Curves That Must Be Retraced

We exhibit a polynomial time computable plane curve ${bf Gamma}$ that has finite length, does not intersect itself, and is smooth except at one endpoint, but has the following property. For every computable parametrization $f$ of ${bfGamma}$ and every positive integer $m$, there is some positive-length subcurve of ${bfGamma}$ that $f$ retraces at least $m$ times. In contrast, every computable curve of finite length that does not intersect itself has a constant-speed (hence non-retracing) parametrization that is computable relative to the halting problem

On approximate and algebraic computability over the real numbers

AbstractWe consider algebraic and approximate computations of (partial) real functions ƒ:Rd ↣ R. Algebraic computability is defined by means of (parameter-free) finite algorithmic procedures. The notion of approximate computability is a straightforward generalization of the Ko-Friedman approach, based on oracle Turing machines, to functions with not necessarily recursively open domains.The main results of the paper give characterizations of approximate computability by means of the passing sets of finite algorithmic procedures, i.e., characterizations from the algebraic point of view. Some consequences and also modifications of the concepts are discussed. Finally, two variants of arithmetical hierarchies over the reals are considered and used to classify and mutually compare the domains, graphs and ranges of algebraically resp. approximately computable real functions

Three topics in the theory of computing: Multi-resolution cellular automata, the Kolmogorov complexity characterization of regular languages, and hidden variables in Bayesian networks

Our work is centered around topics where we provide a new model or approach to a well-known paradigm. We provide a new lens through which to view an area of research, providing access for new researchers and perspectives. After a brief orientation with common terms, we examine computation of real-valued sets, our general multi-resolution cellular automata (MRCA) simulator, how to prove languages are non-regular using Kolmogorov complexity, and how to show hidden variables are valuable in Bayesian networks