Related papers: A Turing Incomputable Coloring Function
We call a subset of an ordinal $\lambda$ recognizable if it is the unique subset $x$ of $\lambda$ for which some Turing machine with ordinal time and tape, which halts for all subsets of $\lambda$ as input, halts with the final state $0$.…
The input of the Maximum Colored Cut problem consists of a graph $G=(V,E)$ with an edge-coloring $c:E\to \{1,2,3,\ldots , p\}$ and a positive integer $k$, and the question is whether $G$ has a nontrivial edge cut using at least $k$ colors.…
We explore in the framework of Quantum Computation the notion of {\em Computability}, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to…
A novel computing model, called \emph{Probe Machine}, is proposed in this paper. Different from Turing Machine, Probe Machine is a fully-parallel computing model in the sense that it can simultaneously process multiple pairs of data, rather…
The problem of sampling edge-colorings of graphs with maximum degree $\Delta$ has received considerable attention and efficient algorithms are available when the number of colors is large enough with respect to $\Delta$. Vizing's theorem…
Image colorization is the process of colorizing grayscale images or recoloring an already-color image. This image manipulation can be used for grayscale satellite, medical and historical images making them more expressive. With the help of…
The production of color language is essential for grounded language generation. Color descriptions have many challenging properties: they can be vague, compositionally complex, and denotationally rich. We present an effective approach to…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
The bandwidth of a signal is an important physical property that is of relevance in many signal- and information-theoretic applications. In this paper we study questions related to the computability of the bandwidth of computable…
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the…
Turing's famous 'machine' model constitutes the first intuitively convincing framework for computing with real numbers. Kleene's computation schemes S1-S9 extend Turing's approach and provide a framework for computing with objects of any…
The diagonal method is often used to show that Turing machines cannot solve their own halting problem. There have been several recent attempts to show that this method also exposes either contradiction or arbitrariness in other theoretical…
There are many extremely challenging problems about existence of monochromatic arithmetic progressions in colorings of groups. Many theorems hold only for abelian groups as results on non-abelian groups are often much more difficult to…
For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While…
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 $\mathcal A$ such that if…
We propose a new type of approximate counting algorithms for the problems of enumerating the number of independent sets and proper colorings in low degree graphs with large girth. Our algorithms are not based on a commonly used Markov chain…
Artificial computing machinery transforms representations through an objective process, to be interpreted subjectively by humans, so the machine and the interpreter are different entities, but in the putative natural computing both…
This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single…
A $t$-tone coloring of a graph $G$ assigns to each vertex a set of $t$ colors such that any pair of vertices $u, v$ with distance $d$ can share at most $d-1$ colors. In this note, we prove several new results on $t$-tone coloring. For…
Classical models of computation traditionally resort to halting schemes in order to enquire about the state of a computation. In such schemes, a computational process is responsible for signalling an end of a calculation by setting a halt…