arXiv subjects
Publications and source records attributed to Martin Ziegler.
We give a new axiomatization of the N-pseudospace, studied in [2] (Tent(2014)) and [1] (Baudisch,Martin-Pizarro,Ziegler(2014)) based on the zigzags introduced in [2]. We also present a more detailed account of the characterization of forking given in [2].
We extend the Theory of Computation on real numbers, continuous real functions, and bounded closed Euclidean subsets, to compact metric spaces $(X,d)$: thereby generically including computational and optimization problems over higher types, such as the compact 'hyper' spaces of (i) nonempty closed subsets of $X$ w.r.t. Hausdorff metric, and of (ii) equicontinuous functions on $X$. The thus obtained Cartesian closure is shown to exhibit the same structural properties as in the Euclidean case, particularly regarding function pre/image. This allows us to assert the computability of (iii) Fréchet Distances between curves and between loops, as well as of (iv) constrained/Shape Optimization.
Memristors based on a double barrier design have been analysed by various nano spectroscopic methods to unveil details about its microstructure and conduction mechanism. The device consists of an AlOx tunnel barrier and a NbOy/Au Schottky barrier sandwiched between Nb bottom electrode and Au top electrode. As it was anticipated that the local chemical composition of the tunnel barrier, i.e. oxidation state of the metals as well as concentration and distribution of oxygen ions, have a major influence on electronic conduction, these factors were carefully analysed. A combined approach was chosen in order to reliably investigate electronic states of Nb and O by electron energy-loss spectroscopy as well as map elements whose transition edges exhibit a different energy range by energy-dispersive X-ray spectroscopy like Au and Al. The results conclusively demonstrate significant oxidation of the bottom electrode as well as a small oxygen vacancy concentration in the Al oxide tunnel barrier. Possible scenarios to explain this unexpected additional oxide layer are discussed and kinetic Monte Carlo simulations were applied in order to identify its influence on conduction mechanisms in the device. In light of the strong deviations between observed and originally sought layout, this study highlights the robustness in terms of structural deviations of the double barrier memristor device.
The massive parallel approach of neuromorphic circuits leads to effective methods for solving complex problems. It has turned out that resistive switching devices with a continuous resistance range are potential candidates for such applications. These devices are memristive systems - nonlinear resistors with memory. They are fabricated in nanotechnology and hence parameter spread during fabrication may aggravate reproducible analyses. This issue makes simulation models of memristive devices worthwhile. Kinetic Monte-Carlo simulations based on a distributed model of the device can be used to understand the underlying physical and chemical phenomena. However, such simulations are very time-consuming and neither convenient for investigations of whole circuits nor for real-time applications, e.g. emulation purposes. Instead, a concentrated model of the device can be used for both fast simulations and real-time applications, respectively. We introduce an enhanced electrical model of a valence change mechanism (VCM) based double barrier memristive device (DBMD) with a continuous resistance range. This device consists of an ultra-thin memristive layer sandwiched between a tunnel barrier and a Schottky-contact. The introduced model leads to very fast simulations by using usual circuit simulation tools while maintaining physically meaningful parameters. Kinetic Monte-Carlo simulations based on a distributed model and experimental data have been utilized as references to verify the concentrated model.
The consistency problem for a class of algebraic structures asks for an algorithm to decide for any given conjunction of equations whether it admits a non-trivial satisfying assignment within some member of the class. By Adyan (1955) and Rabin (1958) it is known unsolvable for (the class of) groups and, recently, by Bridson and Wilton (2015) for finite groups. We derive unsolvability for (finite) modular lattices and various subclasses; in particular, the class of all subspace lattices of finite dimensional vector spaces over a fixed or arbitrary field of characteristic $0$. The lattice results are used to prove unsolvability of the consistency problem for (finite) rings and (finite) representable relation algebras. These results in turn apply to equations between simple expressions in Grassmann-Cayley algebra and to functional and embedded multivalued dependencies in databases.
In this work we report on the role of ion transport for the dynamic behavior of a double barrier quantum mechanical Al/Al$_2$O$_3$/Nb$_{\text{x}}$O$_{\text{y}}$/Au memristive device based on numerical simulations in conjunction with experimental measurements. The device consists of an ultra-thin Nb$_{\text{x}}$O$_{\text{y}}$ solid state electrolyte between an Al$_2$O$_3$ tunnel barrier and a semiconductor metal interface at an Au electrode. It is shown that the device provides a number of interesting features for potential applications such as an intrinsic current compliance, a relatively long retention time, and no need for an initialization step. Therefore, it is particularly attractive for applications in highly dense random access memories or neuromorphic mixed signal circuits. However, the underlying physical mechanisms of the resistive switching are still not completely understood yet. To investigate the interplay between the current transport mechanisms and the inner atomistic device structure a lumped element circuit model is consistently coupled with 3D kinetic Monte Carlo model for the ion transport. The simulation results indicate that the drift of charged point defects within the Nb$_{\text{x}}$O$_{\text{y}}$ is the key factor for the resistive switching behavior. It is shown in detail that the diffusion of oxygen modifies the local electronic interface states resulting in a change of the interface properties of the double barrier device.
