and learning for all the basic models

Spectral functions of low-dimensional quantum systems

The focus of this thesis is set on the calculation of spectral functions for low-dimensional quantum systems. At first the quantum many-body problem is introduced and the related spectral functions of interest are defined. In a next step the applied numerical algorithms are presented: Exact diagonalization & Lanczos algorithm, the numerical renormalization group (NRG) and the density matrix renormalization group (DMRG) in the formulation of matrix product states (MPS). The main focus is set on the calculation of spectral functions with these algorithms. In particular for the DMRG a new algorithm is presented which combines the Lanczos algorithm with matrix product states. The presented algorithms are applied to different problems. The first system denotes a copper crystal with magnetic (cobalt/iron) impurities. These materials are known Kondo-systems, i.e. they show an untypical increase in the resistivity with decreasing temperature. At low temperatures local moments will develop, which will be screened from the surrounding conduction band electrons. The ground state of the system is a long-range strongly correlated many-body state between conduction band electrons and local moments. In the singleparticle impurity spectral function a so-called Kondo resonance at the Fermi energy is formed below the Kondo temperature. In scanning tunneling microscope measurements the spectral function at the surface - the so-called local density of states - of the copper crystals with buried impurities in varying depth was measured. The goal of this study is to verify the existence of long-range Kondo signatures via a comparison of the spectral functions for impurities with different depth below the surface. Therefore in this work numerical simulations are used to calculate the local density of states at the surface. The single impurity Anderson model (SIAM) is applied which shows the Kondo resonance in the impurity spectral function. In order to calculate the local density of states at the surface of the copper crystal the equations of motion of the SIAM and the Dyson equation are used and then solved by a combination of numerical renormalization group and band structure calculations of the copper crystal (linear combination of atomic orbitals). The simulation for impurities in different depth below the surface and the experimental data are in very good agreement and hence support the measurement of a long-range Kondo signature. Furthermore the simulations allow to determine the Kondo temperature of a single impurity. The focus of the second part is set on the one-dimensional Heisenberg model. The spectral function of interest is here the dynamic spin structure factor. The newly developed MPS-Lanczos algorithm within the DMRG is used to calculate the dynamic spin structure factor. The results are compared to analytic results from the Bethe ansatz. In the end the new algorithm is discussed in respect to existing methods within the DMRG to calculate spectral functions

Topics in Programming Languages, a Philosophical Analysis through the case of Prolog

[EN]Programming languages seldom find proper anchorage in philosophy of logic, language and science. is more, philosophy of language seems to be restricted to natural languages and linguistics, and even philosophy of logic is rarely framed into programming languages topics. The logic programming paradigm and Prolog are, thus, the most adequate paradigm and programming language to work on this subject, combining natural language processing and linguistics, logic programming and constriction methodology on both algorithms and procedures, on an overall philosophizing declarative status. Not only this, but the dimension of the Fifth Generation Computer system related to strong Al wherein Prolog took a major role. and its historical frame in the very crucial dialectic between procedural and declarative paradigms, structuralist and empiricist biases, serves, in exemplar form, to treat straight ahead philosophy of logic, language and science in the contemporaneous age as well. In recounting Prolog's philosophical, mechanical and algorithmic harbingers, the opportunity is open to various routes. We herein shall exemplify some: - the mechanical-computational background explored by Pascal, Leibniz, Boole, Jacquard, Babbage, Konrad Zuse, until reaching to the ACE (Alan Turing) and EDVAC (von Neumann), offering the backbone in computer architecture, and the work of Turing, Church, Gödel, Kleene, von Neumann, Shannon, and others on computability, in parallel lines, throughly studied in detail, permit us to interpret ahead the evolving realm of programming languages. The proper line from lambda-calculus, to the Algol-family, the declarative and procedural split with the C language and Prolog, and the ensuing branching and programming languages explosion and further delimitation, are thereupon inspected as to relate them with the proper syntax, semantics and philosophical élan of logic programming and Prolog

The density-matrix renormalization group

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly acquired the status of method of choice for numerical studies of one-dimensional quantum systems. Its applications to the calculation of static, dynamic and thermodynamic quantities in such systems are reviewed. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and non-equilibrium statistical physics, and time-dependent phenomena is discussed. This review also considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by DMRG.Comment: accepted by Rev. Mod. Phys. in July 2004; scheduled to appear in the January 2005 issu