In this work we report on kinetic Monte-Carlo calculations of resistive switching and the underlying growth dynamics of filaments in an electrochemical metallization device consisting of an Ag/TiO2/Pt sandwich-like thin film system. The developed model is not limited to i) fast time scale dynamics and ii) only one growth and dissolution cycle of metallic filaments. In particular, we present results from the simulation of consecutive cycles. We find that the numerical results are in excellent agreement with experimentally obtained data. Additionally, we observe an unexpected filament growth mode which is in contradiction to the widely acknowledged picture of filament growth, but consistent with recent experimental findings.
Lethal Autonomous Weapons promise to revolutionize warfare -- and raise a multitude of ethical and legal questions. It has thus been suggested to program values and principles of conduct (such as the Geneva Conventions) into the machines' control, thereby rendering them both physically and morally superior to human combatants. We employ mathematical logic and theoretical computer science to explore fundamental limitations to the moral behaviour of intelligent machines in a series of "Gedankenexperiments": Refining and sharpening variants of the Trolley Problem leads us to construct an (admittedly artificial but) fully deterministic situation where a robot is presented with two choices: one morally clearly preferable over the other -- yet, based on the undecidability of the Halting problem, it provably cannot decide algorithmically which one. Our considerations have surprising implications to the question of responsibility and liability for an autonomous system's actions and lead to specific technical recommendations.
We give an explicit construction of sharply $2$-transitive groups with fixed point free involutions and without nontrivial abelian normal subgroup.
The computational complexity of the solutions $h$ to the ordinary differential equation $h(0)=0$, $h'(t) = g(t, h(t))$ under various assumptions on the function $g$ has been investigated. Kawamura showed in 2010 that the solution $h$ can be PSPACE-hard even if $g$ is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of $g$ and obtain the following results: the solution $h$ can still be PSPACE-hard if $g$ is assumed to be of class $C^1$; for each $k\ge2$, the solution $h$ can be hard for the counting hierarchy even if $g$ is of class $C^k$.
Nondeterministic polynomial-time Blum-Shub-Smale Machines over the reals give rise to a discrete complexity class between NP and PSPACE. Several problems, mostly from real algebraic geometry / polynomial systems, have been shown complete (under many-one reduction by polynomial-time Turing machines) for this class. We exhibit a new one based on questions about expressions built from cross products only.
The ample hierarchy of geometries of stables theories is strict. We generalise the construction of the free pseudospace to higher dimensions and show that the n-dimensional free pseudospace is ω-stable n-ample yet not (n+1)-ample. In particular, the free pseudospace is not 3-ample. A thorough study of forking is conducted and an explicit description of canonical bases is exhibited.
We show that the isometry group of the bounded Urysohn space is simple. This extends previous work by the authors.
The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity.
Quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables. It generalizes, and on 1-dimensional Hilbert space coincides with, Boolean propositional logic. We introduce the weak and strong satisfiability problem for quantum logic terms. It turns out that in dimension two both are also NP-complete. For higher-dimensional spaces R^d and C^d with d>2 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale model of real computation. This provides a unified view on both Turing and real BSS complexity theory; and extends the still relatively scarce family of NP_R-complete problems with one perhaps closest in spirit to the classical Cook-Levin Theorem. Our investigations on the dimensions a term is weakly/strongly satisfiable in lead to satisfiability problems in indefinite finite and finally in infinite dimension. Here, strong satisfiability turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations
We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.
A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f:[0,1]->R is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multivalued functions f:[0,1]=>R (aka relations), Brattka and Hertling (1994) have considered two notions: weak continuity (which is weaker than relative computability) and strong continuity (which is stronger than relative computability). Observing that uniform continuity plays a crucial role in the Weierstrass Theorem, we propose and compare several notions of uniform continuity for relations. Here, due to the additional quantification over values y in f(x), new ways of (linearly) ordering quantifiers arise, yet none of them turn out as satisfactory. We are thus led to a notion of uniform continuity based on the Henkin Quantifier; and prove it necessary for relative computability. In fact iterating this condition yields a strict hierarchy of notions each necessary, and the omega-th level also sufficient, for relative computability.
We study and compare in two degree-theoretic ways (iterated Halting oracles analogous to Kleene's arithmetical hierarchy and the Borel hierarchy of descriptive set theory) the capabilities and limitations of three models of analytic computation: BSS machines (aka real-RAM) and strongly/weakly analytic machines as introduced by Hotz et. al. (1995).